Population Ecology: A Unified Study of Animals and Plants

CHAPTER 5: PREDATION 141

therefore, is an extreme example of a widespread
phenomenon: the predators' ill-effectsare aggregated,
tending to stabilize the interactionsbetween them and
their prey.

Fig. 5.22 Aggregative responses of (a)the braconid 5.8.5 'Even' distributions
Apanteles glomeratus to plants of different Pieris brassicae
density (Hubbard, 1977), and (b)the ichneumonid Parenthetically within this component of the interac-
Diadromas pulchellus to different densities of leek moth tion, it must be stressed that not all 'predator-prey'
pupae per unit area (Noyes, 1974). (After Hassell, 1978.) distributions are clumped. Indeed, Monro (1967)pro-
vides a striking example of a herbivore going to the
plants without eggs are indeed protected. Moreover, other extreme. Figure 5.23 shows the distribution of
the interaction is also stabilized by the death of larvae ovipositionsby the trypetid fruitfly Dacus tvyoni in ripe
on plants with too many egg-sticks(roughly more than loquat fruit, and also provides an 'expected' random
two per plant). These plants become 'overloaded': they distribution for comparison. It is obvious that the fly
are completely destroyed by the dense aggregation of spreadsits ill-effects much more evenly than expected,
sedentary larvae, but the larvae themselves then have so that there are relatively few fruits that escape and
insufficient food to complete development. This, few that are overcrowded. Once again, however, as in
most of the clumped examples, we can see that the
fundamental basis of the pattern is that it is advanta-
geous to the consumer-in this case as a result of the

Table 5.6 Aggregation in a plant-herbivore interaction.
Comparison of observed distributions of Cactoblastis egg-
sticks per Opuntia plant with corresponding Poisson

distributions. (After Monro, l 9 67.)

*Egg-sticksmore clumped than expected for random Fig. 5.23 An 'even' distribution: the observed distributions
of oviposition stings made by the fruitfly Dacus tryoni on
oviposition. loquat fruit (- - -) compared to an expected random
tEgg-sticks not distributed differently from random. distribution (-) (After Monro, 1967.)

142 PART 2: INTERSPECIFIC INTERACTIONS

reduction in intraspecific competition experienced by response to encounters with prey items. In particular,
each larva. there is often an increased rate of predator turning
immediately following an intake of food, which leads
5.8.6 Underlying behaviour to the predator remaining in the vicinity of its last food
item. Increased turning causes predators to remain in
There are various types of behaviour underlying the high density patches of food (where the encounter-
aggregative responses of predators, but they fall into and turning-rates are high), and to leave low density
two broad categories:those involved with the location patches (where the turning-rate is low). Such behav-
of prey patches, and those that represent the response iour has been demonstrated in a number of predators,
of a predator once within a prey patch. Within the first and is illustrated in Fig. 5.25 for coccinellid larvae
category we can include all examples of predators feeding on aphids (Banks, 1957).
perceiving, at a distance, the existence of heterogene-
ity in the distribution of their prey. Rotheray (1979), The third type of behaviour is demonstrated by the
for instance, found that the parasitoid Callaspidia data in Table 5.7 (Turnbull, 1964), referring to the
defonscolombei was attracted to concentrations of its web-spinning spider Archaearanea tepidariorum preying
syrphid host by the odours produced by the syrphids' on fruit flies in a large experimental arena. The spiders
own prey: various species of aphid. tend, simply, to abandon sites at which their capture-
rate is low, but remain at sites where it is high. In this
Within the second category-responses of predators case, therefore, the spiders modify their leaving-rate
within prey patches-there are three distinct types of (rather than their turning-rate) in response to prey
behaviour. The first is illustrated in Fig. 5.24: a female encounters, but the result, once again, is that preda-
predator of one generationtends to lay her eggs where tors congregate in patches of high prey density.
there are high densities of prey so that her relatively
immobile offspring are concentrated in these profit-
able patches. This response, as far as the predatory
individuals themselves are concerned, is essentially
passive. By contrast, the second type of behaviour
involves a change in a predator's searching pattern in

Fig. 5.24 Distribution of Syrphus eggs in relation to the Fig. 5.25 Search paths of hungry, fourth-instar coccinellid
number of tests of the psyllid Cardiaspina albitextura per leaf larvae before and after capture of a prey (small circle on the
surface (Clark, 1963).(After Hassell, 1978.) path). The rate of turning markedly increases after prey

capture (Banks, 1957). (After Curio, 1976.)

CHAPTER 5: PREDATION 143

Table 5.7 Site occupation and feeding-rate of spiders 5*8*7 'Hide-and-seek'
feeding on Drosophila in an experimental arena with sites of
varying suitability for Drosophila. (After Turnbull, 1964.) We have seen, then, that the distributions of predators
and prey can have important effects on predator-prey
dynamics, because predators tend to concentrate on
profitable patches of prey. There is, however, in some
cases, another perspective from which this behaviour
can be seen: predators and prey can appear, in effect,
to play 'hide-and-seek'. The most famous and illustra-
tive example of this is provided by the experimental
work of HufTaker (1958) and H d a k e r et al. (1963).
Their laboratory microcosm varied, but basically con-
sisted of a predatory mite, Typhlodromus occidentalis,
feeding on a herbivorous mite, Eotetranychus sexmacu-
latus, feeding on oranges interspersed amongst rubber
balls in a tray. In the absence of its predator, Eotetrany-
chus maintained a fluctuating but persistent popula-
tion (Fig. 5.26a);but if Typhlodromus was added during

Fig. 5.26 Predator-prey interactions
between the mite Eotetranychus

sexmaculatus (a)and its predator

Typhlodromus occidentatis (0).
(a)Population fluctuations of
Eotetranychus without its predator,
(b)a single oscillation of predator
and prey in a simple system: and
(c)sustained oscillations in a more
complex system (HufTaker, 1958).
(After Hassell, 1978).

144 P A R T 2: INTERSPECIFIC INTERACTIONS

the early stages of prey population growth, it rapidly the starfish disperse much less readily. They tend to
increased its own population size, consumed all of its stay wherever the clumps are, and there is a time-lag
prey and became extinct itself (Fig. 5.26b). The inter- before they leave an area when the food is gone. The
action was exceedingly unstable, but it changed when parallel with Huffaker's mites is quite clear: patches of
HdTaker made his microcosm more 'patchy'. He mussels are continually becoming extinct, but other
greatly increased its size, but kept the total exposed clumps are growing prior to the arrival of starfish. 'The
area of orange the same; and he partially isolated each starfish show aggregative behaviour, concentratingon
orange by placing a complex arrangement of Vaseline large, profitable clumps, and allowing the initially
barriers in the tray which the mites could not cross. small, protected clumps to become large and profit-
However, he facilitated the dispersal of Eotetranychus able themselves. 'Hide-and-seek' and aggregative be-
by inserting a number of upright sticks from which haviour are, therefore, essentially indistinguishable,
they could launch themselves on silken strands car- and both illustrate the important stabilizing effects of
ried by air currents. The overall result was a series of aggregation on the predator-prey interaction.
sustained and relatively stable predator-prey oscilla-
tions (Fig. 5.26c), probably generated by the following We have one further, important aspect of this
mechanism. In a patch occupied by both Eotetranychus 'aggregative behaviour' component to consider, but
and Typhlodromus,the predators consume all the prey before we do so it is necessary to examine another
and then either disperse to a new patch or become component: the interactions that occur between pre-
extinct. In a patch occupied by the predators alone, dators.
there is usually death of the predators before their
food arrives. But in patches occupied by the prey 5.9 Mutual interference
alone, there is rapid, unhampered growth accompa- amongst predators
nied by some dispersal to new patches. Dispersal,
however, is much easier for the prey than it is for the Predators commonly reserve particular aspects of
predators. The global picture is, therefore, a mosaic of their behavioural repertoire for interactionswith other
unoccupied patches, prey-predator patches doomed predators of the same species: herbivorous (nectar-
to extinction, and thriving prey patches; with some feeding) humming-birds, for instance, actively defend
prey and rather fewer predators dispersing between rich sources of food (Wolf, 1969); badgers patrol and
them. While each patch is ultimately unstable, the visit the 'latrines' around the boundaries between
spatially heterogenous whole is much less so. Once their territories and those of their neighbours; and
again, therefore, patchiness has conferred stability; females of Rhyssa persuasoria, an ichneumonid parasi-
and, in the context of this section, we can see that this toid of wood wasp larvae,will threaten and, if need be,
example illustrates the effect of temporary 'temporal fiercely drive an intruding female from the same area
refuges'. of tree trunk (Spradbery, 1970). On a more quantita-
tive level Kuchlein (1966) has shown that an increase
A rather similar, and ultimately more satisfying in the density of the predatory mite Typhlodromus
example (since it comes from the field), is provided by longipilus (and thus an increase in the number of
the work of Landenberger (1973;in Murdoch & Oaten, predator-predator encounters)leads to an increase in
1975) who studied the predation by starfish of mussel the rate of emigration from experimental leaf discs
clumps off the coast of southern California. Clumps containing prey mites; and a similar situation is shown
which are heavily preyed upon (or are simply too for a parasitoid of leek moth pupae in Fig. 5.27. In
large) are liable to be dislodged by heavy seas so that both qualitative and quantitative examples, then, the
the mussels die: the predators are continually driving essential effect is the same: the time available to the
patches of prey to extinction. Yet the mussels have predator (or herbivore or parasitoid)for 'prey'-seeking
planktonic larvae which are continually colonizing is reduced by encounters with other predators. And
new locations and initiating new clumps. Conversely, the importance of this effect increases with predator

CHAPTER 5: PREDATION 145

Fig. 5.27 The effect of female parasitoid density on consumes less than it would otherwise do. Con-
parasitoid emigration from an experimental cage; the sumption-rateper predator will, therefore, decline with
ichneumonid Diadromus pulchellus (Noyes, 1974).(After increasing predator density. Thus, if we ignore the fact
Hassell, 1978.) that search time is reduced by mutual interference as
predator density increases (i.e. assume that search
density, because this increases the rate of predator- time remains constant),it will appear as if searching(or
predator encounters. attacking) eficiency is declining. Mutual interference
can then be demonstratedby plotting apparent attack-
The precise characteristics of such mutual inteder- ing efficiency (calculated from data on consumption-
ence will vary from species to species, but all examples rates on the assumption of random search; see section
can be reduced to a common form in the following 5.10) against predator density. This has been done in
manner (Hassell & Varley, 1969; Hassell, 1978). The Fig. 5.28, on logarithmicscales. As expected, the slope
end-result of mutual interferenceis that each predator in all cases is negative, and may be denoted by - m,
where m is termed the coeficient of interference. The
general form of this relationship is probably repre-
sented by Fig. 5.28a & 5.28b, in which m remains
constant at high and moderate predator densities, but
decreases at low densities. This indicates that appar-
ent attacking efficiency cannot continue to rise as
predators become increasingly scarce (moving from
right to left). As Fig. 5.28~-e shows, however, the
coefficient of interference often remains constant
throughout the range of densities actually examined.

Fig. 5.28 Interference relationships between the searching Loxostege stricticalis (Ullyett, 1949a).(d) Coccinella
septempunctata feeding on Brevicoryne brassicae (Michelakis,
efficiency (log scale)and density of searching parasitoids or 1973).(e) Phytoseiulus persimilis feeding on deuteronymphs
predators. (a)Encarsia formosa parasitizing the whitefly of Tetranychus urticae (Fernando, 1977).(After Hassell,
Trialeurodes vaporariorum (Burnett, 1958).(b)Chelonus 1978.)

texanus parasitizing eggs of Anagasta kiihniella (Ullyett,

1949b).(c) Cryptus inornatus parasitizing cocoons of

146 PART 2: INTERSPECIFIC INTERACTIONS

This important observation will be utilized in section by the presence of other parasites. Once again, there-
5.13. fore, the general effect is to stabilize the interaction
(see Anderson & May, 1978).
We have just seen that as a result of mutual
interference amongst predators, attacking efficiency 5.10 Interference and pseudo-
decreases as predator density increases. There is, in interference
other words, a density-dependent reduction in the
consumption-rate per individual, and thus a density- We can return now to aggregative behaviour, and
dependent reduction in predator fitness, which will consider an important alternative approach to its
have a stabilizing effect on the predator-prey interac- effects. This was introduced in a mathematicalway by
tion. The coefficient of interference,m, is a measure of Free et al. (1977); but for our purposes their ideas can
this stabilizing effect. be discussed verbally. By concentrating on profitable
patches, a single predator increases the number of
5.9.1 A similar effect amongst parasites prey it eats per unit time; the same predator searching
Interestingly, although parasites are not subject to the randomly would eat less. This means that the appar-
same sort of mutual interference as predators, herbi- ent attacking efficiency (calculated from consumption-
vores and parasitoids, they are affected by another rates on,theassumption of random search)is higher in
process which has a rather similar result. Immunolog- the 'aggregated' predator. Yet the same predator, by
ical responses by hosts can play an important role in removing prey from the profitable patches, will nnar-
parasite mortality, and the strength of the response is ginally reduce the profitability of those patches. This,
often directly related to the size of the parasite burden. in time, will make aggregated search more like ran-
Thus, the probability of host-induced parasite mortal- dom search, and the apparent attacking efficiency will
ity will increase with greater parasite density, as decline. And if there are many predators, all concen-
Fig. 5.29 illustrates. There is no direct interference, of trating on profitable patches, then this decline may be
course, but an individual parasite's fitness is reduced immediately perceptible. Thus, in general, we can
expect the apparent attacking efficiency to decrease as
Fig. 5.29 The relationship between parasite death-rate and predator density increases. Yet this is the very rela-
parasite density within individual hosts for chickens tionship that we saw resulting from mutual predator
infected with the fowl nematode Ascaridia lineata (Ackert interference in the previous section. In the present
et al., 1931).(After Anderson & May, 1978.) case, therefore, this consequence of aggregative behaviour
may be described as 'pseudo-interference'(Free et al.,
1977).

The importance of pseudo-interference, along with
the nature of mutual interference itself, is amply
illustrated in the following example(Hassell, 1971a,b,
1978) in which larvae of the flour moth Ephestia
cautella were parasitized by Venturia canescens. The
larvae were confined in small containers at densities
ranging from four to 128 per container, and exposed
to one, two, four, eight, 16 or 32 parasitoids for 24
hours. Line A in Fig. 5.30a was obtained in the
manner of the previous section (by counting the total
number of hosts parasitized and assuming there was
random search): there was, apparently, a high degree
of mutual parasitoid interference (m large). In fact,

CHAPTER 5: PREDATION 147

Fig. 5.30 (a)Relationships between
searching efficiency and density of
searching Venturia canescens;

(b)relationship between (i)the
proportion of time spent searching
and Venturia density; and (ii)the
percentage of time spent at different
host densities with (c)one
parasitoid, (d)two parasitoids, and
(e)four, eight, 16 and 32
parasitoids. (After Hassell, 1978.)

