Population Ecology: A Unified Study of Animals and Plants

CHAPTER 2: INTRASPECIFIC COMPETITION 41

Fig. 2.9 Seed yield and yield

component responses to density in
determinate and indeterminate

growth forms of Vicia faba. (Data
from Pilbeam et al., 1991.)

(line A in Fig. 2.10). During ear formation (17 weeks) This mortality occurred during flowering, and clearly
and subsequently through to maturity (26 weeks) the if it had been greater a more pronounced parabolic
number of tillers changed little and the number of yield curve would have occurred.
fertile tillers was a constant fraction of the total.
However, the effects of density were reflected in Whilst intraspecific competition may act to reduce
changes in ear weight (Fig. 2.10, lines D and E). These plant size and reproductive output in relation to the
two phases in density adjustment, therefore, resulted developmental schedule of a plant, it is very com-
in a constant individual seed weight at harvest and in monly observed that the number of seeds produced by
an asymptotic yield response per unit area up to 100 a plant bears an exponential relationship to its size
plants m-2. Conversely, the depression of yield at the whether measured by mass or indirectly in the case of
highest sowing density in this experiment was a herbs by stem diameter (Fig. 2.11). (This relationship
does, however, break down where plants show an
consequence of density-induced mortality (note the age-dependentfecundity schedule as in the case of the
end-points of lines C, E and G are displaced to the left). trees.)

42 PART 1 : SINGLE-SPECIES POPULATIONS

Fig. 2.10 Yield-density relationships in Tviticum aestivum. resulting populations display a few large individuals
(Data from Puckridge & Donald, 1967.) A, tiller number per and a great many small ones. Similar distribution
plant after 14 weeks; B, tiller number per plant after 17 patterns have been observed in the growth of single-
weeks; C, number of fertile tillers per plant after 26 weeks; species populations of'crop plants, mixed tree planta-
D, individual ear weight after 17 weeks; E, individual ear tions and in animals (Begon, 1984).
weight after 26 weeks: F, total grain yield at 26 weeks;
G, individual grain weight after 26 weeks. There are two partially interrelated causes of skew-
ness in the size distributions of populations in the
2.5.3 Individual variability absence of the death of plants. Even in plant popula-
tions sown at the same time, individuals will germi-
Characterizing plant populations by an average re- nate at slightly different times, perhaps due to
sponse as we have done up to now masks the variation in the 'local' environment for germination or
mechanism by which compensation actually occurs. because of differences in seed size. These subtle
We must now consider the fates of individuals within differences in time of birth and size at birth will then
the population. become exaggerated as the growth of individuals
proceeds. In addition, however, competition may
Obeid et al. (1967)sowed Linum usitatissimum (flax) further exaggerate skewness. Larger individuals which
at three densities and harvested at three stages of will often be those that emerge first will be compara-
development, recording the weight of each plant tively unaffected by interference from smaller (and
individually. Figure 2.12 makes it quite clear that the later) neighbours. As a consequence they will grow
frequency distribution of weights is skewed towards quickly. Conversely, small plants and late emergers
the left and that the skewing is increased by the not only have to compete with a number of other,
passage of time and by an increase in density. The generally larger individuals, but have to do so on
unequal terms. Intraspecific competition accentuates
initial differences in size: large (early)individuals are
least affected by competition and grow larger still;
small (late) individuds are most &ected, and lag
further and further behind. Competition, then,.serves
to exaggerate size differences that initially may be
determined by environmental chance or variation in
initial starting capital: seed size.

The inevitable consequence of this process is the
development of a population in which size differences
will be exaggerated and a left-handed skew will
develop. A 'hierarchy of exploitation' will result in a
few dominant individuals with high growth rate and
disproportionate share of resources whilst 'the most
common type of plant in experimental (and natural)
plant populations is the suppressed weakling' (Harper,
1977).Experimental support for this view comes from
the work of Ford (1975)on sitka spruce (Picea sitchen-
sis) (Fig. 2.13). The greatest relative growth rate oc-
curred in trees with the largest girth size, these trees
being in the minority.

Finally we can consider the influence of limiting
resources at the level of the modular growth of an

CHAPTER 2: INTRASPECIFIC COMPETITION 43

Fig. 2.11 The relationship between fecundityand size in able influences the size of the module population and
seed plants. Lines show the relationship, on logarithmic hence hdividual plant size. If this limitation arises
scales, between observed sizes of individual plants and through individuals interfering with resource gather-
seeds produced per plant. In (a) size is expressed as basaI ing by one another (intraspecific competition) then
stem diameter for herbs (1-9) and a shrub (10) and as differential reductions in relative growth rate amongst
plants will occur and be reflected in the absolute size
diameter at breast height for trees (11-13). In (b) size is distribution of the population.
measured as above-ground biomass for perennial (1-3) and
annual (4-8) species. (After Watkinson & White, 1985.)

organism. Porter (1983b)grew single Fuchsia plants in 2.5.4 Self-thinningin plants
three different soil volumes and examined the accu-
mulation of modules with time. His results (Fig. 2.14) Certain aspects of these results on Linum are fre-
show that whilst growth was initially rapid in all quently repeated: the effects of intraspecific competi-
volumes, the onset of a stationary plant size-the tion on a growing plant population often become
plateau phase in the curves-was determined by the accentuated with the passage of time. These effects
amount of soil available. Limiting the resources avail- most commonly concern plant density (which de-

44 PART 1: SINGLE-SPECIES POPULATIONS

Fig. 2.12 Frequency distributions of plant weights in
populations of flax, Linum usitatissimum,sown at three
densities. (Data from Obeid et al., 1967; after Harper,
1977.)

CHAPTER 2: INTRASPECIFIC COMPETITION 45

Fig. 2.13 Relative growth rates
(RGR)and frequency distributions of
size (12 equal intervals of girth
classes) in Picea sitchensis, 29 years
after planting at three different
densities. (From Ford, 1975.)

creases with time), and mean plant weight (which

increases with time); these two parameters appear to

be closely related. 'Self-thinning' refers to the dynam-

ics of density-dependent mortality that occurs pro-

gressively in cohorts as individuals grow in size. It has

been studied most often in plant populations in

monocultures. The process is well illustrated in an

experiment by Kays and Harper (1974) in which they

sowed a series .of populations of Lolium perenne,

perennial ryegrass, at densities ranging from 330 to

10 000 seeds m-2, and harvested their populations at

subsequent occasions over a period of 180 days.