For further discussion, sex text.

continuous observation showed explicitly that the time number of such encounters was observed and noted;
spent by parasitoids on host containers did decrease and recalculating the apparent attacking efficiency
with increasing parasitoid density (Fig. 5.30b). In this time yields line C in Fig. 5.30a.
other words, mutual interference caused parasitoids
to leave host patches, and thus spend less time Lines A-C have negative slopes (i.e. coefficients of
searching. The true searching time, then, is not 24 interference) of 0.67, 0.45 and 0.27, respectively.
hours, but the proportion of 24 hours indicated by Thus, 33% of the total interference [(0.67-0.45)/
Fig. 5.30b; and taking this into account increases the 0 . 6 7 1 ~100 is accounted for by parasitoids leaving
apparent searching efficiency and yields line B in host patches, and a further 27% by parasitoids simply
Fig. 5.30a. Yet there is clearly some interference interrupting their searching. However, this leaves 40%
remaining after these effects have been removed. of the total unaccounted for by actual mutual interfer-
ence. The precedng argument (Free et al., 1977)would
A partial explanation of this is provided by the suggest that this is pseudo-interference, and the ag-
existence of an additional aspect of behavioural inter- gregative behaviour shown in Fig. 5.30~-e certainly
ference. Observation showed that parasitoid- supports this. Much more persuasive, however, is line
parasitoid encounters often led not to departure from D in Fig. 5.30a. This has been obtained by abandoning
a patch, but simply to an interruption of probing the assumption of random search, and using instead
(lasting about 1min). This, too, can be used to reduce the observed data on search pattern from Fig. 5.30~-
the 'true' searching time appropriately, because the e. The slope, - 0.03, indicates that there is no longer

148 P A R T 2: INTERSPECIFIC INTERACTIONS

any significant amount of interference. The interfer- tend to maximize their 'profits'. They can be brack-
ence of line C, in other words, was purely a conse- eted together as studies of 'optimal foraging'. This is a
quence of the aggregative behaviour. It was, indeed, subject which is reviewed in some detail by Stephens
pseudo-interference. and Krebs (1986), but we can underline the impor-
tance of optimal foraging between patches by means
Perhaps the initial conclusion to be drawn from this of the following example.
work is that, in the absence of the detailed observa-
tional data, it would have been impossible to partition Cook and Cockrell (1978) fed individual, fourth-
the total interference of line A into mutual interfer- instar mosquito larvae to adult water-boatmen, No-
ence and pseudo-interfence. This is not, however, as tonecta glauca, and as Fig. 5.31a shows, the rate at
disappointing a conclusion as it might at first appear. which nutrients were extracted from a single prey
In any natural situation, mutual interference (if it item declined sharply with time. This occurred not
exists at all) is bound to increase with increased because the water-boatmen were satiated, but be-
aggregation,since aggregation (by definition)leads to a cause the food became increasingly difficult for them
higher probability of encounter. Yet the general effect to extract. Without bending the rules too much, we
of mutual interference will be to drive predators from can treat each prey item as a 'patch'. From this
dense aggregates (where encounters are most fre- viewpoint we can see that the profitability of patches
quent). This will tend to reduce the level of aggrega- to water-boatmen declined rapidly the longer they
tion, and ultimately reduce the mutual interference stayed 'in' them. We can also see that to maxirnize
itself. There is, in short, a complex and dynamic their profits they would, at some time, have to leave
interaction-which it might be extremely difficult to depleted patches and find new, highly profitable ones,
disentangle-between aggregation (leading to pseudo- i.e they would have to drop the old prey and catch a
interference) and behavioural interference; and it new one. Yet this in itself is a costly process: the
may, therefore, be convenient to encapsulate the predators take in no food while they are expending
effects of both in a single parameter: the coefficient of energy searching for, capturing and subduing (i.e.
interference.Plots like the ones in Fig. 5.28 and line A handling) the next prey item. Obviously, if they are to
in Fig. 5.30a, in other words, are able to capture the forage optimally, the water-boatmen must maximize
combined, density-dependent, stabilizing effects of their profits when they set such costs against the
aggregative behaviour and mutual predator interfer- eventual gain from the new patch, and the longer the
ence; and there will be many natural situations in handling time the greater these costs will be. We
which one or both of these effects are extremely would, therefore, expect optimally foraging predators
significant. Both effects, of course, represent strategies to spend relatively long periods at a patch (where the
adopted by the predators to increase their own fitness. rate of profit, even if low, is at least positive) when the
handling time is high; and this is precisely the result
5.11 Optimalforaging obtained by Cook and Cockrell (Fig. 5.3lb). The water-
boatmen's feeding times served to increase tlheir
One further conclusion that can be drawn from this profits (Fig. 5.3lc(i)). Prolonged stays (Fig. 5.3lc(ii)) or
discussion, however, is that the movement of preda- repeated changes of patch (Fig. 5.3lc(iii)) would
tors between patches is itself worthy of detailed study clearly have been less profitable.
whether this movement is a consequence of interfer-
ence, or of patch depletion, or is even a means of patch It must not, of course, be imagined that the water-
assessment. Such study would be closely related to boatmen consciously weigh up pros and cons and act
investigations of diet width among predators (section accordingly. Rather, it is natural selection that has
5.3), and of the way in which predators distribute favoured individuals that adopt a strategy appropriate
their effort amongst patches (section 5.7), because all for foraging in a patchy environment with marked
three are concerned with the ways in which predators patch depletion. Nor must it be imagined that animals
always adopt the strategy which actually maximizes

Fig. 5.31 (a)The cumulative dry weight of food extracted (After Cook & Cockrell, 1978.) (c)The foraging profit of (i)
an optimal forager (660 arbitrary units), compared to one
from an individual mosquito larva by Notonecta as a with a long stay-time (ii)(376 units) and one with a short
stay-time (ii) (175 units).
function of time spent feeding: as time increases,

fdeiemdiinnigshtiimnger(e+tuSrEn)sasnedt in. (b) The relationship between
handling time (intercatch interval).

their foraging profits. As Comins and Hassell (1979) burden all tend to have a stabilizing influence on the
point out, animals must spend (i.e. 'waste') time interaction; while time-lags, the increased effects of
sampling and learning about their environment; and multiple infections in parasites,the existenceof 'main-
the strategies they adopt will be influenced not only by tenance thresholds' and 'type 1' and 'type 2' func-
foraging itself, but also by predator avoidance, and so tional responses all tend to have a destabilizing
on. Nevertheless, we can expect natural selection to influence. We have also seen that predators and prey
'exert evolutionary pressures towards an optimal forag- exert a greater influence on the dynamics of one
ing strategy; and the recognition of optimal foraging is another's populations the more specific(less polypha-
bound to play an increasingly crucial role in our gous) the predator is; that some prey have relatively
understanding of predator-prey interactions.Interest- little fitness to lose by their death; that the quality as
ingly, Comins and Hassell (1979) suggested, from an well as the quantity of food can exert important
analysis of models, that although optimal foraging has influences on predator populations; and that, in many
important consequences for predator fitness, its effects respects, predators tend to forage optimally.
on predator-prey dynamics are rather similar to those
resulting from a much simpler, fixed pattern of aggre- Nevertheless, having examined these components
gation. we remain essentially ignorant of their relative impor-
tance, and of the patterns of abundance that we might
Our view of the 'simplest, abstracted, two-species expect them to give rise to, either alone or in combi-
predator-prey system' is, by now, somewhat less nation. In order to go some way towards dispelling
naive than it was. We have seen that intraspecific this ignorance, we turn, once again, to mathematical
competition amongst prey, 'type 3' functional re- models.
sponses, aggregated distributions of ill-effects (and, as
special cases, spatial and temporal refuges), mutual 5.13 Mathematicalmodels
interference amongst predators, and the increase in
host immunological response with increased parasite Of the models that we might consider, Lotka (1925)
and Volterra (1926) constructed a classical, simple,
continuous-time predator-prey model, which Rosen-
zweig and MacArthur (1963) developed further in

150 PART 2: INTERSPECIFIC INTERACTIONS

graphical form: Caughley and Lawton (1981)reviewed -Et = AP,
modelsof plant-herbivore systems;and Crofton (1Wl), H,
Anderson and May (1978) and May and Anderson
(1978) constructed models of the host-parasite inter- ( A can, alternatively, be thought of as the parasitoid's
action (reviewed by Anderson, 1981). However, be- searching efficiency; or the probability that a given
cause of the range of factors which they explicitly and parasitoid will encounter a given host; or, indeed, the
successfully incorporate, we shall concentrate here on 'area of discovery' of the parasitoid, within which it
difference equation models of host-parasitoid systems encounters all hosts.) Remember that we are dealing
(cf. section 3.2) These are described in much greater with parasitoids. This means that a single host can be
detail by Hassell (1978). Then, in section 5.l3.3, we encountered several times, but for the most part only
shall turn briefly to a model of grazing systems (Noy- the first encounter leads to successful parasitization;
Meir, 1975).The conclusionsthat we are able to draw predators, by contrast, would physically remove their
throughout will throw important light on predator- prey, and thus prevent re-encounters.
prey interactions generally.
If encounters occur in an essentially random fash-
5.13.1 Host-parasitoid models ion, then the proportions of hosts encountered zero,
one, two, three or more times are given by the
We begin by describing a very simple model, which successive terms in the appropriate 'Poisson distribu-
can be used as a basis for further developments tion' (described in any basic textbook on statistics).
(Nicholson, 1933;Nicholson & Bailey, 1935).Let H, be The proportion not encountered at all, p,, is given by:
the number of hosts, and P, the number of parasitoids
(in generation t): r is the intrinsic rate of natural where e m ( - E,/&) is another way of writing e-"~'~'.
increase of the host, and c is the conversion rate of Thus the proportion that is encountered (one or more
hosts into parasitoids, i.e. the mean number of para- times) is l -p,, and the number encountered (or
sitoids emergingfrom each host. If H, is the number of
hosts actually attacked by parasitoids (in generation t), r ( :,)lattacked)is:
then, clearly:
H,=Ht(l-po)=Ht l-exp - -

In other words, ignoring intraspecificcompetition, the And substituting this expression for H, into equation
hosts that are not attacked reproduce, and those that 5.1 gives us:
are attacked yield not hosts but parasitoids. For
simplicity we shall assume that each host can support This is the Nicholson-Bailey model of the host-
only one parasitoid (c = 1).Thus, the number of hosts parasitoid interaction. Its simplicity rests on two
attacked in one generation defines the number of assumptions:
1 that parasitoid numbers are determined solely by
,parasitoids produced in the next (P, + = H,). the rate of random encounters with hosts; and
To derive a simple formulation for H, we proceed as 2 that host numbers would grow exponentially but
follows. Let E, be the number of host-parasitoid for the removal of individuals by random encounter
encounters (or interactions) in generation t. Then if A with parasitoids.
is the proportion of the hosts encountered by any one
parasitoid:

and

CHAPTER 5: PREDATION 151

As Hassell (1978) makes clear, an equilibrium tion in the absence of parasitoids. Figure 5.33 illus-
combination of these two populations is a possibility, trates the patterns of abundance resulting from this
but even the slightest disturbance from this equilib- revised model in terms of r and a new parameter, q
rium leads to divergent oscillations (Fig. 5.32). Thus, ( = H*/K) where H* is the equilibrium size of the host
our simple model, although it produces coupled oscil- population in the presence of parasitoids. For a given
lations, is highly unstable. Nevertheless, it is clearly a value of r and K, q depends solely on the parasitoids'
formal restatement of the naive expectation expressed efficiency,A. When A is low, q is almost 1(H* = K), but
in section 5.1: when a single predator and a single at higher efficienciesq approaches zero (H*c< K).
prey interact in the simplest imaginable way, coupled
oscillations are the result. It is clear from Fig. 5.33 that intraspecific competi-
tion amongst hosts can lead to a range of abundance
Since most observed patterns of abundance are patterns. Moreover, intraspecific competition is obvi-
considerably more stable than those produced by the ously a potentially important stabilizing factor in
Nicholson-Bailey model, we must be particularly host-parasitoid systems. This is particularly so for low
interested in modifications to it which enhance stabil- and moderate values of r, and high and moderate
ity. The most obvious modification we can make is to values of q; but even with low values of q (high
replace the exponential growth of hosts with density- parasitoid efficiency), the fluctuations are not alto-
dependent growth resulting from intraspecificcompe- gether unlike those observed in natural populations.
tition (section 5.5). Following Beddington et al. (1975), Note, however, that with high reproductive-ratesand
this is done by incorporating a term like the one used low parasitoid efficiencies we return to the chaotic
in the logistic equation (section 3.3), giving behaviour characteristic of the single-species popula-
tions in section 3.4.1.
where K is the carrying-capacity of the host popula-
The next modification we can make to the
Nicholson-Bailey model is to consider explicitly the
parasitoids' functional response to host density (sec-
tion 5.7). In the present context, this is described by

Fig. 5.32 Population fluctuations
from an interaction between the
greenhouse whitefly Trialeurodes
vaporariorum (0)and its chalcid
parasitoid Encarsia forrnosa (0)T.he
thin lines show the estimated

outcome from a Nicholson-Bailey

model (Burnett, 1958). (After

Hassell, 1978.)

152 PART 2: INTERSPECIFIC INTERACTIONS

Fig. 5.34 Type 2 functional responses generated by
equations 5.7. (a) Nasonia vitripennis parasitizing Musca
domestica pupae (DeBach & Smith, 1941) a = 0.027;
T, = 0.52. (b)Dahlbominus fuscipennis parasitizing Neodiprion

sertifer cocoons (Burnett, 1956; a = 0.252; T,, = 0.037. (After
Hassell, 1978.)