Growth in this species occurs through the accumula-

tion of ramets (shoots or tillers) on the genet (estab-

lished plant). Throughout the experiment (Fig. 2.15a)

there was continual death of genets. Initially,this was

most marked at the highest sowing density but latterly

it assumed a rate independent of sowing density.

During the early part of the experiment the number of

ramets per unit area increased but then declined to

give a similar number per m2 after 125 days regard-

less of initial genet starting density. This constancy

was repeated after 180 days but at a lower density. Fig- 2-14 The production of modular units by ~uchsia
Compensatoryprocesses within the population result- (cultivar Royal Velvet), in different soil volumes. The solid

ing in this constancy of tiller number derive from line shows the total number of modules assuming

of genets and rates of unrestricted exDonential rate. Plants received
birth and death of tillers on them (Fig. 2.15a). Plotting
abundant water but no additional nutrients. (From Porter,
the average genet size (mean above-ground plant 1983b.)

46 PART 1: SINGLE-SPECIES POPULATIONS

Fig. 2.15 Intraspecific competition affects tillers more than We have introduced two quite separate descriptions
genets. (a) Changes in tiller and genet density in of the manner in which plant populations respond to
populations of ryegrass, Lolium perenne, sown at a range of density: (i) yield-density or competition-density (as
densities. (b)The change in genet density and mean genet they were originally termed by Kira et al., 1953)effects
weight over the course of five harvests (H,-H,) for the same where mortality within the population does not occur;
ryegrass populations. Arrows indicate progression of time and (ii)the relationship between plant size and popu-
and dashed l i e s link populations harvested at the same lation density as self-thinning occurs.
time. (Data from Kays & Harper, 1974; after Harper, 1977.)
In Fig. 2.7b at each harvest time, the reciprocal
biomass) against the density of surviving genets relationship between mean plant biomass (W)and the
(Fig. 2.15b) we can examine the process of plastic density of plants (N)can be modelled as follows.
response intertwined with mortality. The essence of
studying self-thinning is to examine not a range of W = wm(l+ a ~ ) ~ (2.1)
initial densities all at one time, but the time course of a
single initial density as the individuals grow in size. where W, can be interpreted as the yield of a plant
The time courses show that the 'average' genet in a isolated from competing neighbours and a and b are
population grows, i.e. increases in biomass (points parameters that describe the competition effect. b has
move upwards in the graph) at a rate diminishingwith been variously interpreted as the rate at which the
density (dotted lines curve down), until at a critical effects of competition change with density (Vander-
size, depending on genet density, further biomass meer, 1989). or the efficiency of resource utilization
increases can only be achieved with a concomitant (Watkinson, 1984). These interpretations are, at best,
loss of genets (points in the diagram shift to the left). attempts to associate biological meaning to the para-
Eventually the time courses appear to progress along a meter b in an empirical model and we will return to
line with a gradient of - 1.5. their return to their functional interpretation in
Chapter 3. When b = 1,there is exact weight-density

CHAPTER 2: INTRASPECIFIC COMPETITION 47

compensation and a plot of log mean weight against where W and N are as defined above. However, it is
log density has a slope of - 1. Values of all three crucial to emphasize that this description is only
parameters (W,, a and b) will increase during the appropriate when the population is undergoing density-
course of plant growth and the change in a and b will dependent mortality. Originally, Yoda et al. (1963)
proposed that the exponent k had a fixed value of
be more rapid when the total availability of resources - 1.5 and the relationship was described as the
self-thinning rule or the ' - 312 power law'. The
is least. Figure 2.16 shows competition-densityeffects implication of this rule is that the self-thinning line
in Vulpiafasiculata populations grown at three fertilizer represents a boundary condition that is common to all
levels; note that values of b change more quickly when plant species. This implication and the interpretation
aftd biological significance of k and c has been a matter
plants are grown at low fertilizer levels. of strong debate (see Westoby, 1984; Lonsdale, 1990).
There are two major strands of evidence that support
The relationship between average plant size and the the view that there is an inverse relationship between
mean plant weight and density with a slope of
number of survivors in a self-thinning population can approximately - 312. The first is that the relationship
be described by exists for a great many species widely differing in
overall size and form (Fig. 2.17).
log (W) = log c - k log N (2.3)
The second is based around the explanation of the
Fig. 2.16 The influence of density and low (a),medium (Q) value of 312 on physical grounds. The biomass of a
plant will be proportional to its volume and in turn the
or high (0n)utrient regime on the shoot dry weight per volume occupied will be proportional to the spatial
plant of Vulpia fasiculata at three successive harvests. Note area in which the plant is growing. This area is
inversely related to density. Biomass will therefore be
how the lines become steeper as mean plant size increases. inversely related to density according to a function
determined by the volume to area ratio. As volume is
(From Watkinson, 1984.) a cubic and area is a squared dimension the power
will be 312. Thus as individuals grow in a self-thinning
population their 'areas' increase by a power of 2, their
densities decrease by a power of 112 but their mean
biomass increases by a power of 3. Burrows (1991)
provides a detailed geometric explanation of the - 312
thinning rule based on considerations of canopy
volume.

From a number of studies it has been found that the
values of k fall in the range between 1.3 and 1.8 on
log-log plots of mean plant biomass against density of
survivors (White, 1980). The universality of the - 312
thinning rule, therefore, has been questioned by a
number of workers on the grounds that some species
significantly deviate from the 312 relationship (Zeide,
1987; Weller, 1990) and that the line represents a
boundary rather than a course along which self-
thinning populations will progress (White, 1985). In a
careful review, Lonsdale (1990) concluded that there
were too many examples in which a slope of 312 was

48 PART 1: SINGLE-SPECIES POPULATIONS

Fig. 2.17 The relationship between average plant weight Watkinson (1982) grew Lolium perenne at densities at
and surviving plant density in self-thinning populations of which self-thinning occurred and observed the trajec-
tories of populations at both fdl and low (17%)light.
trees and herbs. Each line represents a different species in a At low light, the maximum yield attainable was
total of 30. (After White, 1980.) reduced and interestingly populations self-thinned
with a trajectory of - 1.In contrast, under high light a
slope of - 3/2 was observed. It thus seems possible
that self-thinning may occur at two different rates but
no plant population yet has been observed to make the
switch.

The ' - 312 law' remains a matter of debate amongst
plant ecologists and development of our understand-
ing will probably only come with a deeper knowledge
of the mechanisms of competition and greater atten-
tion to variation in the performance of individuals.
(Charles-Edwards,1984; Firbank & Watkinson, 1985;
and Hara, 1988 provide launch points for the inter-
ested reader.)

In summary, we have seen how work on intraspe-
cific competition in plants has established that the
responses of actual populations are far more complex
than was envisaged for 'scramble' and 'contest'. There
is little uniformity of response from plant to plant, and
the quality, as well as the quantity, of surviving
individuals is affected. As a consequence, there is no
sudden threshold of response. Plants show these
features particularly clearly; it is appropriate, now, to
examine some zoological examples.

not found to justify the claims of universality for the 2.5.5 Competition in Patella cochlear
rule. It has been shown (Osawa & Sugita, 1989) that
plant mortality often occurs in the approach to the Branch (1975) studied the effects of intraspecific
self-thinning line (see data points H, and H, in competition on the limpet Patella cochlear, making
Fig. 2.15b) and it may well be that this phenomenon observations on natural populations in South Africa
can explain measurement of slopes of steeper than varying in density from 125 to 1225 m-2. This species
- 3/2. Finally, we must note that a slope of - 3/2 feeds mainly on the alga, Lithothamnion, which grows
indicates that the increase in mean biomass of a plant not only on the limpets' rocky substrate, but also on
in a self-thinning population is faster than the reduc- the limpet shells themselves. Thus, the total amount of
tion in density due to mortality. However, yield cannot food (based on the total surface area) remains approx-
indefinitelyincrease, since in every environmentthere imately constant. As density rises, however, the juve-
will be a maximum total yield that a species may nile limpets increasingly live and feed on the shells of
achieve. In this situation we might expect the thinning adults, so that competition for space is largely elimi-
line to have a slope of - 1 indicating that the total nated. But there is ever more intense competition for
yield per unit area remains constant. Lonsdale and food. Some of Branch's results are illustrated in
Fig. 2.18.

CHAPTER 2: INTRASPECIFIC COMPETITION 49

Fig. 2.18 Intraspecific competition in a limpet. (a) Maximum ties there are comparatively few (large) adults. Once
length (0)and biomass ( X ) of the limpet, Patella cochlear, in again, the effect of intense intraspecific competition is
relation to density. (b)Size distributions of the limpets at a population dominated (numerically) by suppressed
three densities. (c)The effect of intraspecific competition on weaklings.
the limpets' gametic output. (Data from Branch, 1975.)
Branch had previously established that larger ani-
Figure 2.18a shows that as density increases, there mals produce proprtionally more gametes. Thus, the
is a compensatory reduction in limpet size, leading to changing size distribution of the population, despite
a stabilization of the total biomass at around 125 g the stabilization of biomass, would be expected to lead
mP2for all densities in excess of around 450 m-2. The to a reduced gametic output as density increased. This
regulatory properties of intraspecific competition and is confirmed in Fig. 2.18c, where once again the use of
the plasticity of individual response are both readily k-values has proved instructive. At low densities
apparent. It is clear from Fig. 2.18b, however, that (where there is little evidence of a sudden threshold)
consideration of only the mean or the maximum size the reduction in size does not compensate for the
of limpets would be very misleading; the size distribu- increase in density (b < l), and the total gametic
tion undoubtedly alters as density increases. The output increases. However, at high densities there is
reason appears to be that at low densities there is little increasing overcompensation (b > l), and the total
juvenile mortality, most individuals reach the adult gametic output decreasesat an acceleratingrate. Once
stage, and (large) adults come to dominate the popu- again, there is a moderate density (around 430 mP2),
lation. As density and intraspecific competition in- at which b = l and the gametic output is at its
crease, however, there is increased juvenile mortality maximum. The ultimate, regulatory effects of in-
and decreased rates of growth, so that at high densi- traspecific competition are acting on the contributions
of limpets to future generations.

50 P A R T l : SINGLE-SPECIES POPULATIONS

2.5.6 Competition in the fmit fly few eggs. As Fig. 2.19 shows, with increasing larval
As a second zoological example, we will consider the density there is a decrease in the size of adults
experimental work of Bakker (1961) on competition produced; this will lead to a decrease in the number of
between larvae of the fruit fly, Drosophila melanogaster. eggs contributed to the next generation. Hence, even
Bakker reared newly hatched larvae at a range of below the threshold for the larval mortality, intraspe-
densities. Some of his results are illustrated in cific competition is exerting a density-dependent reg-
Fig. 2.19. As far as larval mortality is concerned, ulatory effect on the D. melanogaster~larvae.But it is
intraspecific competition appears to approach pure the quality, not the quantity, of larvae which is
scramble. There is a sudden threshold at a density of affected, and, ultimately, their contributionto the next
around 2 larvae mg-l yeast, and mortality thereafter generation which is reduced. Above the mortality
very quickly reaches 100%. But the simplicity of this threshold, the larvae are so reduced in size that they
situation-and its similarity to pure scramble-is very are not even large enough to pupate; when this
largely an illusion. Up to the threshold density, com- happens they eventually die.
petition has little or no effect on larval mortality; but
the growth-rate of the larvae is very much affected, as Clearly, in animals as in plants, intraspecificcompe-
is their final weight at pupation. Moreover, it is well tition is very much more complicated than scramble
known that in D. melanogaster small larvae lead to or contest. Moreover, the complications-lack of a
small pupae, which lead to small adults; and that sudden threshold, individual variability and so on-
these small adults, if female, produce comparatively are very largely the same in both major kingdoms.

Fig. 2.19 The effects of intraspecific competition for food in 2.6 Negativecompetition
the fruitfly Drosophila melanogaster. For adult weight loss, a
variable number of larvae competed for a constant amount Finally, we need to consider an interaction which is
of food. For larval mortality, a constant number of larvae beyond the spectrum of intraspecificcompetitiondealt
were given a variable amount of food. (Data from Bakker, with so far. It is an example of a situation in which
1961; after Varley et al., 1975.) fecundity increases (or mortality decreases)with rising
density. Birkhead (1977) studied breeding success in
the common guillemot (Uria aalge) on Skomer Island,
South Wales. The birds breed there in several sub-
colonies of differingdensity. Female guillemots lay just
one egg, and a pair of birds can be considered
successful if they rear their chick until fledging. By
visiting the various sub-colonies at least once a day
throughout the breeding season, Birkhead was able to
make careful observations on the losses of eggs and
chicks, and could compute the percentage of pairs
breeding successfully. His results are shown in
Fig. 2.20. It is quite clear that as density increases,
breeding success, and thus fecundity, also increases.
This is a case of inverse density-dependence. Compe-
tition is actually negative, and might more properly be
called cooperation. Indeed, cooperation does appear to
be the explanation for these results. Great black-
backed and herring gulls are both important predators
of the eggs and chicks of guillemots on Skomer, and
denser groups are less susceptible to predation,

CHAPTER 2: INTRASPECIFIC COMPETITION 51

Fig. 2.21 The 'AIIee effect' (Allee, 1931). Density-dependent

birth at moderate and high densities leads to a stable
carrying-capacity, K, but the change to inversely density-
dependent birth at IOWdensities leads to an unstable

equilibrium, U, below which the population will decline to
extinction (see also Fig. 2.2).