Fig. 5.33 Stability boundaries for the parasitoid-host model So that the total number of hosts handled in genera-
with intraspecifichost competition, as in equation 5.6. tion t by each parasitoid is now aTsHt.If we consider
(After Beddington et al., 1975.) For further discussion, see that 'total available search time' is equal to 'total time'
text. minus 'total handling time', then:

the relationship between Ha/P, (the mean number of T, = T - ThaTSH,,
hosts attacked per parasitoid) and H, ; i.e. the relation-
ship between the number of hosts attacked by a which, by rearrangement, gives:
constant number of parasitoids (Ha with P, constant)
and H,. Until now we have been assuming that this T, = 1+ T
relationship is linear (equation 5.4): aThHt'

Ha = H, {l- exp( - AP,)). so that:

In other words, it is implicit in the Nicholson-Bailey Substituting this into equations 5.4 and 5.5 gives:
model that the 'predation'-rate of the parasitoids
continues to rise indefinitely with increasing host As Fig. 5.34 shows, equation 5.7 generates a type 2
density, i.e. handling time is zero. This is clearly at functional response, in which the maximum possible
variance with the data examined in section 5.7, and is, number of hosts attacked is determined (as expected)
in any case, impossible. Obviously it is important to by T,,/T, and the rate of approach to this asymptote is
replace this linear relationship with the types 2 and 3 determined by a (the instantaneous search-rate).
functional responses which are actually observed. Equation 5.8, therefore, represent the Nicholson-

Dealing first with the type 2 response, we saw in
section 5.7.1 that the essential feature underlying it is
the existence of a finite 'handling time'. Thus, we shall
let T be the total amount of time available to each
parasitoid, Th the time it takes to deal with each host
(handling time), T, the total amount of time available
to each parasitoid for host seeking, and a the parasi-
toids' instantaneous rate of search (or attack-rate, see
section 5.7). Then, by definition:

CHAPTER 5: PREDATION 153

Bailey model with a type 2 functional response incor- Equation 5.9 generates a type 3 functional response,
porated into it. Their dynamic properties were and equations 5.10a and b are, therefore, the
examined by Hassell and May (1973)and, as expected, Nicholson-Bailey model with a type 3 response incor-
the inverse density-dependence makes this model less porated into it.
stable than the Nicholson-Bailey (recovered from this
model when Th = 0). However, increased instability is We have already noted (in section 5.7.4) that the
relatively slight as long as ThIT << 1,and we can see in density-dependent aspect of this response is likely to
Table 5.8 (Hassell, 1978)that this is, indeed, generally have an essentially stabilizing effect, but, as with the
the case, This model indicates, therefore, that the type 2 response, examination ofthe appropriate model
destabilizing tendencies of type 2 functional responses allows us to make an interesting qualification to this
are unlikely to be of major importance in nature. informal conclusion. In particular, Hassell and Comins
(1978) found that in the situation we have been
In order to model the type 3 functional response, we dealing with-one host and one parasitoid coupled
shall follow the pragmatic approach of Hassell (1978), together in a discrete-generation model-a type 3 re-
and assume first that only the instantaneous search- sponse alone is incapable of stabilizing the interaction.
rate, a, varies with host density, and second that it Conversely, there are at least two alternative situa-
does so in the simplest way compatible with the data tions in which it becomes a much more potent
examined in section 5.7.3, namely stabilizing force. The first is when the time delay of
discrete generations is removed and replaced by the
where X and y are constants. Substituting this into instantaneous reaction of continuous breeding (Mur-
equations 5.7 and 5.8 (and rearranging) gives us: doch & Oaten, 1975). The second is when the parasi-
toid is polyphagous,and the type 3 response is a result
[ (H, =JHEt lh- e:xpx z1+ of parasitoid 'switching'. Hassell and Comins (1978)
believe that this behaviour would essentiallyallow the
(Ht+l=Htexpr - xTH,Pt (5.1Oa) parasitoid to maintain itself at a constant density, so
(5slob) that:
1+y~~+ XT~H:
pt=pt+~=P.
[ (=Ht l -exp l +2yx?hH:)]
And if this equation is merged with equation 5.10a,
then the patterns of abundance are as summarized in
Fig. 5.35.

Table 5.8 Estimated values of handling time Th from equation 5.7 for a selection of parasitoids. The values of Th/T are based
on conservative estimates of longevity.(After Hassell, 1978.)

154 PART 2 : INTERSPECIFIC INTERACTIONS

Fig. 5.35 Stability boundaries for the parasitoid-host model
with constant parasitoid density (parasitoid switching)
and the host as described in equation 5.10a. (a)y = 0;

(b)g = 2 m ) . (After Hassell & Comins, 1978.)

Overall, therefore, these models make it clear that Fig. 5.36 The effects of refuges on stability: (a) a constant
the apparently simple consequences of the different proportion refuge (equations 5.11). and (b)a constant
'predator' functional responses are subject to signifi- number refuge (equation 5.12). (After Hassell & May,
cant qualifications. 1973.)

We turn now to the important question of heteroge- butions and efforts, and argued simply that the
neity (section 5.8), and consider first the special, distribution of host-parasitoid encounters was not
extreme case of a host 'refuge'. If there is a constant random but aggregated. In particular, he assumed
proportion refuge such that only a proportion of the (with some justification-see May, 1978a) that this
hosts, y, are available to the parasitoids, then equa- distribution could be described by the simplest and
tions 5.5 can be replaced simply by: most general of the appropriate statistical models, the
negative binomial. In this case, the proportion of hosts
Conversely, if there is a constant number refuge in not encountered at all is given by:
which H, hosts are always protected, the appropriate
modification is: p',= [l

H, + = Hoer+ (H, - H,) exp{r- M , ) (5.12) where k is a measure of the degree of aggregation:
maximal aggregation at k = 0,minimal aggregation at
Pt+1=(H,-Ho)(l - e m - M , ) ) . k = oo (recovery of the Nicholson-Bailey model). The
appropriate modification of p, in equations 5.6 gives
The results of these modifications are summarized in us a model that incorporatesboth aggregated encoun-
Fig. 5.36 (followingHassell & May, 1973). It is clear ters and intraspecific competition amongst hosts:
that, while both modifications have a stabilizing effect
on the interaction,the constant number refuge is by far )AP, -k
the more potent of the two. This is no doubt due to the
density-dependent effect of having a greater propor- Pt+l=Ht(l-[l+T]
tion of the host population protected as host density
decreases.

By contrast, a very much more general approach to
heterogeneity has been taken by May (1978a). He set

aside the precise nature of host and parasitoid distri-

CHAPTER 5: PREDATION 155

The patterns of abundance generated by this model ship, therefore, incorporates mutual parasitoid inter-
are summarized in Fig. 5.37, from which it is clear ference and aggregation.
that the already moderately stable system described
by equation 5.6 is given a marked boost in stability by The appropriate model is clearly:
the incorporation of significant levels of aggregation
(k S 1). Of particular importance is the existence of and the patterns of abundance resulting from these
stable systems with very low values of q. Of the equations are illustrated in Fig. 5.38. Not surprisingly,
stabilizing factors so far considered, encounter aggte- since aggregation of encounters is being incorporated,
gation (i.e. heterogeneity)is obviouslythe most potent. 'total interference' is shown to be an extremely potent
stabilizing force.
We consider next a topic that was established in
section 5.9 as being closely connected with the aggre- 5.13.2 Heterogeneity in host-parasitoid interactions
gation of encounters: mutual parasitoid interference.
Following Hassell and Varley (1969),we shall adopt a We have already seen in the previous section that the
simple, empirical approach and derive a form for the tendency for parasitoids to aggregate in areas of high
searchingefficiency,A, which conforms to the log-log host density can be a powerful stabilizing force in
plots of Fig. 5.28, i.e. host-parasitoid systems. The models of that section
(Hassell& May, 1973, 1974; Murdoch & Oaten, 1975;
where log Q and - m are the intercept and slope of the May, 1978a) showed that the stability observed was
plots, and, in particular, m is the 'coefficient of the result of directly density-dependent mortality.
interference'. Note that, because this relationship is However, it is now clear that other types of mortality
empirical, it serves to describe not only interference contribute to stability in host-parasitoid interactions.
proper but pseudo-interference as well. The relation- Spatial variation in parasitism that is independent of
host density and that which shows inverse density-
dependent mortality can also contribute in a signifi-

Fig. 5.37 Stability boundaries for the parasitoid-host model Fig. 5.38 StabiIity boundaries for the parasitoid-host model
with parasitoid aggregations as in equations 5.13: (a) k = CO; with mutual interferenceas in equations 5.14. (After
(b)k = 2; (c) k = l ; (d)k = 0.(After HasselI, 1978.)(See Hassell & May, 1973.)
Fig. 5.33.)

156 PART 2: INTERSPECIFIC INTERACTIONS

cant way to stability (Chesson & Murdoch, 1986; such instances, those hosts that occur in patches into
Pacala et al., 1990; Hassell et al., 1991). which parasitoids have aggregated are more likely to
be parasitized than those in patches which have low
The concept that underlies how these different parasitoid numbers. The second is if they occur in
density-related mortalities can stabilize host- patches of high host density, into which parasitoids
parasitoid interactions is the 'relative risk of parasit- aggregate in response to host density. In particular if
ism' for a host individual (Chesson & Murdoch, 1986) the relationship between parasitoid density and host
and is illustrated in Fig. 5.39. This 'risk' refers to a density between patches is an accelerating one (i.e. a
heterogeneous distribution across hosts of the proba- power function),then hosts in the highest host density
bility of encounters with parasitoids -all hosts are not patches are more likely to be parasitized than in any
equally likely to be parasitized (except in the unusual lower host density patches (Fig. 5.39~).In each of
circumstance shown in Fig. 5.39a). There are essen- these two scenarios the converse is, of course, also
tially two ways in which hosts have a high risk of important. In the first example, patches in which
being parasitized. The first is if parasitoids aggregatein there are many hosts but few parasitoids will be
patches independently of host density (Fig. 5.39b). In

Fig. 5.39 Four possible relationships between parasitoid patch lie above the dashed line and correspondinglya low
and host density. In (a)there is no aggregation of risk, with risk of parasitism when parasitoid numbers are below this
the same ratio of parasitoids to hosts in each patch. (b) This line. In (d) a data set typical of those reported in field
is a host-density-independent (HDI)model in which studies is shown. The solid line of best fit to the data is
identical to that shown in the HDD model of (c). Parasitoids
parasitoids are aggregated, but not in relation to host are again seen to be aggregating in patches of high host
density. There is a high risk of parasitism when parasitoid density, but in this example the match between parasitoid
numbers in a given patch lie above the dashed line (= 'no density and host density is not perfect. CV2 is calculated as
aggregation of risk line' of part (a))and correspondingly a follows: divide parasitoid numbers by host numbers in each
patch to give 'risk of parasitism'; calculate the mean 'risk of
low risk when parasitoid numbers are below this line. The parasitism' and its variance; divide the variance by the
solid line in (c) represents a host-density-dependent (HDD)
model in which parasitoids are once again aggregated, but square of the mean.
this time in response to host density. Once again there is a
high risk of parasitism when parasitoid numbers in a given

CHAPTER 5: PREDATION 157

ref~~gfeosr the hosts. In the second example, low host the CV2 rule was a good indicator of stability. How-
density patches effectively become refuges from para- ever, the models examined did all make some
sitism. assumptions-namely (i) that interactions between
parasitoid and host were coupled and synchronized;
We imagine that parasitoids frequently aggregate in (ii)that there were discrete host and parasitoid gener-
patches of high host density, but that the matching of ations; (iii) that exploitation of hosts within a patch
host and parasitoid density is far from perfect was random; and (iv) that there was no interference
(Fig. 5.398). The two scenarios we have presented between parasitoids or competition between hosts.
represent opposite ends of a continuum. Hassell et al.
(1991) refer to models that are closest to the first A criterion such as the CV2rule is only of any use to
scenario as host-density-independent heterogeneity a population ecologist if the parameters necessary to
models (HDI) and those that resemble the second as calculate CV2 can be estimated from field data. The
host-density-dependent heterogeneity models (HDD). rule, Cv2> 1is in terms of the distribution of search-
ing parasitoids, but such data are rarely availablefrom
The aggregation of risk of parasitism can be mea- natural populations. Data from host-parasitoid inter-
sured by the ratio of the standard deviation of the risk actions are usually in the form of a relationship
of parasitism per host divided by the mean risk of between percentage parasitism and host density per
parasitism, or the coefficientof variation (CV).For some patch; some examples are shown in Fig. 5.40. Pacala
discretetime models of host-parasitoid interactionsthe and Hassell (1991)explain how the parameters neces-
criterion for stability of the host-parasitoid interaction sary to calculate Cv2 can be estimated from typical
is that Cv2 > 1(Pacalaet al., 1990;Hassellet al., 1991).
Stability through heterogeneity in the probability of field data.
encounters is achieved through strengtheningdensity- Hassell and Pacala (1990) analysed 65 field studies
dependencies, specifically through pseudo-inter-
ference (see section 5.10 and Taylor, 1993). Pseudo- that reported per cent parasitism and local host
interferencerefers to reduction in parasitoid efficiency density per patch and from which estimates of CV2
resulting from an increasingnumber of host-parasitoid could be made. In 18 of these 65 studies CV2> 1
encounters involving previously parasitized hosts. By which suggests that the heterogeneity present, if it was
increasingthis pseudo-interference, aggregationof risk repeated from generation to generation, would be
makes host mortality less dependent on parasitoid sufficient to stabilize the interaction between the host
density and conversely reduces per capita parasitoid and parasitoid populations. In 14 of the 18 examples
recruitment. The result is that parasitoids have a lesser in which Cv2 > 1, the component of heterogeneity
impact on host density and oscillations in parasitoid that was independent of host density was greater than
density are damped (Taylor, 1993). Stability is thus that which was dependent on host density, indicating
promoted. that HDI heterogeneity contributes most to the total
heterogeneity, and therefore most towards stability,
By partitioning Cv2 into components, it is possible more often than HDD heterogeneity. This finding goes
to show that both density-dependent and density- against the conventional wisdom that stability can
independent patterns of parasitoid distribution con- only be promoted by density-dependent factors alone.
tribute to stability in the same way. What is We shall return to this topic in Chapter 6.
particularly interesting and surprising about this ap-
proach is the finding that density-independent hete- The spatial distributions of two tephritid flies (Uro-
rogeneity can contribute so much to stability. Hassell phora stylata (F.) and Terellia sewatulae L.) attacking
et al. (1991) and Pacala and Hassell (1991) examined thistle flower heads and the levels of parasitism from
the contribution of HDI and HDD to population six associated parasitoids were examined over a
stability by applying the CV2 criterion to five host- 7-year period in the field by Redfern et al. (1992).
parasitoid models which made different assumptions These data provide a rare opportunity to seek both
about host and parasitoid distributions. In each case temporal density-dependentparasitism between aver-
age parasitoid density and host density per generation

158 PART 2: INTERSPECIFIC INTERACTIONS

Fig. 5.40 Three examples from field studies of different as well as any spatial patterns that may be operating
patterns of parasitism from patch to patch; (a) direct within the same system, and to assess their relative
contributions to stability using the approach outlined
density-dependent parasitism of the scale insect Fiorinia above. Figure 5.41 shows the relationship between
externa by the eulophid parasitoid Aspidiotiphagus citrinus on total percentage parasitism and the density of the
the lower crown of 30 hemlock trees (McClure, 1977); tephritid hosts per flower head. There is significant
(b)inverse density-dependent parasitism of gypsy moth density-dependent mortality between total parasitism
Lymantria dispar eggs by the encyrtid parasitoid Ooencyrtus of T. sewatulae and host density per year (Fig. 5.41a),
kuwanai (Brown & Cameron, 1979);(c) density-independent which is the result of the combined parasitism of the
parasitism of gall midge Rhopalomyia californica by the two main parasitoids, Pteromalus elevatus and Tetmsti-
toryrnid parasitoid Torymus baccaridis (Ehler, 1987). (After chus cirsii. The comparable analysis for Urophora
Hassell et al., 1991.) stylata shows no such relationship (Fig. 5.41b). The
spatial analysis of parasitism is summarized in
Fig. 5.42, which shows the temporal variation in CV2
and its HDI and HDD components for the total
parasitism of each host species. C v 2 > 1in 5 of the 6
years in which parasitism was observed for U. stylata.
In each case this stability is brought about largely as a
result of HDI components of heterogeneity
(Fig. 5.42a). C v 2 > 1 in only 1 of 5 years in which
parasitism was observed for Terellia sewatulae, but once
again HDI was responsible rather than HDD
(Fig. 5.42b). The results of this study indicate how
parasitism may be regulating two tephritid species.
There appears to be conventional temporal density-
dependence contributing to the stability of the inter-
action of T. sewatulae and its parasitoids with little
evidence for stabilizing heterogeneity. In contrast,
there is no evidence of temporal density-dependence
in this relatively short run of data for Urophora stydata,
but there is considerable potentially stabilizing heter-
ogeneity, which may have a considerable impact on
the dynamics of the host.

How general is the CV2 > 1 rule? Future work is
likely to show that it has been useful in focusing
attention on the role of aggregation of risk in prornot-
ing stability, rather than being absolutely right. While
the rule holds for a range of non-overlapping genera-
tion discrete time host-parasitoid models, of the lype
described by equation 5.1, it is clear that there is a
range of factors not incorporated into the models of
Hassell et al. (1991) which can alter the degree of
aggregation of risk needed for stability. Some of these
factors are common in nature. For example when the
functional response of the parasitoid is type 1 (as

CHAPTER 5: PREDATION 159

Fig. 5.41 Total parasitism from generation to generation in dependent aggregation from zero are destabilizing,
relation to the average density of tephritid hosts per flower and only large amounts of density-dependent parasi-
head for the period 1982-88. Fitted line given by equation toid aggregation are stabilizing-when handling time
reduces parasitoid efficiency in high host density
y = a + b log,, X. (a) Terellia serratulae: a = 56.07 (SE 10.45); patches (Ives, 1992a,b).
b = 9.90 (SE 9.90); 1.2 = 0.75, P < 0.05; (b) Urophora stylata:
The most obviousviolation of the assumptions of the
a = 17.26 (SE 3.46); b = 11.70 (SE 10.49);? = 0.44, C V >~ 1rule in its application to date is that many of
P >. 0.05. (After Redfern et al., 1992.) the field studies analysed, including the one described
above, involve more than one parasitoid-host interac-
assumed in the models examined above), density- tion and two or more competing host species. Others
dependent parasitism is always stabilizing. However, include several parasitoid species whose effect some-
parasitoid functional responses are more typically of
type 2. In such cases initial increases in density-

Fig. 5.42 Values of CV2 for the

period 1982-88 for total parasitism
of (a) Urophora stylata, and (b) Terellia
serratulae. The proportional
contribution of the host-density-

dependent (dark shading) and host-

density-independent (light shading)

components to the total CV2 is also
illustrated. (After Redfern et al.,
1992.)

160 PART 2: INTERSPECIFIC INTERACTIONS

times was lumped in total parasitism. The effects of (see May, 1977 for a review).
additional species vary. When two specialized parasi- We have already seen (in section 5.5.1) that plant
toids share a host species greater aggregation is
required for stability than with a single parasitoid. populations, with the individuals growing as multiply
However, a generalist parasitoid whose parasitism branched units, may not necessarily exhibit a simple
rate is positively correlated with host density provides pattern of population growth. In particular, we argued
additional stability (Taylor, 1993). that the rate of vegetative growth may be conveniently
expressed in terms of photosynthetic assimilation, the
The models considered in this section have been in net assimilation rate being the rate of biomass growth
discrete time with non-overlapping generations. The having accounted for respiratory losses. Figure 5.43
role of aggregation by parasitoids in a heterogeneous demonstrates that if the rate of growth in biomass is
environmentin continuous time models with overlap- plotted against the leaf area index (the ratio of total
ping generations is controversial. Murdoch and leaf area to horizontal ground beneath the canopy),
Stewart-Oaten (1989)found that aggregation indepen- then there is an intermediate, optimum index which
dent of host density did not affect stability and that maximizes growth-rate-a result of the shifting bal-
aggregation to patches of high host density was ance between photosynthesis and respiration as bio-
typically destabilizing. Ives (1992a,b) concluded, and mass and shading increase. Not surprisingly, since
we can only agree with him, that there is such a total biomass and leaf area index are so closely
wealth of dynamic complexity in continuous-time associated, there is also a humped, curved relationslhip
models that it may be impossible to apply the conclu- between growth-rate and biomass. The actual shape
sions from any model to a real parasitoid-host system of such curves (e.g. Fig. 5.44) will depend on the
without detailed observations and experiments on the interaction of many factors (see Blackman, 1968 for a
hosts and parasitoids in question. review), but in all cases an optimum biomass will
exist, yielding a maximum growth-rate.
5.13.3 A model of grazing systems
Conversely, the removal of vegetation biomass oc-
We turn now to a model of grazing systems developed curs as a result of grazing, and we have seen that the
by Noy-Meir (1975). It is typical of a range of models, rate of herbivore consumption depends on a variety of
all of which essentially incorporate the 'Allee effect' factors. In broad terms, however, consumption-rateis
(section 2.6); and all such models indicate that preda- likely to follow the saturation curve of a 'type 2' or,
tors and their prey (in this case grazers and their food more rarely, a 'type 3' functional response (section
plants) can coexist at more than one stable equilibrium 5.7). Of course, the total rate of herbivore consumption
will increase with herbivore density. This will result in

Fig. 5.43 (a) Idealized relationship
between crop growth-rate and leaf
area index. (b)Actual relationship
found in subterranean clover by
Davidson and Donald (1958).(After
Donald, 1961.)

CHAPTER 5: PREDATION 161

Fig. 5.44 Plant growth in New Zealand ryegrass-clover a family of curves, all of similar shape, but with the
height of the maximal, saturation consumption-rate
pastures as a function of biomass (Brougharn. 1955. 1956). itseIf increasing with herbivore density (see, for in-
(After Noy-Meir, 1975.) stance, Fig. 5.45a).

We now have two families of curves; and they can
be used to model the outcome of grazing by superim-
posing them on one another, because the difference
between the rates of growth and consumption gives
the net change in the growth-rate of vegetation bio-
mass. In Figs 5.45a, 5.46a, 5.47a and 5.48a, the
growth- and consumption-rates are plotted against
biomass for four different sets of conditions. These
allow the net growth-rates to be inferred for various
combinations of herbivore density and biomass, and
this information is summarized in the corresponding
Figs 5.45b, 5.46b, 5.47b and 5.48b. These illustrate
the positions of the biomass zero isoclines, separating
circumstances of positive and negative biomass
growth.

The simplest grazing model is shown in Fig. 5.45.
For each type 2 herbivore consumption curve
(Fig. 5.45a), biomass increases below point A (because

Fig. 5.45 (a)Rates of plant growth and herbivore herbivore density which lead to biomass increase (arrows
consumption (at a range of herbivore densities)plotted left to right) from combinations which lead to biomass
against plant biomass. A,, A, and A, are stable equilibria decrease (arrows right to left). (After Noy-Meir, 1975.)For
at which growth- and consumption-ratesare equal. (b) The further discussion, see text.
isocline dividing combinations of plant biomass and

162 PART 2: INTERSPECIFIC INTERACTIONS

Fig. 5.46 Similar to Fig. 5.45 except that in (a), as a result point B; while between points A and B, growth
of altered consumption-ratecurves, there is an unstable exceeds consumption,and biomass increases. Point A
equilibrium at intermediate herbivore densities, B,; and in is, therefore, still an essentially stable equilibrium,but
(b)there is a shaded area close to the isocline, indicating point B is an unstable turning point. A biomass slightly
combinations of plant biomass and herbivore density at less than B will decrease to extinction, driven by
which minor changes in either could alter the outcome overconsumption; a biomass slightly greater than B
from stable equilibrium to plant decline and ultimate will increase to the stable equilibrium at point A L . As
extinction, or vice versa. For further discusion, see text. Fig. 5.46b makes clear, then, there is an intermediate
range of herbivore densities at which either equilib-
growth exceeds consumption), but decreases above rium or extinction is possible, and small changes in
point A (consumption exceeds growth). Each point A herbivore density or vegetation biomass occurring
is, therefore, a stable equilibrium. As herbivore den- close to the isocline in this region (hatched in
sites increase, however, (and total consumption in- Fig. 5.46b) can obviously have crucial effects on the
creases) the level of stable biomass reduces (shown by outcome of the interaction.
the shape of the isocline in Fig. 5.45b), and at herbi-
vore densities exceeding some critical value, consump- In Figs 5.47 and 5.48 two even more complex
tion is greater than growth for all levels of biomass and situations are illustrated, but in general terms the
the plant population is driven to extinction; this is outcomes are the same in both. Figure 5.47 represents
obviously not unreasonable. a plant population that maintains a reserve of material
which is not accessible to grazers (underground stor-
A slightly more complex situation, in which the age organs, for example, or plant parts which are
consumption curve reaches saturation more suddenly, inedible). The origin of the growth-rate curve is,
is shown in Fig. 5.46. The outcome is unchanged at therefore, displaced to the left of the origin of the
low herbivore densities (equilibrium) and at high consumption-ratecurve. Figure 5.48 represents a her-
herbivore densities (extinction); but at intermediate bivore population exhibiting a 'type 3' functional
densities, the consumption-rate curve crosses the response. In both cases, the outcome is unchanged at
growth-rate curve twice (points A and B). As before, low herbivore densities; there is a single, stable
biomass decreases above point A (because consurnp- equilibrium (point A).
tion exceeds growth), but this is now also true below

CHAPTER 5: PREDATION 163

Fig. 5.47 Similar to Fig. 5.46 except that there is a 'reserve' herbivore density at which minor changes in either could
of ungrazable plant biomass. As a consequence there are in alter the outcome from low biomass stable equilibrium to
(a)low biomass stable equilibria at high and intermediate high biomass stable equilibrium,and vice versa. For further
herbivore densities (C2and C,). The shaded area in (b), discussion, see text.
therefore, indicates combinations of plant biomass and
above it. These high density cases are, therefore, also
Pit high herbivore densities, however, the plant characterized by a single, stable equilibrium. How-
population is no longer driven to extinction (un- ever, while the low herbivore density equilibria were
grazable reserves and type 3 responses stabilize the maintained at a high biomass by the self-regulatory
interaction). Instead there is a point, C,, at which the properties of the plants, these high herbivore density
two curves cross, such that growth exceeds consump- equilibria are maintained at a low biomass by either
tion below C,, while consumption exceeds growth the ungrazable reserves of the plants or the functional
response of the herbivore.
Fig. 5.48 Similar to Figs 5.45 and 5.46 except that the
herbivores exhibit a sigmoidal, type 3 functional response. Moreover, at intermediate herbivore densities in
In other respects the figure is like Fig. 5.47. For further Figs 5.47a and 5.48a, the consumption curve crosses
discussion, see text.

164 PART 2: INTERSPECIFIC INTERACTIONS

the growth curve at three points (A,, B, and C,). As management of grazing stock is to delay the start of
before, point A is a high biomass stable equilibrium, grazing until a minimum of vegetation growth has
point C is a low biomass stable equilibrium, and point been exceeded. It is clear that for a given herbivore
B is an unstable turning point. Populations in the density this will tend to prevent extinction in Fig. 5.46,
region of point B, therefore, might increase to point A while in Figs 5.47 and 5.48 it will favour a high
or decrease to point C, depending on very slight, biomass equilibrium, where the plant growth-rate and
perhaps random changes in circumstances. These thus the food available to the grazers will also be high.
plant-herbivore systems have alternative stable states;
and a small change in the size of either the plant or the Finally, it must be admitted that this 'one plant/one
grazer population in the hatched regions of Figs 5.47b grazer' model is an unrealistic abstraction of the real,
or 5.48b can shift the system rapidly from one stable multi-speciesworld. Nevertheless,it provides a sirnple
state to the other. The crucial point, therefore, is that and effective framework for the comparison and
systems with ungrazable reserves or type 3 functional analysis of plant-herbivore dynamics, and also indi-
responses may undergo sudden drastic changes in cates the complex, indeterminate, multi-equilibrial
population levels, which are nonetheless grounded in behaviour of predator-prey systems generally.
the regulatory dynamics of the interaction.
5.14 'Patterns of abundance'
It is difficult to evaluate this model's predictions reconsidered
critically, because much of the data on grazing is not
amenable to this form of analysis. Yet the available This chapter began, after some preliminary defini-
evidence for Australian pastures certainly suggests tions, with an outline survey of the patterns of
that combinations of observed growth and consump- abundance associated with predation; and, since 'pre-
tion curves may well lead to situationswith alternative dation' was defined so inclusively, this was effectively
stable states (Fig. 5.49); and as Noy-Meir (1975) has a survey of abundance patterns generally. Now, hav-
argued, the predictions of the model are borne out by ing examined many of the components of 'predatory'
the experiences of agronomists and range managers. interactions,we ought to be in a position to reconsider
In particular, one practice often recommended in the these patterns. But we must remember that no
'predator-prey system' exists in a vacuum: both
Fig. 5.49 Experimentallyobtained curves of growth, G species will also be interacting with further species in
(ryegrassand subterranean clover) and consumption, C(by the same or different trophic levels. Thus, Anderson
sheep), standardized to maximum rates of 100%.(After (1979) considered that parasites probably often play a
Noy-Meir, 1975.) complementary role in the regulation of host popula-
tion~,increasing their susceptibility to predation
(though evidence for this is mainly anecdotal). Simi-
lary, Whittaker (1979)concluded that quite moderate
levels of grazing will often produce a severe effect on a
target plant when (and because) they are combined
with (otherwise ineffective)interspecific plant compe-
tition. Lawton and McNeill (1979) concluded that

.plant-feeding insects stood between the 'devil' of their

natural enemies, and the 'deep blue sea' of '.. food

that, at best, is often nutritionally inadequate and, at
worst, is simply poisonous'. Finally, Keith (19163)
suggested because of hare cycles on a lynx-free island
that the (supposedly)classical snowshoe hare/Canada
lynx predator-prey oscillations (see Fig. 5.la) are

CHAPTER 5 : PREDATION 165

actually a result, at least in part, of the hare's Finally, in general, we have seen that very similar
interaction with its food. It is clear, in short, that to principles apply to all those interactions included
truly understand the patterns of abundance exhibited within the blanket term 'predation'; and in all cases,
by different types of species, we must consider, there has been considerable progress in understand-
synthetically, all of their interactions. ing the details underlying the relationships. Overall,
however, it must be admitted that while there is
Nevertheless,it is apparent from this chapter that we nothing in the observed patterns of abundance that
have at our disposal a wealth of plausible explanations should surprise us by being essentially inexplicable,
for the patterns of abundance that occur. Individually, we are rarely in a position to apply specific explana-
each of the components can give rise to a whole range tions to particular sets of field data.
of abundance patterns, and in combination their po-
tentialities are virtually limitless. What we lack, regret- 5.15 Harvesting
tably, is the sort ofdetailed field informationthat would
allow us to decide which explanations apply in which In this section, we are concerned with examining the
particular situations. We are forced, simply, to con- dynamics of populations that regularly suffer loss of
clude-and this is especially apparmt in the mathe- individuals as a consequence of the deliberate atten-
matical models-that observed patterns of abundance tions of mankind, through cropping or harvesting. In
reflect a state of dynamic tension between the various all predator-prey interactions the predator will profit
stabilizing and destabilizing aspects of the interactions. by maximizing the crop taken while ensuring that the
prey does not become extinct; and we have already
Beyond this, we can point to food aggregation, and seen, in this chapter, that in natural populations the
the 'aggregativeresponses of consumers to this aggre- survival of a certain number of prey individuals will
gation, as probably the single most important factor tend to be ensured as a by-product of certain features
stabilizing predator-prey interactions. Indeed, Bed- of the interaction. In the instances involving man as
dington et al. (1978)noted that the biological control of the harvester, however, the problem of conscious
insect pests is characterized by a persistent, strong harvest optimization remains, i.e. the problem of
reduction in the pest population following the intro- ensuring neither over-exploitation (hastening extinc-
duction of a natural enemy (i.e. stability at low q);and tion) nor under-exploitation (cropping less than the
they suggested, from the analysis of mathematical prey population can sustain). It is with this in mind
models, that the mechanism which is most likely to that many ecologists have devoted considerable atten-
account for this is the differential exploitation of pest tion to the manner in which plant and animal
patches in a spatially heterogeneous environment. populations can be exploited for the benefit of man-
(Even more important than this is the fact that they kind.
were able to marshal several field examples in support
of this suggstion.)In brief, it appears that aggregation 5.15.1 Characteristics of harvested populations
is of very general significance in maintaining prey
populations at stable, low densities. From the outset we can see, from common sense
alone, that the first, immediate consequence of har-
A further insight, stemming from mathematical vesting is to reduce the size of the population, and this,
models, is that predator-prey systems can exist in in turn, will generally affect the life expectancy and
more than one stable state. Indeed the model in the fecundity of the survivors in the harvested population.
previous section is only one of a range that possess Nicholson (1954a), for instance, cultured Australian
such properties (see May, 1977, 1979 for reviews), sheep blowflies (Lucilia cuprina) under conditions that
and this range covers predator-prey, host-parasitoid, restricted adult food supplies but provided larvae with
plant-herbivore and host-parasite interactions. We a food excess (ensuringvery little larval mortality). His
therefore have a ready explanation for parasite epi-
demics, pest outbreaks, and sudden, drastic alter-
ations in density generally.