Fig. 2.20 Cooperation: the effect of density on breeding with density while death-rate remains constant. Small
success in subpopulations of the common guiIlemot Uria populations get smaller still (because death-rate ex-
aalge on Skomer. (Data from Birkhead, 1977.) ceeds birth-rate), while larger populations increase in
size (because birth-rate exceeds death-rate). However,
because a number of guillemots are able (together)to it is likely that these destabilizing tendencies will
deter gulls by lunging at them. Thus, our spectrum of disappear and then be reversed as population size
competition (and density-dependence) must for com- increases and the canrying-capacity is approached.
pleteness, be extended to 'cooperation' and inverse The situation over all densities, therefore, is likely to
density-dependence. Not surprisingly,just as there is a be as illustrated in the whole of Fig. 2.21, and is
regulatory tendency associated with density- known as the 'Allee effect' (Allee, 1931). The regula-
dependence, there is a destabilizing tendency associ- tory density-dependent effects to the right of the figure
ated with inverse density-dependence. This is are easy to image. The destabilizing inversely density-
illustrated diagrammatically for the guillemots on the dependent effects to the left of the figure could result
left-hand side of Fig. 2.21, in which birth-rate rises from cooperation (as with the guillemots), or from
certain other problems associated with low density
(mate-finding, for instance). We shall consider the
Allee effect again in section 5.13.3.

Chapter 3

3.l Introduction the natural and experimental worlds;
2 an aid to enlightenment on aspects of population
It would be easy to follow one example of intraspecific dynamics which had previously been unclear; and
competition with another; each would, in some way, 3 a system which can be easily incorporated into
be different from the rest. However, science progresses more complex models of interspecific interactions.
by not merely accumulating facts but by discovering
patterns. We must, therefore, try to pin-point those We will begin by imagining a population with
features of the organism and its environment which discretegenerations, i.e. one in which breeding occurs
are common to many, or even all, of the special cases. in particular seasons only. Having developed a model
We can then concentrateon these essential features in for this situation, we can return to populations with
our attempts to understand the dynamics of popula- continuous breeding to developan analogousmode1in
tions in general. We may even be able to discover how a similar fashion.
best to control the distribution and abundance of
organisms, or how to predict the responses of 3.2 Populationsbreeding
populations to proposed or envisaged alterations in at discrete intervals
their environment. But in order to do this-or even to
make a start in doing this-we require a general 3.2.1 The basic equations
conceptual framework on which each special case can
be fitted. We require a system that embodies every- Suppose that each individuals in one generation gives
thiig that the special cases have in common, but rise to two individuals in the next. If we begin with 10
which, by manipulation of its parts, can be made to individuals in the first generation, then the series of
mirror each of the special cases in turn. In short, we population sizes in succeeding generations will obvi-
require a model. ously be: 20, 40, 80, 160 and so on. This factor by
which population size is multiplied each generation is
As Levins (1968) has suggested, our perfect model commonly called the reproductive-rate, and we can
would be maximally general, maximally realistic, denote it by R, i.e. in the above example R = 2.
maximally precise and maximally simple. However, in
practice, an increase in one of these characteristics Note immediately that we are assumingthat a single
leads to decreases in the others. Each model is, figure can characterize reproduction for a whole (pre-
therefore, an imperfect compromise, and it is in this sumably.heterogeneous) population on all occasions.
light that models of population dynamics should be Note also, that since 'reproduction' is equivalent to
seen. Mostly they sacrifice precision and a certain 'birth minus death', we have avoided dealing with
amount of realism, so as to retain a high degree of birth and death separately. R is, therefore, a 'net rate
generalityand simplicity. They are usually in the form of increase' or 'net reproductive-rate'.
of an equation or equations illustrated by graphs, but
may, occasionally, be in the form of graphs without We can denote the initial population size (10 in our
accompanying equations. example)by No, meaning the population size when no
time has elapsed. Similarly when one generation has
To be useful,and therefore successful, our model of elapsed the population size is N,, when two genera-
single-speciespopulation dynamics should be:
1 a satisfactory description of the diverse systems of tions have elapsed it is N, and, generally, when t

generations have elapsed the population size is Nt.
Thus:

CHAPTER 3: MODELS OF SINGLE-SPECIES POPULATIONS 53

No = 10, in the population, and the net reproductive-rate (R)

N1 = 20, ,does not require modification. It is still true, therefore,

that is: that N, + = N,R, or, rearranging the equation:

Nl=NoxR As population size increases, however, there is more
and more competition, and the actual net reproduc-
and, for instance: tive-rate is increasingly modified by it. There must
presumably come a point at which competition is so
N,=N~XR=N~XR~RXR=N,R~=~~. great, and net reproductive-rate so modified, that the
population can do no better than replace itself each
In general terms:
,generation.In other words, N, + is merely the same as
,N, + = N, R (a difference equation) ,N, (no greater),and N, IN, + equals 1.The population

and sizeat which this occurs,as we have alreadyseen,isthe
caving-capacity of the population, denoted by K. This
N, = NoRt is the justification for point B in Fig. 3.l.

It is plain to see, however, that these equations lead ,It is clear, then, that as population size increases
to populations which continue to increase in size
indefinitely. The next, obvious step, therefore, is to from point A to point B the value of N, IN, + must also
make the net reproductive-ratesubject to intraspecific rise. But it is for simplicity's sake, and only for
competition. To do this we will incorporate intraspe- simplicity's sake, that we assume this rise follows the
cific competition into the difference equation. straight line in Fig. 3.1. This is because all straight

Consider Fig. 3.1. the justification for point A is as lines are of the simple form: y = (slope)X +(intercept).
follows: when the population size (N,) is very, very
small (virtually zero), there is little or no competition The value for the 'intercept' is clearly 11R.That for the
'slope', considering the portion between points A and
B, is (1 - 1lR)K. The equation of our straight line is,
therefore:

which, by simple rearrangement, gives:

It is probably simpler to replace (R - 1)lK by a and
remember:

The new reproductive-rate (repacing the unrealisti-
cally constant R) is, therefore:

Fig. 3.1 The inverse of generation increase (NJN,+ ,) rising
with density (N,). For further discussion, see text.