166 PART 2: INTERSPECIFIC INTERACTIONS

Table 5.9 Effects produced in populations of the blowfly, Lucilla cuprina, by the destruction of different constant percentages
of emerging adults. (After Nicholson, 1954a.)

results (Table 5.9) show that, whilst the adult popula- high and largely unchanged by the removal. More-
tion declined with increasing exploitation, both pupal over, the population size prior to harvest and the rate
production and adult emergence increased, resulting in of subsequent regrowth are also related to the size of
an overall increase in the 'birth'-rate. Coincident with repeated harvests taken after a unit period of recovery
this rise, moreover, was a decrease in the rate of (Figs 5S O and 5.51). At low population sizes, succes-
natural adult death, and this led to extension of the sive harvests are small (Fig. 5S l a ) ,but with increasing
mean adult life span. The reduction in population size population size they increase steadily to a maximum
resulting from the act of harvesting, therefore,brought and then decline; and as Fig. 5.51b shows, there is a
about two changes: (i)increased fecundity of surviving parabolic relationship between harvest size and popu-
adults; and (ii) reduced adult mortality; and, indeed, lation size. There is, therefore,an optimal size at which
we might have expected this from our knowledge of a population can be maintained, which, on repeated
the effects of intraspecfic competition (Chapter 2). harvesting, ensures a maximum sustainable harvest;
Reduction in population density in a resource-limited and repeated harvesting at this size is also followed by
environment tends to benefit the individual survivors. the rapid recovery of the population because its rate of
regrowth is maximal. Note that in the case of Nichol-
The act of harvesting also has an important effect on son's blowflies (see Table 5.g), harvest-size ('adults
the rate of regrowth of the population. This can easily destroyed per day') continued to rise as the exploita-
be seen by considering a population undergoing tion level increased and the mean population size
density-dependentregulation and following a logistic decreased. There was also a consistent rise in the rate
growth curve (Fig. 5.50). At a point in time, t , (when of regrowth (compounded from an increased number
the population has reached a size, N,) we remove of 'adults emerged' and a decreased rate of 'natural
some individuals from the population, i.e. we crop it. adult death'). It seems, therefore, that these popula-
The population will continue to grow, but from the tions were towards the right-hand side of the appro-
reduced size (H,) that it had already reached at time priate parabola, where intraspecific competition was
t,. Clearly, the rate of growth of the population after fairly intense. For the logistic equation itself, the
harvesting (the slope of the curve) will depend on the optimum population size is in fact K12 (where K is the
time at which the harvesting occurred. If removal carrying-capacity)and we can show this mathemati-
takes place early (Fig. 5.50a), then the rate of subse- cally by differential calculus. (Readers familiar with
quent growth will be low, and indeed reduced by calculus will know that 'finding the maximum of a
harvesting. If removal takes place late (Fig. 5.50c), function' involves differentiating the appropriate
then the rate of growth, though increased,will also be equation-the logistic in this case-setting the deriva-
low. If, however, it takes place when the population is tive to zero, and solving for N.) The important biolo-
growing most rapidly (Fig. 5.50b), then the rate will be

CHAPTER 5 : PREDATION 167

Some justification for this type of model is provided
by studies of harvested fish populations. Since it is
impossible to estimate the sizes of oceanic fish popu-
lation numerically, stock sizes (or total biomass)can be
used instead. The equation describing the changes
must be:

Sycar = Sycar + Rec + G - D - C

where S is stock size, and Rec is recruitment of stock, G
growth, D loss by natural mortality and C stock caught
by fishing. If stock size remains constant from one
year to the next then:

Fig. 5.50 The effects of harvesting at (a) early, (b) middle In other words, at any stock size, if the stock is in a
ancl (c)late stages of population growth with the population steady state, then the gains from recruitment and
growing logistically. For further discussion, see text. growth will be exactly offset by the losses from natural
mortality and fishing. So, if the stock is growing
gical point to realize, however, is that the maximum logistically,and we wish to maintain steady-state condi-
sustainable harvest is not obtained from populations tions, then we must exploit the stock according to the
at the carrying-capacity, K, but from populations at harvest parabola. Figure 5.52a shows the catch sizes
lower, intermediate densities, where they are growing of yellow fin tuna from 1934 to 1955 plotted against
fastest. The exact shape of the hawest parabola depends fishing intensity (a measure recorded by the fishing
on the growth function, and the harvest parabolas of industry that may easily be related to stock size).
different sigmoidal growth curves will vary accord- Interestingly, the actual recorded catches do seem to
ingly. The 'logistic' parabola is symmetrical because lie on the estimated harvest parabola; and indeed the
the logistic curve itself is symmetrical. size of the catch from 1934 to 1950 was increased by
100 million pounds without destroying the fish popu-
lation, and from 1948 onwards large catches could be
sustained. A fairly similar situation is shown for a
lobster fishery in Fig. 5.52b. It does appear, therefore,
that at least some fisheries are regulated in a density-
dependent and possibly even logistic fashion.

It would be gratifying to be able to claim that the
management and conservation of all exploited fish
populations are based on detailed knowledge and
careful modelling. Yet, while substantial research in
the fishing industry has deepened our understanding
of the factors influencing the stability and yield of
exploited populations (Beverton & Holt, 1957; Gul-
land, 1962),for three main reasons we are still unable
to fully comprehend their dynamics. The first reason is
a genuine ignorance of population dynamics: obtain-
ing the data for long-lived species demands extensive
and often expensive study over many seasons. The

168 PART 2 : INTERSPECIFIC INTERACTIONS

Fig. 5.51 (a) Growth increments in unit periods of time at fish were spawned (Fig. 5.53b), there are other envi-
different stages of growth of a population growing ronmental variables that play a highly significant role.
logistically. (b) The parabolic relationship between these The movements of ocean currents within the coastal
increments (or, alternatively, the size of repeated harvests) nursery grounds, for instance, are often seasonally
and population size. erratic, and young planktonic fish may be swept into
the deep ocean where they are unlikely to survive.
second reason is that many populations are struc- Temperature, currents and other factors, then, are all
tured, and we shall consider this problem presently. likely to blur any relationship between the size of the
The third reason is environmental variability. The breeding population and the number of offspring born
environment, particularly climate, frequently affects to it. However, Radovich (1962)has argued that the
natural populations independently of density, and recruitment relationship is in fact curvilinear with a
environmental fluctuations may, therefore, create recognizable optimum. Thus, in Fig. 5 . 5 3 ~three
considerable difficulty in the evaluation of the effects curves have been drawn: I1 gives the best statistical fit
of harvesting on population size. This is illustrated by to the data, while I and I11 have been drawn by
data on the Pacific sardine, Sardinops caemlea, off the inspection to encompass the variability observed. I
coast of California, where the annual catch exhibits represents the relationship we might expect when
extreme fluctuations (Fig. 5.53a). There has been environmental factors combine to give the 'best'
considerable controversy over the relationship be- possible conditions, and I11 the relationship for the
tween the population sizes of: (i) the 'spawning stock'; 'worst' conditions. The clear consequence of this
(ii)the mature breeding fish (2 years old and over);and underlying curvilinearity is that recruitment tends to
(iii) the 2-year-old recruits; and although Murphy be little affected by changes in spawning stock size
(1967) has established that the number of recruits is near the optima,but is more drastically affected at the
partly determined by the water temperature when the extremes. The utility of these curves stems from the

CHAPTER 5: PREDATION 169

Fig. 5.52 (a)The relationship between fishing intensity and
yield of yellow fin tuna in the eastern pacific between 1934
and 1955. The curved line is the estimated harvest
parabola (Schaefer, 1957).(After Watt, 1968).(b) Similar
relationship for the western rock lobster Palinurus cygnus
(Hancock, 1979).(After Beddington, 1979.)
ability they give us to assess the levels of fishing that
safeguard against extinction even under the poorest
circumstances. If for instance we wish to ensure a
consistent recruitment X (Fig. 5.53c), then the bare
Fig. 5.53 (a)The annual catch of the Californian sardine,
and (b) the effect of sea temperature on the recruitment of
the sardine (Watt, 1968).(c)The average parabola relating
the number of recruits to the size of the spawning stock,
together with the maximum and minimum parabolas
(Radovich, 1962). For further discussion, see text. (After
Usher, 1972.)

170 PART 2: INTERSPECIFIC INTERACTIONS

minimum of stock that must be left after harvesting (Daphnia pulicaria), which breed continuously
under the 'average' conditions is S,. The difference, throughout the year and possess a prodigious capabi-
S, - S,, represents the additional stock that should not lity for population increase by virtue of the very large
be harvested if we wish to guarantee recruitment number of eggs they lay. They are also particullarly
when recruitment is lowest, in the 'poorest' environ- convenient for harvesting studies because they have a
ment. Overall, the problems of incorporating environ- short life span; and in their population structure we
mental variability into a predictive harvesting model can recognize young, adolescent and adult foirms.
are readily apparent (but see lles, 1973 for a more Figures 5.54a and 5.54b show the consequences of
sophisticated attempt to overcome these problems). taking harvests every 4 days from Daphnia populations
which were maintained with a constant supply of food
We are now in a position to review some of the at a constant temperature for about 9 months. Har-
consequences of harvesting. First, it is quite clear that vesting was aimed specifically at the smallest size-class
populations can be systematically exploited with a which (apart from a few young adolescents)contained
consequent reduction in population size. It is also Daphnia which were less than 4 days old. The popu-
clear that harvests can be optimized (i.e. the size of lation structure in the absence of harvesting is shown
repeated harvests maximized) by harvesting at some in Fig. 5.54a,%%':the numbers per class decline to a
intermediate density where the population growth-rate minimum at,the large adolescent stage (LA), and the
is greatest. However, such harvesting is only sustain- adult class is the most abundant (the data are pre-
able at an exploitation-rate that allows sufficient time sented on a logarithmic scale). Increasing the propor-
for the replacement of cropped individuals. This tion of young fleas harvested had two main effects.
period of time will depend on the fecundity and The first was to reduce consistently the total popula-
generation time of the species in question. It is easy to tion size (Fig. 5.54a & 5.54b). The second was to alter
imagine an intensity of harvesting (an exploitation- the population structure (Fig. 5.54a): as the harvesting
rate) in excess of the replacement-rate, such that the of small Daphnia increased from 25 to 90% of their
population declines to extinction ('stepping down' the number, the discrepancy between young, small and
curve in Fig. 5.51a). We can also conclude (from the large adolescent frequencies diminished. We may note
Lucilia population, and from a consideration of in- in addition, however, that 90% exploitation did not
traspecific competition)that the fitness of survivorsin a push the population towards extinction, and indeed,
harvested population is often increased. there was a consistent rise in total yield as
exploitation-rate increased (Fig. 5.54b). This occurred
5.15.2 Harvesting in structured populations because the yield per individual rose quickly enough to
more than compensate for the decline in population
In practice, many harvesting procedures deliberately size (Fig. 5.54b); and this consistent rise in the fitness
select individuals to be cropped. In fishing, for in- of survivors (cf. Nicholson's blowflies) was also1 re-
stance, many nets permit small fry to escape so that flected by a steady increase in life expectancy aver-
large (and possibly older) individuals are captured. In aged over all size-classes (Fig. 5.54a). Such increases
other situations (seal culling for instance) small or were associated, moreover, with a consistent rise in
young members are the principal object of the harvest. the proportion in the population of the exploited,
We must therefore consider the age and size composi- youngest class prior to harvesting (Fig. 5.54c), i.e. a
tion of populations being exploited. further indication of the changing population struc-
ture. This presumably occurred because exploitation
To investigate the properties of such structured reduced the numbers available to enter the older
populations in the field under particular harvesting classes, and also reduced intraspecific competition,
regimes is an immensely arduous task, and, in conse- leading to an increased per capita production of the
quence, many workers have utilized relatively simple youngest class such that its proportion in the poyula-
laboratory systems. A good example is the work of
Slobodkin and Richman (1956) using water fleas

CHAPTER 5: PREDATION

Median life expectancy
of survivors (days):

YSALAA YSALAA YSALAA YSALAA YSALAA

Exploitation rate 0% 25% 50% 75% 100%

Fig. 5.54 (a)The effect of harvesting on the

size structure of a Daphnia population, 0 25 50 75 90100
Exploitation rate of young (%)
where the size-classes are: Y, young; SA,

small adolescent, LA, large adolescent; and

A, adult. The shaded halves of the

colu~mnsrepresent those removed and the

open halves those left after harvesting.

(b)The effect of harvesting on mean

population size before harvest (- - -),

mtoetaclmypieoldpu(-la-ti.o.n-) size after harvest (-),
and yield per individual
(- .--). (c)The effect of harvesting on the

proportion of the youngest class in the

total population (all Slobodkin & Richman, 20 40 60 80 100
Exploitation rate (%)
1956). For further discussion, see text.

(After Usher, 1974.)

tion increased. For the exploitation-rates examined tant to determine whether 'yield' should be measured
here, these effects apparently more than compensated in terms of number or biomass, since the two are not
for the losses due to harvesting, in that there was a necessarily equivalent. Figure 5.5 5 (Usher et al., 1971)
consistent increase in yield. Beyond some higher illustrates this problem in populations of an inverte-
point, however, increases in productivity would be brate herbivore, the collembolan Folsomia candida. The
unable to compensate for reductions in numbers, and populations were subjected to regular harvesting over
the population would decline to extinction. In the all size-classes at rates of 0, 30 and 60%of the total
present case, this point is apparently between 90 and population every 14 days, and the animals were
100% exploitation. always given excess food. Harvesting reduced the
numerical size of the Folsomia populations far less
Parenthetically, we should note that once we have than it reduced their biomass. Moreover, the crop
accepted that populations are structured, it is impor-

172 PART 2: INTERSPECIFIC INTERACTIONS

Fig. 5.55 The effects of exploitation

on a population of the collembolan
Folsomia candida; control, 0%

exploitation(-1; 30% exploitation
(- --); 60% exploitation (. . a).