54 P A R T 1: SINGLE-SPECIES POPULATIONS

We can now examine the properties of this equation sirable, in as much as its shows our population grad-
and the net reproductive-rate within it, to see if it is ually approaching the carrying-capacity (K) first
satisfactory; and in some respects it certainly is. mentioned in Chapter 2, equation 3.3 is only one of
Figure 3.2 shows a population increasing in size from many that would lead to such a pattern. We chose
very low numbers under the influence of equation 3.3. equation 3.3 because of its simplicity.
In the first place, when numbers are very low, the
population increases in the same, 'exponential' fashion At the top of the curve, N, approaches K, so that:
as a population unaffected by competition (equation
3.1). In other words, when: Thus,

N, approaches 0, and

1+ UN,approaches 1, N t = K = N t +=l N t + z = N t + e3t,c.

and If, however, the population is perturbed such that Nt
exceeds K, then:
1+RaNt approaches R.
l+-(R - 1)Nt > R.
During this initial phase, therefore, the rate at which K
the population increases in size is dependent only on
the size of the population and the potential net and
reproductive-rate(R) of the individuals within it.
W+, < N .
The larger N, becomes, however, (and thus the more
In other words, the population will return to K. It will
competition there is) the larger 1+ UN,becomes, and also do so if perturbed to below K. The carrying-
capacity is a stable equilibrium point to which
the smaller the actual net reproductive-rate populations return after perturbation. Our model,
therefore, exhibits the regulatory properties classically
[R/(l + UN,)]becomes. This situation, in which the net characteristic of intraspecific competition.

reproductive-rateis under density-dependentcontrol,
is responsible for the 'S-shaped' or sigmoidal nature of
the curve shown in Fig. 3.2. It is important to note,
however, that while such a sigmoidal increase is de-

Fig. 3.2 Exponential(dashed) and sigmoidal increase in 3.2.2 Incorporation of a range of competition
density (N,) with time, in a population with discrete
generations. It is clear, from the type of population behaviour that
it leads to, that equation 3.3 can describe situationsin
which a population with discrete generations repro-
duces at a density-dependentrate. What is not clear is
the exact nature of the density-dependence, or the
exact type of competition that is being assumed. This
can now be examined.

Each generation, the potential production of off-
spring is obviously N, R, i.e. the number of offspring
that would be produced if competition did not inter-
vene. The actual production (i.e. the number that

survive) is N,W(1 + UN,).We already know, however,

CHAPTER 3 : MODELS OF SINGLE-SPECIES POPULATIONS 55

that the %-value' of any population process is given the k versus log,& graph) varying from zero to
by: infinity.

k = loglo (no, produced)- log,, (no. sumi~ng). One such general model (and there are others), was
originally proposed by Hassell (1975). It is a simple
In the case of our equation, therefore: modification of equation 3.3:

We also know, from Chapter 2, that the type of As Fig. 3.4 indicates, plots of k against logloNt will
intraspecific competition can be determined from a now approach b rather than 1, irrespective of the
graph of k against loglON,.Figure 3.3 shows a series of values of a and No. This equation can, therefore,
such plots for a variety of values for a and NO.In each
case the slope of the graph approaches 1. In other describe populations reacting in a whole range of ways
words, in each case the density-dependence begins by
undercompensating,and approachesexact compensa- to the effects of competition, and b is the parameter
tion at higher values of N,. Our model, therefore, lacks which measures this degree of under- or overcom-
the generality we would wish it to have. But such pensation. This whole range is only apparent at higher
aensities (competition gradually increasing in intensity
generality is easily incorporated, All we require is a as N, increases); but when we remember the actual
examples considered in Chapter 2, this seems if any-
model that can give us, potentially, b-values (slopes of
thing, a positive advantage of the model. Note, how-
ever, that we have still not specified whether it is
births or deaths or both which are being affected: to do

this would require an even more complex model.

Fig. 3.3 The intraspecific competition inherent in Fig, 3.4 The range of intraspecific competition which can

equation 3.3. The final slope (of k against logloN,) is unity be expressed by equation 3.4. At higher densities, the slope
(exact compensation), irrespective of the starting density No, of k against log,,N, reaches the value of the constant b in
the equation, irrespective of No and a.
or the constant a ( = (R - 1)lK).

56 PART l : SINGLE-SPECIES POPULATIONS

Nevertheless, provided we remember that we are
dealing with a net rate of increase, equation 3.4 can
be a useful, simple model.

3.2.3 Models for annualplants

The same model can also be applied to annual plants
with discrete generations. A convenient starting point
is the law of constant final yield (see Fig. 2.6), the
underlying relationship between biomass and density
being an inverse one (see Fig. 2.7). Early workers
expressedthis in a variety of mathematicalmodels (see
Willey & Heath, 1969 for a review) but we will
concentrate on one recent development. Watkinson
(1980)used a generalized expressionthat can describe
both asymptotic and parabolic yield responses to
density and also allows a biological interpretation of
the constants involved. It expresses average plant size
in the populations, E, as both a function of plant size
in the absence of competition wm and population
density, N, after the action of competition (i.e. at
harvest);

In this, there is clearly more than a passing similar- Fig. 3.5 The two general yield-density responses in plant
populations expressed on a unit area basis (a)and on an
ity to equation 3.4. The term (1+ UN)here acts in a individual plant basis (b),according to equation 3.5. The

number identical to that in eauation 3.4. while b dotted line indicates a constant proportional relationship
between yield and sowing density. (1)Exact com~ensation.
allows the incorporation of a of c o m ~ens at o ~ b = 1;(2) overcompensation, b = 1.5. In both cases,
responses; wm 9 like NR in equation 3.4 is 'what is
= 100 and a = 0.1. (b)the Valuesof the terms in
achieved' in the absence of competition.
equation 3.5 are indicated for the two yield responses. (The
To appreciate the features of this model we will value of 1.5 in curve (2)has no particular significance to

explore the hypothetical data presented in Fig. 3Sa, the power term of the self-thinning line.) Plant mortality is
which shows the two general forms of the yield absent.

response curve that we have already discussed (the

range of units on the axes of the graph are arbitrary).