(After Usher et al., 1971.)

taken with 30% exploitation yielded a higher biomass sharply, and at 75% it led to the extinction of the
than with 60% exploitation, while the actual numbers population. Thus, we have confirmation that the
of arthropods taken were approximately equal in the maximum sustainablebiomass yield must be attained
two regimes. at some intermediate level of exploitation-probably
close to or just below 25%. The second feature is that
It is legitimate at this point to ask whether field harvesting had a discernible effect on both the size-
populations of vertebrates behave in a similar way to structure of the guppy population (since the mean
these laboratory populations, and, regrettably, study- length of the harvested fish declined noticeably with
ing fish populations in the same detail presents almost increasing exploitation), and the age-structure
intractable experimental problems. Silliman and Gut- (Fig. 5.57). This, like the data on Daphnia, illustrates
sell (1958),however, have succeeded in modelling the the general conclusion that the structure ofpopulations
process of commercial fishing practices, with density- changes when specific classes are exploited.
regulated populations of guppies (Lebistes reticulatus).
Selective fishing, permitting small fish to escape, was The guppy data also illustratethe process of overex-
applied every 3 weeks (correspondingto the reproduc- ploitation, i.e. of harvesting at a level in excess of the
tive periodicity of the fish), and the catch size was maximum which is sustainable. In contrast to the
carefully controlled but changed after 40-week peri- Daphnia data (see Fig. 5.54c), there appears to be a
ods, so that the effects of a range of exploitation from consistent decline in the proportion of the exploited,
10 to 75%could assessed. Although the results are by adult class between 25 and 75% exploitation (see
no means as clear as those we have examined so far, Fig. 5.57). These higher exploitation-rates apparently
two points of interest are reinforced (Fig. 5.56). First, result in a lowering of the adult proportion to a level
as we might expect, the size of the catch, measured as that is unable to replenish the fish removed by
numbers or total biomass, was related to the level of harvesting. Increases in the fitness of survivors are
exploitation;it was higher (though decreasing slightly) insufficient to compensate for the decreases in num-
at 25% than it was at 10010,but at 50% it decreased bers, and extinction would inevitably follow if harvest-

CHAPTER 5: PREDATION 173

Fig. 5.56 Exploitation of guppy

populations: biomass of the catch

(-)I number of fish caught (- .-.),

average length of harvested fish

.(. . . a). (After Silliman & Gutsell,

1958.)

juveniles are being exploited, because some of these
juveniles would fail to reach maturity anyway. To
generalize, we might suggest that maximal yields can
be obtained from structured populations by removing
those individuals that are likely to contribute least to

the production of progeny in the future.

Fig. 5.57 Adult proportions in Silliman and Gutsell's (1958) 5.13.3 Incorporating population structure:
guppy populations at different rates of exploitation, with matrix models of harvesting
the oresumed relations hi^ interoolated between the ~oints.
In the logistic model of harvesting, we did not distin-
ing at these levels (i.e. overexploitation) were to guish between age-classes of individuals;and this may
continue. justifiablybe seen as a neglect of the significanceof

Comparisons between animals as different as Daph- population structure and of age-specificfecundityand
nia and guppies must obviously be carried out with survival. Age-determined parameters can be incorpo-
caution. Nevertheless, it is interesting to note that rated into a model of a harvested population, however,
when a very juvenile portion of the Daphnia popula- by use of the matrix model from Chapter 3, and Usher
tions is exploited, the maximum sustainable yield is (1972)has illustrated the practical application of this
obtained at a very high rate of exploitation (greater approach in an examination of the potential of the
than 90010);but when adult guppies are exploited, the blue whale (Balaenoptera musculus) for harvesting. He
peak in sustainable yield occurs at a much lower collated data into a transition matrix describing the
exploitation-rate. It appears that in a structured pop- fecundity and survival of female blue whales, but as
ulation, higher exploitation-rates can be sustained if Table 5.10 shows, the paucity of these data necessi-
tated the structuring of the population into only six
2-year age-classes up to the age of 12 and a single

age-class thereafter (12 + ).

As explained in section 3.5, the subdiagonal ele-
ments of the matrix are measures of the probability of

174 PART 2: INTERSPECIFIC INTERACTIONS

Table 5.10 Transition matrix for a population of female 1: 1.The value for the 12 + class has ben reduced to
blue whales (After Usher, 1972.)
0.45 to take account of irregular breeding in older
survivalfrom one class to the next (0.77 for each of the whales.
first six 2-year periods). Similarly, the lower right-
hand corner element of the matrix is the 2-yearly Using the iterative procedure outlined previously
survival-rate for females over 12. Its value of 0.78 (section 3.5), we can calculate the net reproductive
rate of the population (R)when it has achieved a stable
gives individuals entering the 12 + class a mean life age structure. This equals 1.0072, which, being very
close to 1, indicates that the whale population can
expectancyof 7.9 years; female blue whales can live to grow at only a very slow rate. We can now calculate
ages of between 30 and 40 years. The top row of the level of exploitation that the population can
values in the matrix give the fecundity terms for withstand, without entering a decline. In percentage
females: they reflect the fact that breeding does not terms this is {(R - 1)lR)X 100, or 0.71°/o. In other
start until the fifth year, and that full sexual maturity
does not arrive until females are 7 years old. The .words, the sustainable yield of the population every
fecundity terms of the 8-9 and 10-11 year classes are
equal to 0.5, because females produce only one calf year is approximately 0.35Oo/ of every age-class
every 2 years and the sex ratio of the population is Harvesting-rates in excess of this would lead to losses
that the species could not counteract unless homeo-
static mechanisms acted to alter the fecundity and
survival values in the matrix. In view of this, and the
intensity with which whaling has been carried out, it
is not surprising that blue whale numbers declined
significantly in the early 1930s; and the species has
been threatened ever since.

Further models, taking into account the effects of
differentially exploiting different classes of a struc-
tured population, are beyond the scope of this book.
The interested reader can consult Law (1979) and
Beddington (1979).

Part 3

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Chapter 6

6.1 Introduction 6.2 Nicholson's view

There have, in the past, been contrasting theories to A.J. Nicholson, an Australian ecologist, is usually
explain the abundance of animal and plant popula- credited as the major proponent of the view that
tion~I.ndeed, the interest has been so great, and the density-dependent, biotic interactions (which Nichol-
disagreement often so marked, that the subject has son called 'density-governingreactions') play the main
been a dominant focus of attention in population role in determining population size (Nicholson, 1933,
ecology throughout much of this century. The present 1954a, b, 1957, 1958).In his own words: 'Governing
view, however, is that the controversy has not so reaction induced by density change holds populations
much been resolved, as recognized for what it really is in a state of balance in their environments', and
(like so many other controversies): a product of the
protagonists taking up extreme positions and arguing '...the mechanism of density governance is almost
at cross purposes. In this chapter we shall confine
ourselves to a brief summary of the main arguments in always intraspecific competition, either amongst the
the controversy (sections 6.2-6.5). This should pro- animals for a critically important requisite, or amongst
vide the necessary background, but will avoid our natural enemies for which the animals concerned are
making too much use of historical hindsight. It would
be unwise (and unfair) to analyse the arguments in too .requisites'. Moreover, although he recognized that
great a detail, since most were propounded between
1933 and 1958 when understanding of population '. . factors which are uninfluenced by density may
ecology was even less sophisticated than it is now.
More inclusive reviews may be found in Bakker pthraotdtuhceeypornolfyoudniddseoffbeyct's.u..pmonoddifeynisnigtyt'h, eheprcoopnesritdieesreodf
(1964),Clark et al. (1967),Klomp (1964)and Solomon
(19164). In section 6.6 we shall illustrate how the the animals, or those of their environments, so influ-
problem seemed to have been resolved by the exten- encing the level at which governing reaction adjusts
sive life table analysis of a Colorado beetle population. population densities'. Even under the extreme influ-
In section 6.7 we shall look at the resurrection of the
problem in the 1980s, by exposing just what life table ence of density-independent factors '...density gover-
analyses do not tell us, particularly with respect to
spatial density-dependence. We shall end this section nance is merely relaxed from time to time and
by suggesting how life-table data will have to be subsequently resumed, and it remains the influence
collected in the future, and report some encouraging which adjusts population densities in relation to envi-
results from the most extensive set of census data yet ronmental favourability' (all Nicholson, 1954b). In
published, which reaffirms the importance of density- other words, Nicholson may be taken to represent the
dependent population regulation. Then, in sections
6.8-6.11, we shall deal with a heterogeneous se- view that density-dependentprocesses '... play a key
quence of topics, all of which have a bearing on the
general question of how population sizes are regulated role in the determination of population numbers by
and determined. operating as stabilizing (regulating) mechanisms'
(Clark et al., 1967).

6.3 Andrewartha and Birch's view

Baarnyedc'o..n....trinpaslagt,yetnhneeorvailpe,aworfttmhianint doderetnoesrrimtsyei-ncdoinenpgdeatnrhydeeinmatbppuroonrcdtaeasnnscceees,

of some species' (Clark et al., 1967) is most com-
monly attributed to two other Australian ecologists,

178 PART 3: SYNTHESIS

Andrewartha and Birch. Their view is as follows 6.4 An example: Thrips imaginis
(Andrewartha & Birch, 1954):
Davidson and Andrewartha (1948a, b) studied pcrpu-
The numbers of animals in a natural population lation changes in a phytophagous insect, the apple-
may be limited in three ways: (a) by shortage of blossom thrips Thrips imaginis which lives on roses
material resources, such as food, places in which throughout southern Australia. They obtained esti-
to make nests, etc.; (b)by inaccessibility of these mates of abundance for 81 consecutive months by
material resources relative to the animals' counting the number of thrips on a sample of 20 roses
capacities for dispersal and searching; and (c)by on each of approximately 20 days each month
shortage of time when the rate of increase r is (Fig. 6.1); and then, for a further 7 years, they
positive. Of these three ways, the first is obtained similar estimates for spring and early sum-
probably the least, and the last is probably the mer only. In addition, they monitored local tempera-
most important in nature. Concerning(c),the ture and rainfall throughout the same period. By the
fluctuations in the value of r may be caused by use of a 'multiple regression' analysis (see for instance
weather, predators, or any other component of Poole, 1978), they were able to 'account for' 78% of
environment which influences the rate of the variance in the yearly peak of thrip numbers by
increase. reference to four climatic factors: (i) the suitability of
temperature for development up to 31 August; (ii)the
Andrewartha and Birch, therefore '. .. rejected the suitability of temperature for development in Septem-
ber and October; (iii)the suitability of temperature for
traditional subdivision of environment into physical development in August of the previous season; and
and biotic factors and 'density-dependent' and (iv) the rainfall in September and October. In other
'density-independent' factors on the grounds that words, knowledge of these four factors in any one
these were neither a precise nor a useful framework year would allow the size of the peak in thrips
within which to discuss problems of population ecol- numbers to be estimated with a good degree of
ogy' (Andrewartha & Birch, 1960). The views of statistical accuracy.
Andrewartha and Birch are probably made more
explicit, however, by considering one of their exam- Clearly, the weather (as represented by these four
ples; subsequent discussion of this example will then factors)plays a central and crucial role in the determi-
lead on to a crystallization of the current status of the
controversy.

Fig. 6.1 Mean monthly population
counts of adult Thrips imaginis in
roses at Adelaide, Australia
(Davidson& Andrewartha, 1948a).
(After Varley et al., 1975.)

CHAPTER 6: POPULATION REGULATION 179

Fig. 6.2 The mean monthly logarithm of population size for And when Smith (1961)applied methods that were so
Thrips imaginis and its variance. (After Smith 1961.) designed, he obtained excellent evidence of density-
dependent population growth prior to the peak. He
nation of T. imaginis numbers at their peak. Yet found that:
Andrewartha and Birch (1954)used this, and the fact 1 there were significant negative correlations between
that no 'density-dependentfactor' had been found, to population change and the population size immedi-
conclude that there was 'no room' for a density- ately preceding the spring peak; and
dependent factor as a determinant of peak thrip 2 there was, at the same time, a rapidly decreasing
numbers. The only 'balance' that Davidson and An- variance in population size (Fig. 6.2).
drelwartha (1948b)recognized was 'a race against time
with the increase in density being carried further in Moreover, Varley et al. (1975) were able to suggest
those years when the favourable period lasts longer, the existence of a strongly density-dependentmortality
but never reaching the point where competition factor,following the peak density. They did this by the
begins to be important'. use of two hypothetical examples (Fig. 6.3). In both
cases, the adult population (N,) is given a reproductive
As Varley et al. (1975) point out, however, the rate of 10, but is then subjected to a random (i.e.
multiple regression model is not designed to reveal density-independent, perhaps climatic) factor deter-
directly the presence of a density-dependent factor. mining the (peak) number of larvae (N,). (Note that
this random element, plotted as a k-factor, is the same
in Fig. 6.3a and 6.3b.) In addition, however, Fig. 6.3a
shows a weakly density-dependent mortality factor
acting on the larvae to determine the next genera-
tion's adult population size, while Fig. 6.3b has a
strongly density-dependent factor. As a consequence of
this, the climatic factor accounts for 91010 of the
variation in NLin Fig. 6.3b, but only 32% in Fig. 6.3a.
It appears, in short, as if a fairly strong density-
dependent factor must also have been acting on the
thrip population between the summer peak and the
winter trough; and, indeed, its presence can be

Fig. 6.3 In both (a) and (b)an adult -Q) Range Range
population. NA, is given a ten-fold of NL of NL
3
reproductive rate. A random factor Range Range
(representedby k,) then acts to leave (D of NA of NA
a larval density. N L , which is plotted Generations +
21 2 Generations ---+
on the top line; (a)has a weakly
density-dependent factor (k2);(b)has 2g '
a strongly density-dependent factor.
For further explanation, see text. Q
(After Varley et al., 1975.)
3m