Figure 3.5b shows the corresponding data for mean both depart appreciably from the density-independent

individual plant size (measured perhaps as biomass) line of constant proportionality (dotted)at N = 10 and

plotted against density. Note initially that the maxi- this in fact is the reciprocal of a in equation 3.5. We

mum competition-free plant size (W,) is 100 units of can see from Fig. 3.5b that it is at this density that

biomass. Generally speaking, the pattern of divergence individual plant size begins to be depressed. The

of the two types of response is determined by the reciprocal of this density (i.e. a) is then broadly an

values of the two parameters a and b in equation 3.5. estimate of the space required for one isolated plant to

For convenience, however, the value of a ( = 0.1) is grow to maximum size. In this arbitrary case, it is

common to both curves. In Fig. 3.5, curves 1 and 2 clearly 0.1 units of area. This area has been called the

CHAPTER 3 : MODELS OF SINGLE-SPECIES POPULATIONS 57

'ecological neighbourhood area' (Antonovics & Levin, tion size that can (asymptotically)be reached. We can
1980; Watkinson, 1981). now combine equations 3.5 and 3.6 to give a model

Where exact compensation for density is occurring describing the relationship between i? and N where
at high density-curve 1in Fig. 3.5a-plant biomass is
directly inversely proportional to density. The slope of there are effects both on mortality and growth.
the line is - 1(Fig. 3.5b), i.e. b = l.Overcompensation, In equation 3.5
reflected by values of b > l , results in an enhanced
proportional reduction of plant size with density (the but N is the number of plants at harvest which is given
slope of the line in Fig. 3.5b is steeper) and yield per
unit area is depressed at high density below a maxi- by equation 3.6. Hence, by substitution,
mum occurring at an intermediate one.
Furthermore,to calculate yield per unit area at harvest
The model's most conspicuous and valuable fea- we merely have to multiply i?, (the average plant
tures are that it has the generality to describe the
whole range of yield-density responses in the absence weight) by N (the density at harvest). We can now
of mortality (via the constants a and b) and that the
parameters in the model are biologically interpretable. examine graphically the behaviour of this model in
Fig. 3.7, which shows the yield of a population of
Equation 3.5, however, does not incorporate mor- plants after density-dependent regulation, in relation
tality of plants. A number of studies have shown that to starting population size. In this simulation, as be-
the relationship between density before and after the fore, each plant when grown under isolated conditions
action of density-dependent mortality follows the requires 0.1 units of area (a) to achieve a maximum
graphical form shown in Fig. 3.6. The mathematical size of 100 biomass units (W,). Where there is exact
description of this is another variant of equations 3.3 compensation at high densities (b = l), the effect of
and 3.4: decreasing the maximum population size attainable
(llm) is to lower yield. If, however, there is overcom-
Where Ni is the number of plants sown, N is the pensation (b > 1) the shape of the yield-density curve
depends on the intensity of density-dependent mortal-
number surviving and llm is the maximum popula- ity (governed through m) in relation to the ecological
neighbourhood area (a). Overt overcompensation is
Fig. 3.6 Population sizes before (N,)and after (N) density- greatest when the potential maximum population size
dependent mortality. The curve follows the equation (llm)is highest. Hence, the model has the generality to
encompass the range of density responses which we
N = Ni(l + mN,)-l. know to occur. Furthermore, it incorporatesthe basic
mechanisms of plasticity and mortality that are part of
the response. We must bear in mind though that it is
only a static description of an essentially dynamic
process occurring within plant populations.

Up to now we have been concerned only with the
relationship between plant size after the action of
competition and initial population density (i.e. at
sowing), but it is a relatively easy step to extend our
model to one that describes changes from generation
to generation. To do this we need to know the
relationship between plant biomass at harvest, G , and
the number of seeds prbduced per plant, S. Often this
may be described by

58 PART l : SINGLE-SPECIES POPULATIONS

Fig. 3.7 The relationship between

initial population size Niand yield

per unit area as determined by
equations 3.6 and 3.7. Constants in

the model are a = 0.1, W, = 10,
b = l.0 (solid line), b = 1.5 (dotted
lines). Values of m as indicated.

Where q and p are constants describing the exact form equation 3.4 and will of course display the properties
of the relationship. We may therefore express seed already discussed.
number per plant at harvest as a function of seeds
sown in an identical way to equation 3.7: 3.3 Continuous breedin-g

S =h(l +aNi(l+ m ~ ~ ) - l ) - ~ We can now return to populations which breed con-
tinuously. For our -purp- oses, the essential difference
where h is the number of seeds produced by a plant in between these and ones with discrete generations is
isolated (non-competitive) conditions. Multiplying that population size itself changes continuously rather
both sides of the equation by the number of surviving than in discrete 'jumps'. This is illustrated in Fig. 3.8.
plants gives us the seed yield per unit area, and hence
we have a population model relating the number of
seeds sown to those harvested. Thus

Since Ni and SN are the population sizes in successive
generations, we may replace them by Nt and N, + ,,
remembering that we are dealing with populations of
seeds. Upon rearrangement our model becomes

On inspection, this too is very similar to equation 3.4,

differing only in so far as it includes an additional term Fig. 3.8 Three populations growing exponentially at the

for densit~-de~endent mmt In the absence same rate. A has long, discrete generations; B has short,
discrete generations: C breeds continuously.
SJ,.

of mortality (m = 01, this model contracts to

CHAPTER 3: MODELS OF SINGLE-SPECIES POPULATIONS 59

,Curves A and B are both of the form ZV,+ = Nt R, and the size of the population and the reproductive-rate of
individuals.
represent populations increasing (exponentially) at
essentidy the same rate. ('Rate' in this casemeans 'the We can proceed now in Fig. 3.9 exactly as we did in
amount by which population size (Nt)increases per Fig. 3.1. The rate of increase per individual is unaf-
unit time (t)'.) The difference between them is that fected by competition when N approaches zero, and is,
generation time is much shorter in curve B than curve therefore, given by r (point A). When N reaches K, the
A. The sameprocess has been extendedin curve C until carrying-capacity,the rate of increase per individualis
the straight line segments are so small (infinitesimally 0 (point B). As before, we assume that the line
small)that the graph is a continuouscurve. Neverthe- between A and l3 is straight, and thus:
less, the rate at which the population increases has
remained essentially the same. Curve C describes a and
situationin which, during each and every exceedingly
small time interval, there is the possibility, at least, of This is the so-called Iogistic equation. Its characteris-
birth and death. In other words,breeding and dyingare tics are essentially the same as those of the difference
continuous. Thus, the initial equation we require in equation 3.3 described previously. Minor dissimilari-
order to build our general model of a continuously ties, associated with the change from the difference to
breeding population must retain the essential proper- the differential form, will be discussed later; and, of
ties of N, = NORtb, ut must describea continuouscurve. course, it describes a continuous sigmoidal curve,