0

180 PART 3: SYNTHESIS

inferred from Fig. 6.1 itself, where the maxima are dependent factor' can be isolated (cf. section 2.3). We
spread over an eight-fold range, with the minima must concentrate on the nature of the processes and
covering only a four-fold range. In fact, Davidson and effects which emanate from such interactions.
Andrewartha themselves (1948b) felt that weather 3 Andrewartha and Birch's view that density-
acted as a density-dependent component of the envi- dependent processes play no part in determining the
ronment during the winter, by killing the proportion abundance of some species clearly implies that the
of the population inhabiting less favourable 'situa- populations of such species are not regulated. Yet, as
tions'. (If the number of safe sites is limited and we have seen, even the data they themselves brought
remains roughly constant from year to year, then the forward contradicts this view. Indeed it is logically
number of individuals outside these sites killed by the unreasonable to suppose that any population is abso-
weather will increase with density.) They also felt, lutely free from regulation. Unrestrained population
growth is unknown, and unrestrained decline leading
hdoepweenvdeern, tththaetotrhyi's..d.isdinnceotNficithothlseong(e1n9e3ra3l,,pdpe. n1s3it5y-- to extinction is extremely rare. Moreover, the fluctua-
tions in almost all populations are at least limited
6) clearly excludes climate from the list of possible enough for us to be able to describe them as 'com-
"density-dependent factors".' mon', 'rare', and so on. Thus, we can take it that all
natural populations are regulated to some extent, and
6.5 Some general conclusions are therefore, to some extent, influenced by density-
dependent processes.
There are several conclusions of general importance 4 Furthermore, Andrewartha and Birch's view that
that we can draw from this single example: density-dependent processes are generally of minoir or
1 We must begin by distinguishing clearly between secondary importance,if it is taken to imply that such
the determination of abundance and its regulation. processes are unimportant, is not justified either. Even
Regulation has a well-defined meaning (section 2.3), in their own example of thrips, density-dependent
and by definition can only occur as a result of a processes, apart from regulating the population, were
density-dependent process. Conversely, abundance also crucially important in determining abundance.
will be determined by the combined effects of all the 5 Conversely, we must remember that the weather
factors and all the processes that impinge on a accounted for 78% of the variation in peak thrip
population. Certain other terms, particularly the con- numbers. Thus, if we wished to predict abundance or
trol of abundance, do not have a single, well-defined decide why, in a particular year, one level of abun-
meaning. As Varley et al. (1975)stress, a reader should dance was attained rather than another, then weather
pause and consider carefully what kind of meaning is would undoubtedly be of major importance. By impli-
being attached to such terms. cation, therefore, the density-dependent processes
2 We must also recognize that any dispute as to would be of only secondary importance in this respect
whether climate (or anything else) can be a density- (see also section 6.6).
dependent factor is essentially beside the point. In 6 Thus, it would be unwise to go along with Nichol-
Davidson and Andrewartha's thrips, at the appropri- son wholeheartedly. Although density-dependent pro-
ate time of the year, weather confined the insects to a cesses are an absolute necessity as a means of
limited amount of favourable 'space'. It is irrelevant regulating populations and are generally by no means
whether or not we consider weather or space to be the unimportant in determining abundance, they may
density-dependent factor. What is important is that well be of only minor importance when it comes to
the two, together, drove a density-dependent process, explaining particular observed population sizes. More-
and that the effects of weather on the thrips were, in over, because a11 environments are variable, the posi-
consequence, density-dependent.Weather, like other tion of any 'balance-point' is continually changing. In
abiotic factors, can interact with biotic components of spite of the ubiquity of density-dependent,regulating
the environment so that no single, simple 'density-

CHAPTER 6: POPULATION REGULATION 181

(i.e. balancing) processes, therefore, there seems little population estimates had confidence limits which
value in a view based on universal balance with rare were 1O0Ioor less of the mean; and
non-equilibrium interludes. On the contrary, it is 2 the timing of samples varied from year to year, to
likely that no natural population is ever truly at ensure that the effects of variable climate on the
equilibrium.This, remember, was the essential reason insects' rate of development were allowed for.
for our pragmatic definition of 'regulation' in section
2.3. The sampling programme provided estimates for
seven age-intervals, from which a life table could be
6.6 A life-table analysis of a constructed. These were: eggs, early larvae, late
Colorado beetle population larvae, pupal cells, summer adults, hibernating adults
and spring adults. In addition, one further category
These conclusions can be illustrated most effectively was included, 'females X 2', to take account of any
by analysing some life-table data in a way which has unequal sex ratios amongst the summer adults.
its roots in the work of Morris (1959) and Varley and
Gradwell(1960),and is discussed more fully by Varley Table 6.1 lists these age-intervals and the numbers
et al. (1975). As will become clear, the analysis allows within them for a single season, and also gives the
us to weigh up the role and importance of each of the major 'mortality factors' to which the deaths between
various mortality factors, and also allows us to distin- successive intervals can be attributed. Figures ob-
guish between factors that are important in determin- tained directly from sampling are indicated in bold
ing the total mortality-rate, factors that are important type, the rest were obtained by subtraction. Amongst
in determiningjuctuations in mortality-rate, and fac- the eggs, predation and cannibalization were moni-
tors that are important in regulating a population. In tored directly, since both processes left behind recog-
the present case, the analysis is applied to a popula- nizable egg remains. Conversely, reductions in
tion of the Colorado potato beetle (Leptinotarsa decem- hatchability due either to infertility or rainfall (mud
lineata)in eastern Ontario where it has one generation splash), were assessed by returning samples of eggs to
per year (Harcourt, 1971). the laboratory and observing their progress individu-
ally. Finally, the figure for eggs 'not deposited' was
6.6.1 Life-table data based on the difference between the actual number of
eggs and those expected on the basis of spring-adult
'Spring adults' emerge from hibernation around the number and mean fecundity. These five egg 'mortality
middle of June, when the potatoes are first breaking factors' have, for simplicity, been presented as acting
through the ground. Within 3-4 days oviposition successively in Table 6.1, although, in reality, they
begins, extending for about 1month and reaching its overlap considerably. The loss of accuracy resulting
peak in early July. The eggs are laid in clusters from this is generally slight (Varley et al., 1975).
(average size 34 eggs)on the lower leaf surface and the
larvae crawl to the top of the plant where they feed The principal mortality factor during the first larval
throughout their development passing through four age-interval (from hatching to second instar) was
instars. Then, when mature, they drop to the ground rainfall, since, during heavy downpours, the small
and form pupal cells in the soil. The 'summer adults' larvae were frequently washed from the leaves to the
emerge in early August, feed, and then re-enter the ground where they died in small puddles of water.
soil at the beginning of September to hibernate and This mortality was assessed by taking population
become the next season's 'spring adults'. counts before and after each period of rain. Con-
versely, amongst older larvae the major mortality
Details of the sampling methods can be found in factor was starvation.
Harcourt (1964),but it should be stressed that:
1 ,on each occasion, sampling was continued until Larval mortality due to parasites and predators was
insignificant, and well within the sampling error. In
the pupal stage, by contrast, parasitization was an
important cause of mortality; the numbers of sound

182 PART 3: SYNTHESIS

Table 6.1 Typical set of life-table data collected by Harcourt (1971) for the Colorado potato beetle (in this case, for Merivale
1961-62). Figures in bold type were obtained directly, the rest by subtraction.

Age inteml N u m b per Numbers 'Mortality log,, N k-value (k~,,)
he 96 potato 'dying' factor'
hllb 4.072 0.105 (( kh~,h.))
Fatly larvae -- Not dcpositcd 3.967 0.021
Iatc larvac 11 799 InfcrtiIc 3.946 0.021 (k~d)
PUpt-11cells 2531 Rain$]! O.W
Summer adults 9268 445 Cannibalism 3.925 0.024 4klP)
Fcmalc X L 882 4 408 Predators 3.8hl 0
I Iibcrnating adults 8415 1147 Rainfall 0.237 (k2)
Spring adults 7268 376 Starvation 3.838 0.m2
h892 0 IT. d o r y p h o r ~ 3.838 -0.017 (k(1
6892 3.501 1.312
3170 3722 Sex (52%fcmalc) 33..499 O.05R (k4)
3154 16 Emigration 2,916
3280 1.204 ((kh,) )
- 126 Frost 1.146
Ih (ki)
3264 k ~ a1 /
14
2

pupae and those containing puparia of the parasitic direct sampling in early July. There was no evidence of
tachinid fly, Dorgphorophaga doryphorae were esti- spring migration, nor of generation-to-generation
mated directly from the sampling data. changes in fecundity. (Where the data provided more
than one estimate of the numbers in a particular stage,
Amongst the summer adults, the principal cause of these were integrated into a single figure.)
'mortality' was emigration provoked by a shortage of
food. This was assessed from a series of direct counts As Table 6.1 shows, k-values have been computed
during the latter half of August. Damage by frost, the for each source of mortality, and their mean values
major mortality factor acting on hibernating adults, over 10 seasons for a single population are presented
was assessed by digging up a sample at the end of in the second column of Table 6.2. These indicate the
April. The number of spring adults was estimated by relative strengths of the various mortality factors as

Table 6.2 Summary of the life-table analysis for Canadian Colorado beetle populations (data from Harcourt, 1971). b and a
are, respectively, the slope and intercept of the regression of each k-factor on the logarithm of the numbers preceding its
action; r 2 is the coefficient of determination. (See text for further explanation.)

- _ZC

Mean wgmxlon on b B r=
km1
0.095 0.17 0.27
lims not dcposiicd h,, 0.026 - 0.0241 - 0.05 11.07 0.86
0.m - 0.01 0.00 0.Orl
Kggs infrrlilc km 0.090 - 0.W5 9.12
Rainbll on 0.036 0-00 0.1 5 0.02
1lp.g~wnnibali~cd krr 0.09 1 0.000
-- OS11 - 0.02 0.41
Egg p d a t i o n h, 0.185 - 0.002 0.03 ' - 1.05 0.05
l,an-c 1 IratnfaIl) 0.033 - 0.011 0.03 0.66
~IP -0.012 0.37 0.83
Larvac 2 (starvation) 1.543 0.010 0.37 -0.04
kz 0.170 0.136 0.04
-0.11 - 6.79
k5 - 0.029 0.01 O.H9
I'uP.~c (11. ~ O V / ~ ~ O ~ X P ) k4 2.65 0.13
Ibcqual scx ntio Q.004 0.02
Kmigmtion kg -.-. 0.-W2
0.906
Frost h 0.010

k7 .

CHAPTER 6: POPULATION REGULATION 183

contributors to the total rate of mortality within a ficient of each individual k-value on the total genera-
generation. Thus, the emigration of summer adults tion value, k,,,,. Podoler and Rogers (1975) have
has by far the greatest proportional effect, while the pointed out that a mortality factor that is important in
starvation of older larvae, the frost-induced mortality determining population change will have a regression
of hibernating adults, the 'non-deposition' of eggs, the coefficient close to unity, because its k-value will tend
effects of rainfallon young larvae and the cannibaliza- to fluctuate in line with k,,,,, in terms of both size and
tion of eggs all play substantial roles. direction. A mortality factor with a k-value that varies
quite randomly with respect to k,,,,,, however, will
6.6.2 'Key-factor' analysis have a regression coefficient close to zero. Moreover,
the sum of all the regression coefficients within a
What the second column of Table 6.2 does not tell us, generation will always be unity. Their values will,
however, is the relative importance of these factors as therefore, indicate their relative importance as deter-
determinants of the year-to-year fluctuations in mor- minants of fluctuations in mortality, and the largest
talily (remember sections 6.4 and 6.5). We can easily regression coefficient will be associated with the key
imagine, for instance, a factor that repeatedly takes a factor causingpopulation change (Morris, 1959; Varley &
significant toll from a population, but which, by Gradwell, 1960).
remaining constant in its effects, plays little part in
determining the particular rate of mortality (and thus In the present example, it is clear that the emigra-
the particular population size) in any one year. In tion of summer adults, with a regression coefficient of
other words, such a factor may, in a sense, be 0.906, is the key factor; and other factors (with the
important in determining population size, but it is possible exception of larval starvation)have a neglig-
certainly not important in determining changes in ible effect on the changes in generationmortality, even
population size, and it cannot help us understand why though some have reasonably high mean k-values. (A
the population is of a particular size in a particular similar conclusion could be drawn, in a more arbitrary
year. This can be assessed, however, from the third fashion, from a simple examination of the fluctuations
column of Table 6.2, which gives the regression coef- in k-values with time (Fig. 6.4). Note, moreover, that

Podoler and Roger's method, even though it is less

Fig, 6.4 The changes with time of 66-67 67-68 67-6868-69
the various k-values of Canadian
Colorado beetle populations. Note Merivale Ottawa Richmond
that there are two quite separate
scales on the vertical axis and that

k, is therefore undoubtedly the 'key
factor'. (Data from Harcourt, 1971.)

184 PART 3: SYNTHESIS

Fig. 6.5 (a) Density-dependent emigration amongst respectively, the slopes, intercepts and coefficients of
Canadian Colorado beetle summer adults which is determinationof the various regressionsof k-values on
overcompensating (slope = 2.65). (b) Inverse density- their appropriate 'log,o initial densities'. Three factors
dependence in the parasitization of Colorado beetle pupae seem worthy of close examination.
(slope = - 0.11).(c) Density-dependence in the starvation of
Colorado beetle larvae (straight line slope = 0.37; final slope The emigration of summer adults (the key factor)
of curve, based on equation 3.4 = 30.95). (Data from appears to act in an overcompensating density-
Harcourt, 1971.) dependent fashion, since the slope of the regression
( = 2.65) is considerably in excess of unity (Fig. 6.5a).
arbitrary than this graphical alternative, still does not (Once again, there are statistical difficulties in assess-
allow us to assess the statistical significance of the ing the significance of this regression coefficient, but
regression coefficients, because the two variables are these can be overcome (Varley & Gradwell, 1963;
not independent of one another.) Varley et al., 1975;Vickery & Nudds, 1Wl), and in the
present case density-dependence can be established
Thus, while mean k-values indicate the average statistically.) Thus, the key factor though density-
strengths of various factors as causes of mortality in dependent does not so much regulate the population
each generation, key-factor analysis indicates their
relative strengths as causes of yearly changes in as lead, because of overcompensation, to violent
generation mortality, and thus measures their impor- fluctuations in abundance (actually discernible from
tance as determinants of population size. the data: see Fig. 6.7). Indeed, the potatolColorado
beetle system is only maintained in existence by
6.6.3 Regulation of the population humans, who prevent the extinction of the potato
population by replanting (Harcourt, 1971).
We must now consider the role of these factors in the
regulation of the Colorado beetle population. In other The rate of pupal parasitism by D. doryphorae
words, we must examine the density-dependence of (Fig. 6.5b) is apparently inversely density-dependent
each of these factors. This can be achieved most easily (though not significantly so, statistically), but because
by using the method established in Chapter 2, of the mortality-rates are small, any destabilizing effects
plotting k-values for each factor against the common this may have on the population are negligible. Never-
logarithm of the numbers present before the factor theless, it is interesting to speculate that at low pop-
acted. Thus, columns 4, 5 and 6 in Table 6.2 contain, ulation levels presumably prevalent before the crea-
tion of potato monocultures by man, this parasitoid

CHAPTER 6: POPULATION REGULATION 185

could act as an important source of beetle mortality present case, is to use the observed, empiricalrelation-
(Harcourt, 1971). ships between beetle numbers and mortality-rates, as
summarized in the regression equations of columns 4
Finally, the rate of larval starvation appears to and 5 of Table 6.2. This approach literally uses what
exhibit undercompensating density-dependence has happened in the past to predict what is likely to
(though statistically this is not significant). An exami- happen in the future but it neglects any detailed
nation of Fig. 6Sc, however, indicates that the rela- consideration of the interactions occurring at each of
tionship would be far better reflected, not by a linear the stages. (Of course, hybrid models, using different
regression, but by a curve of the type discussed and approaches at different stages, can also be con-
examined in seciton 3.2.2. If such a curve is fitted to structed.)
the data, then the coefficient of determination rises
from 0.66 to 0.97, and the slope (b-value)achieved at In the present case we can argue from Table 6.2
high densities is 30.95 (though it is, of course, much that since, on average:
less than this in the density range observed). Hence, it
is quite possible that larval starvation plays an impor- k,, = 0.27 - 0.05 log,, (total potential eggs)
tant part in regulating the population, prior to the
destabilizing effects of pupal parasitism and adult or,
emigration.
log,, (total potential eggs)- log,, (eggsdeposited)
6.6.4 A population model = 0.27 - 0.05 log,, (total potential eggs)

This type of analysis of life table data allows us to then,
examine the role and important of each of the various
mortality factors acting on a population. It also illus- log,, (eggs deposited)
trates the differences between factors that are impor- = 1.05 log,, (total potential eggs)- 0.27.
tant in determining year-to-year changes in mortality-
rate, and factors that are important in regulating (or Similarly,
even destabilizing)a population. The final logical stage
in such an analysis is to construct a synthetic model log,, (fertile eggs)
that will allow us to predict: = 1.O1 log,, (eggs deposited)- 0.07,
1 the probable future progress of a given population;
and and so,
2 the consequences to the population of natural or
enforced changes in any of the mortality factors. log,, (fertile eggs)
= 1.01 { l.OS log,, (total potential eggs) - 0.27) - 0.07.
Such a model should have as many 'steps' as there
are 'stages' in the original analysis, and for each step If this is repeated for each stage, it allows us, eventu-
there are two forms that the model could take. Ideally, ally, to predict log,, (spring adults) from log,, (total
the mortality-rate at a particular stage should be potential eggs). In other words, we have constructed a
estimated directly from data on the mortality factor model that will allow us to predict the level of
itself. Thus, the mortality-rate of early larvae, for infestation in any one year given the number of eggs
instance, should be estimated from rainfall data, and laid the previous year. Alternatively, since we know
that of pupae from data on the numbers of D. that each female lays, on average, 1700 eggs, we can
doryphorae. Yet, the construction of these specific use the number of spring adults in any one year to
submodels for each of the stages requires extensive predict the numbers in the various stages during the
collection of data. An imperfect but simpler alterna- following year. In the present case, the model's
tive, and the one to which we shall be restricted in the predictions are illustrated in Fig. 6.6 (in which 'total
potential eggs' is used to predict 'spring adults' in each
of the 10 generations studied) and Fig. 6.7 (in which
the number of eggs present in 1961 at a single site is
used to generate an adult population curve for the

186 PART 3: SYNTHESIS

Fig. 6.6 The correlation between the observed number of Fig. 6.7 The observed and estimated numbers of Colorado
Canadian Colorado beetle spring adults and the number beetle spring adults at Merivale, based on the numbers
estimated on the basis of the k-value model. For further observed in 1961. For further explanation,see text. (Data
explanation,see text. (Data from Harcourt, 1971.) from Harcourt, 1971.)

following six seasons). In both cases the predictions effects are strong enough to outweigh the stochastic,
are compared with the observed figures, and, given density-independent effects (Vickery & Nudds, 1991).
that in Fig. 6.7 any errors are bound to accumulate, Two main methods have been used for investigating
the fit of the model is very satisfactory. This is a and detecting density-dependence. The first method
pleasing reflection of the amount that we can learn has been the detection of density-dependence in
from such analyses of life-table data; and it serves to particular mechanisms, such as specific parasitoids,
re-emphasize the need to consider carefully the inte- predators or competitors, from life-table data, the
grated effects of all factors, both biotic and abiotic, technique described in section 6.6 for the Colorado
potato beetle. The second method has had as its
wpohpeunlawtieonsese.k to understand the abundance of natural objective the detection of density-dependence of popu-
lation change from census data, counts made in
6.7 The problem re-emerges successive annual generations at the same stage in the
life cycle. Several statistical techniqueshave been used
In recent years a density-dependence debate has in order to achieve this end (Williamson, 1972;
resurfaced, but is now focused on the detection of Bulrner, 1975; Slade, 1977; Vickery & Nudds, 1984,
density-dependence rather than on the roles of 1991; Pollard, et al., 1987; Turchin, 1990). These
density-dependent and density-independent factors in techniques are a substitute for more extensive analy-
population regulation. As many populations persist for ses such as key-factor analysis. Each of these two main
long periods within some bounds there can be no methods are now seen to have problems which are
doubt that they experience some form of density- considered in sections 6.7.1 and 6.7.2, respectivelly.
dependent limitation. However, the difficulty is in
trying to establish whether the density-dependent 6.7.1 Life-table analyses

Dempster (1983)challenged the view that populations
of temperate Lepidoptera were regulated by density-
dependent mortalities from a study of 24 sets of

CHAPTER 6 : POPULATION REGULATION 187

life-table data. By plotting k-values against the loga- among patches of different host density, and concen-
rithm of the population density upon which the trate their efforts in high host density patches, then
mortality operates (see section 6.6), he found no individuals within high density patches may be at a
evidence for density-dependence in eight studies, greater risk than those in low density patches. Thus,
density-dependence due to intraspecific competition in high density patches there could be density-
in a.further 13 studies,and density-dependence due to dependent mortality, and in low density patches
natural enemies in just three. Dempster even ques- inverse density-dependent mortality; patches of inter-
tioned whether the density-dependent mortality from mediate density would show density-independent
natural enemies was sufficient to regulate the three mortality. The average mortality over all patch types
populations in question. If Dempster's results were might be density-independent, which is what would
representative for reasonably well-studied groups like be detected by key-factor analysis. The population of
Lepidoptera, they could also be true for other groups the host would nevertheless be regulated due to
ancl have wider implications for our views of popula- density-dependent mortality in the high host density
tion regulation in general. The theme introduced by patches. Such a phenomenon has been called spatial
Dempster was carried further by Stiling (1988), who density-dependence.
found that in 26 of the 58 insect species studies he
reviewed, there were no obvious density-dependent The problem of detecting density-dependence from
factors operating and that natural enemies were life tables when there are spatial effects can be
acting in a density-dependent manner in just 13 illustrated by reference to the work of Southwood and
Reader (1976)and Hassell et al. (1987)on the popula-
studies. This led him to conclude, like Dempster, that tion dynamics of the viburnum whitefly (Aleurotracheus
population densities of insects commonly fluctuate jelinekii). Viburnum tinus, an evergreen shrub, is the
between some lower floor and upper ceiling set by main British host for Aleurotracheus jelinekii. Whiteflies
limiting resources, rather than being maintained
around some more fixed equilibrium. What these appear at the end of May or early June and show little
studies illustrate is that either density-dependent fac- migratory activity so that the study bush (B in Hassell
tors, particularly involving natural enemies, are not et al., 1987)is effectively a closed system. The life cycle
operating as frequently as was earlier supposed in is as follows. Up to 30 eggs are laid per female in
insect populations, or the methods of detecting wax-coated clusters on the undersides of leaves. The
density-dependence from life-table data are not as eggs hatch after about 4 weeks, leaving the egg cases
sensitive as could be hoped. attached to the leaves. The larvae spend their lives,
after some wanderings in the first instar, on the lower
One of the principal problems in detecting density- leaf surfaces. By November most individuals have
dependence from life-table data is that the methods reached instar four in which they remain until feeding
used, and described in section 6.6, assume that the resumes again the following spring. Over 90% of
important mortality rates are functions of average leaves normally-survive into a second season. Two
population sizes in successive generations. Conse- sampling methods were used. An annual census was
taken prior to adult emergence in spring, from which
quently, these methods are not designed to test for the data for key-factor analysis were derived. Further
density-related processes that act differentially be- data were collected from more detailed studies on
labelled leaves of 30 cohorts per generation which
tween different fractions of the population within a were followed from egg to emerging adult. Thus
generation. This kind of heterogeneity within a gener- conventional temporal data is provided from the
ation can occur when there is differential survival annual censuses and within-generation mortality data
between patches of different population density. We are provided from the studies on individual leaves.
have already seen that such heterogeneity can have Key-factor analysis on 16 generations of data fails to
far-reaching consequences for competitive (section detect any evidence of density-dependent mortality.
4.16) and predator-prey (section 5.13) interactions.
For example, if parasitoids search non-randomly

188 PART 3: SYNTHESIS

However, in 8 of the 9 years in which within-leaf example, natural enemies, will be indistinguishable
studies were made, density-dependent mortality was from those apparently displaying no density-
detected. The failure to detect density-dependence dependent effects. Such populations could have been
from the conventional life-table data is the direct prominent in the studies of Dempster (1983) and
result of the failure to sample on the scale at which Stiling (1988).
this density-dependence operates, probably in con-
junction with the inevitable stochastic elements Second, many of the data sets analysed by Stiling
present. were not sufficiently long to make the detection of
density-dependence very likely. Hassell et al. (1989)
It is important, then, in undertaking life-table stud- analysed the same life tables as Stiling and found that
ies, to sample at a range of spatial scales. Heads and the proportion of studies in which density-dependence
Lawton (1983) have done this in their study of the was detected by the original authors increased mark-
natural enemies of the holly leaf-miner (Phytomyza edly with the number of generations available for
ilicis), an agromyzid fly. They measured leaf-miner analysis. The result was particularly clear in 28 life
densities along a holly hedge, and mortalities imposed tables from univoltine insects. This is an encouraging
by a guild of specific parasitoids and birds, using result when set against the problems of spatial
nested quadrats varying in size from 0.03 to 1m2.The density-dependence described above.
mortalities imposed by three species of pupal parasi-
toid were independent of density at all four spatial 6.7.2 Single-species time series
scales, but the larval parasitoid, Chrysocharis gemma,
revealed a range of responses to host density. In areas Several statistical tests have been devised with which
of high host density it aggregated demonstrably, and to analyse annual census data. The simplest ones
this effect was most notable at the lowest quadrat size.
As the data were pooled into increasingly large ,involve analyses of plots of X,+ against X,, where
sampling units the effect became progressively , ,X, = log, N, and X, + = log, N, + , N, being the popula-
weaker, and was undetectable at the largest quadrat +,tion size at time t and N, population size at time
size. The only real solution to the problem is to look
for more detailed information from each generation, t + l (e.g. Slade, 1977). Other tests are a little more
and to design a sampling programme that is stratified
in space (Hassell, 1985). complicated.For example Bulmer's (1975)autoregres-
sion technique specifically contrasts the density-
There are two additional very different and very
important reasons why analyses of life-table data have ,independent null hypothesis of population change,
failed to detect density-dependent mortalities. First,
Turchin (1990) point out that conventional life-table X, + = X, + E,, where E, is a normally distributed error
analyses fail to detect delayed density-dependence
because they are simply not designed to detect it. He +,term, against the density-dependent alternative hypo-
evaluated the evidence for density-dependence in the
population dynamics of 14 species of forest insects, thesis, X, = aX,+ r + E,, where a and r are constants.
and assessed the effect of regulation lags on the
likelihood of detecting direct density-dependence. In The randomization method of Pollard et al. (1987)uses
five of these 14 cases there was clear evidence for the correlation coefficient between the change in
direct density-dependence, but seven of the nine population size and the population size itself to
apparently non-regulated populations, were, in fact, compare the observed data with randomized sets of
subject to delayed density-dependent regulation. The annual abundances, which define the null hypothesis
implication of this work is that populations character- of density-independence. None of these tests is perfect
ized by delays in regulation brought about by, for and there are unresolved problems associated with
them. For example, none of the tests takes account of
measurement errors, which are thought to lead to an
increased likelihood of detecting density-dependence
where none exists. Most authors attemptingto analyse
census data for density-dependence have used several
methods and taken a consensus view of the results.

Some analyses of annual census data have clearly

CHAPTER 6: POPULATION REGULATION 189

Fig. 6.8 Percentage of time series with significant ( S 0 ) degree of bias as other data in that outbreak species,
density-dependence in (a) aphids, and (b) moths detected typically those of greatest interest to applied entomol-
ogists, are not disproportionately represented; the
--using three methods: (-) Bulmer; (- ---) Ricker; (- -) species chosen select themselves by getting caught in
standard traps. Woiwod and Hanski (1992) analysed
Pollard et al.; (-----) all methods. (After Woiwod & Hanski, 5715 time-series of annual abundances of 447 species
1992.) of moths and aphids in the UK. This was more than
two orders of magnitude greater than the number of
been deficient on several grounds, most notably, as time-series analysed in comparable studies. They
with the life-table data, on sample size. Many of the found that the incidence of significant density-
statisticaltests employed to detect density-dependence dependence increased with the length of the time
have low statistical power with the samples sizes series in the range 10-24 years, a similar finding to
frequently used. This problem does not affect the most analyses based on life-table data (Fig. 6.8). Density-
extensive study of density-dependence on insect pop- dependence was detected more frequently in species
ulation~by Woiwod and Hanski (1992)which uses the which did not show a significant trend in abundance.
large data base collected by the Rothamsted Insect This was thought not to be a statistical artefact;
Survey. This survey includes time-series data for populations showing a systematic change in size
huindreds of species of moths and aphids collected through time may indeed exhibit less density-
simultaneously from light traps (moths) and suction dependence than those showing no such change.
traps (alate aphids) throughout the UK. In 1976 there
were 126 light traps run by 460 volunteers (Taylor,
1986). The data set does not suffer from the same

190 PART 3: SYNTHESIS

Woiwod and Hanski used three different techniques to that this extensive work should finally lay to rest any
detect density-dependence. In what they considered remaining doubts about the extent of density-
the most sensitive statistical test, due to Bulmer dependence in natural populations, of insects, at least.
(1975), on data sets longer than 20 years, density-
dependence was detected in 79%of moth and 88%of 6.7.3 Population regulation in vertebrates
aphid time-series. The analysis also revealed that
noctuid moths and aphids showed highly significant The preceding'sectionsare dominated by insect exam-
differences between species, indicating that the ples because there are more data for insect popula-
density-dependence detected was a repeatedly mea- tions than for any other group of animals,and because
surable characteristic of species. In other words, insect pest problems have driven the need to under-
density-dependence was consistently detected in some stand insect population ecology. Table 6.3 gives the
species and not others. There was rather less system- numbers of studies reporting different causes of
atic variation between sites, but not surprisingly density-dependence in vertebrates. The table is de-
species did vary between sites if environmentalcondi- rived from Sinclair (1989) from which the source
tions, to which they may be sensitive, differed greatly. references can be obtained.
It should be remembered that previous studies had
often been concerned with unreplicated time-series Some patterns are very clear from this table. Large
from which it was often difficult to know what value terrestrial mammals seem to be regulated through
should be placed on the results. Finally, Hanski and their food supply, a conclusion based on studies in
Woiwod (1991)looked at evidencefor delayed density- which food supply either has been measured directly
dependence. Their results support Turchin's (1990) or inferred from its effects on fertility. This may appear
view that delayed density-dependence may be rela- a somewhat surprising conclusion given the number
tively common in forest moths, which often have of studies that have demonstrated an increase in prey
cyclic or outbreak dynamics (woodlandnoctuid moths numbers following predator removal, but as Sinclair
showed more delayed density-dependence than non- points out, most studies of this kind merely demon-
woodland noctuids), but in the majority of insects strate that predators limit prey populations, not that
Hanski and Woiwod (1991) found no more delayed they regulate them. In small mammals the biggest
density-dependence than would be expected by single cause of regulation is density-dependent exclu-
chance. The good agreement between the methods sion of juveniles from breeding colonies, but it is
used by Woiwod and Hanski (those of Ricker, 1954; difficult to know how much credence to give to this
Bulmer, 1975; Pollard et al., 1987) indicates strongly conclusion since predation is so rarely measured in
field studies of small mammal populations. In birds

Table 6.3 Number (Olo) of reports of separate populations recording cause of density-dependence.(After Sinclair, 1989.)

~itorsF ~ U I ' ' paisites -' -Dks~ire Wai ~ o d anto.

populaliorts

Large 0 6(hO) 4(40) 0 0 0 10
marine
mammals 1(1) 71 (991 0 0 2(3) 0 72
0
tape 14 (67) 5(24) 4(19) 1 (6) 0 14(67) L1
tcnmZriaI 5 (33) 8153) 0
mammals 0 T ($7) 15

, SrnaIl mammals
Birds