Diferentiation is a mathematical process which is
specifically designed to deal with changes occurring
during infinitesimally small intervals of time (or of any
other measurementon the x-axis). Those familiar with
differential calculus will clearly see (and those unfa-
miliar with it need merely accept) that differentiating

N, = hRt by t we get:

and

This is the equationwe require, dN/dtis the slope of the Fig. 3.9 The rate of increase per individual dNdt 1/N
curve, defining the rate at which population size falling with density (N).For further discussion,see text.
increaseswith time. log, R is usually replaced by r, 'the
intrinsic rate of natural increase' or 'instantaneous
rate of increase', but the change from R to r is simply
a change of currency: the commodity dealt with
remains the same (essentially 'birth minus death').

+,The diferential equation dNldt = rN, just like the differ-

ence equation N, = N,R, describes a situation of
exponential population increase dependent only on

60 P A R T 1: SINGLE-SPECIES POPULATIONS

rather than a series of straight lines. Note once again, tion may be examined has been set out and discussed
moreover, that there are many other equations which by May (1975). It is a method which uses fairly
would lead to sigmoidal increase; the justification of sophisticated mathematical techniques, but we can
the logistic is the simplicity of its derivation. Note, too, ignore these and concentrate on May's results. These
that the logistic equation is based on exact compensa- are summarized in Fig. 3.10. Remember that
tion, in just the same way as its discrete generation Fig. 3.10a refers to equation 3.4. It describes the way
analogue. In the case of the logistic, however, it is by in which populations fluctuate (Fig. 3.10b) with dif-
no means easy to incorporate a factor, such as b, ferent values of R and b inserted. (The value of a, i.e.
which will generalize the model to cover all types of (R - 1)/K, determines the level about which popula-
competition. Our equation for a continuously breeding tions fluctuate, but not the manner in which they do
population must, therefore, remain comparatively so.)As Fig. 3.10a shows, low values of b andlor R lead
imperfect. to populations which approach their equilibrium size
without fluctuating at all. Increases in b andlor R,
Drawing these arguments together then, we are left however, lead firstly to damped oscillations gradually
with two equations: equation 3.4 as an apparently approaching the equilibrium; and then to 'stable limit
satisfactory general model for discrete-generation in- cycles' in which the population fluctuates around the
crease and equation 3.8 as a somewhat less satisfac- equilibrium level, revisiting the same two, or four, or
tory model for the behaviour of a continuously even more points time and time again. Finally, with
breeding population. large values of b and R, we have a population
fluctuating in a wholly irregular and chaotic fashion.
3.4 The utility of the equations
The biological significance of this becomes apparent
3.4.1 Causes of population fluctuations when we remember that our equation was designed to
model a population which regulated itself in a density-
We can now proceed to examine the models' utility. dependent fashion. We can see, however, that if a
The extent to which they can be incorporatedinto more population has even a moderate net reproductive-rate
complex models of interspecific interactions will be- (and an individual leaving 100(= R) offspring in the
come apparent in Chapters 4 and 5. We shall begin next generation in a competition-free environment is
here by determining whether our models can indeed not unreasonable), and if it has a density-dependent
throw important new light on aspects of population reaction which even moderately overcompensates,
dynamics. To do so we shall examine the question of then far from being stable, it may fluctuate in numbers
fluctuations in the sizes of natural, single-speciespopu- without any extrinsic factor acting. Thus, our model
lation~T. hat some degree of fluctuation in size is shown system has taught us that we need not look beyond the
by all natural populations hardly needs stressing. intrinsic dynamics of a species in order to understand
the fluctuations in numbers of its natural populations.
The causes of these fluctuations can be divided into It is worthwhile stressing this. Our intuition would
two groups; extrinsic and intrinsic factors. Amongst probably tell us that the ability of a population to
the extrinsic factors we include the effects of other regulate its numbers in a density-dependent way
species on a population, and the effects of changes in should lend stability to its dynamics. Yet our model
environmentalconditions. These are topics which will shows us that if this regulation involves a moderate or
be discussed in later chapters. For now, we will large degree of overcompensation, and at least a
concern ourselves with intrinsic factors. Our approach moderate reproductive-rate, then the population's
will be to examine our models to see which values of numbers may fluctuate considerably because of those
the various parameters, and which minor alterations very 'regulatory' processes. As May, himself, con-
to the models themselves, lead to population fluctua-
tions; and to see what type of fluctuation they lead to. cludes: '...even if the natural world was 100010predict-

The method by which our discrete-generation equa- able, the dynamics of populations with "density-

CHAPTER 3: MODELS OF SINGLE-SPECIES POPULATIONS 61

Fig. 3.10 The effect of intraspecificcompetition on ,N,+ = N,X reproductive-rate
population dynamics. (a) The range of population
fluctuations (themselvesshown in (b))generated by In other words, there is a time-lag in the popula-
equation 3.4 with various combinations of b and R inserted. tion's response to its own density, caused by a time-lag
For further discussion, see text. (After May, 1975.) in the response of its resource. The amount of grass in
a field in spring being determined by the level of
dependent'' regulation could nonetheless in some grazing the previous year is a simple but reasonable
circumstances be indistinguishable from chaos' (see example of this, the behaviour of this modified equa-
also section 6.12). tion, as shown by computer simulations, is as follows:

In examining this general model, the special, exact- R < approx. 1.3 :exponential damping
compensation case (b = l ) has itself been covered.
Irrespective of R, the population will reach K without R approx. 1.3 : damped oscillations.
overshooting; and with R-values greater than about
10 this will take little more than a single generation. In comparison, the original equation, without a time-
(The analogous differential equation-the logistic- lag, led to exponential damping for all values of R. The
behaves very similarly to this: always approaching K time-lag has provoked the fluctuations. Once again,
with exponential damping.) This special case can be therefore, examination of our model has taught us
modified, however, to incorporate an additional fea- which intrinsic features of a species' dynamics can
ture. We have assumed until now that populations lead to fluctuationsin its population density.
respond instantaneously to changes in their own den-
sity. Suppose, on the contrary, that the reproductive- In fact, there are other types of time-lag and they all
rate is determined by the amount of resource available tend to provoke fluctuations in density. Consider again
to the population, but that the amount of resource is the difference between a population with discrete
determined by the density of the previous generation. generations and one with continuous breeding. In
This will mean that the reproductive-rateis dependent both cases, the population responds throughout each
on the density of the previous generation. Thus, since:

62 PART 1: SINGLE-SPECIES POPULATIONS

'time interval' to the density at the start of that time blance between this and the pattern predicted by any
interval. With continuous breeding the time intervals of our models. Nevertheless, the initial rise in num-
are infinitesimally small, and the response of the bers, decelerating as it approaches a plateau (albeit a
population is, therefore, continually changing; con- fluctuating one), is reminiscent of the pattern gener-
versely with discrete generationsthe population is still ated by our difference equation model with b close to
responding at the end of a time interval to the density 1. If, furthermore, we take into account the fluctua-
at its start. In the meantime, of course, this density has tions in environmentalconditions, the interactions of
altered: there is a time-lag. Thus, the difference the sheep with other species, and the undoubted
equation model is more liable to lead to fluctuations imperfections of the sampling method, then the re-
than the differential logistic, and it is the time-lag semblance between Fig. 3.11b and our model begins
which accounts for the difference. Moreover, some to look more significant. This is the nub of our
organismsmay respond to density at one point in their problem in deciding how satisfactory our models are.
life cycle, and actually reproduce some time later; this On the one hand, the field data, unlike some data
'developmental time' between response and reproduc- collected under controlled conditions, do not follow
tion is also a time-lag. In all cases, the population is our models' behaviour exactly. On the other hand,
responding to a situation that has already changed, there is nothing in the field data to make us reject our
and an increased level of fluctuation is the likely models. We could claim, in fact, that our models
result. represent a very satisfactory description of the popu-
lation dynamics of sheep in Tasmania, but that the
There are two important conclusions to be drawn complexities of the real world tend to blur some of the
from this discussion. The first is that time-lags, high edges. Similar conclusions could be drawn from
reproductive-rates and highly overcompensating Fig. 3.11~-e, although in each case the model con-
densiQ-dependence (either alone or in combination) cerned might be slightly different; perhaps a diierent
are capable of provoking all types of fluctuation in value for a and b, or the addition of some sort of
population density, without invoking any extrinsic time-lag. Figure 3.llc, for example, would be quite
cause. The second conclusion is that this is clear to us adequately described by our diierence equation with
only because we have studied the behaviour of our R = 20 and b = 2; the pattern in Fig. 3.11d could be
model systems. accounted for, simply, by environmental fluctuations;
and the fluctuations in numbers of Daphnia
3.4.2 The equations as descriptions (Fig. 3.11e) do appear, quite genuinely, to be caused
by a time-lag.
Finally, we must examine the ability of our models to
describe the behaviour of natural and experimental The most reasonable conclusion seems to be this.
systems, There are two aspects of this. The first is Our models seem to be satisfactory, in as much as they
concerned with the way in which single-species popu- are capable of describing the observed patterns of
Iations increase or fluctuate in size; the second with increase and fluctuation as long as environmental
the precise way in which intraspecific competition fluctuations, interspecific interactions and the imper-
affects the various facets of fecundity and survival. fections of sampling are taken into account. Con-
versely, the discrepancies between our models and
A number of examples of population increase and our data could be due, quite simply, to the essential
fluctuation are illustrated in Fig. 3 .ll. Figure 3.11a inadequacies of our models, particularly the unrealis-
describes the continuous increase of a population of tic simplifying assumptions they make. To examine
yeast cells under laboratory conditions (Pearl, 1927): their utility more critically, we must consider their
the resemblance to the logistic curve is quite striking. ability to describe the effects of intraspecific competi-
Figure 3.11b, conversely, describes the year-by-year tion on fecundity and survival. It should also be
change in the size of the sheep population of Tasmania stressed, however, that a true test of a model's ability
(Davidson, 1938).There is no obvious detailed resem-

CHAPTER 3: MODELS OF SINGLE-SPECIESPOPULATIONS 63

Fig. 3.11 Observed population fluctuations. (a)Yeast cells. In considering the ability of our models to describe
(AfterPearI, 1927.)(b)Tasmanian sheep. (After Davidson, the effects of intraspecific competition, we must con-
1938.)(c) The stored-productbeetle Callosobmchus fine ourselves to the discrete-generationequation. The
macuiatus. (After Utida, 1967.)(d) The great tit Parus major logistic model, restricted as it is to perfect cornpensa-
in Holland. (After Kluyver, 1951.)(e)The water flea tion, cannot be expected to describe a range of
Daphnia magna. (After Pratt, 1943.) situations; its utility lies in its simplicity and the ease
with which it can be understood. The capabilities of
to describe population behaviour should not consist our difference equation can be critically assessed,
only of a comparison of numbers; it should also
consider the underlying biological similarities: the because, for any set of data the most appropriate
respective R-values for instance, or the existence-in values for a and b-those giving the 'best fit' to the
both model and fact-of a time-lag. data-can be estimated using a statistical technique

64 PART 1: SINGLE-SPECIES POPULATIONS

Fig, 3.12 Data from competition experiments and their Hassell et al., 1976.)(b)Shepherd's purse, Capsella bursa-
pastoris: a = 0.377, b = 1.085 (Palmblad, 1968).(c)the
description based on equation 3.4. (a) Beetles:
blowfly Lucilia cuprina (Nicholson,1954b): the moth Plodia
CaIlosobruchus chinensis (a)a = 0.00013, b = 0.9 (Fujii, 1968);
interpunctella (Snyman, 1949). (After Hassell, 1975.)
C. maculatus ( X ) a = 0.0006, b = 2.2 (Utida, 1967); C.
maculatus (0)a = 0.0001, b = 2.7 (Fujii, 1967). (After

(followingHassell, 1975 and Hassell et al., 1976).The number of seeds borne per flowering plant declined
curve produced by the model can then be compared with density over the entire range (Fig. 3.l4b).
with the original data points. This has been done for
several examples (including two from Chapter 2), and Fitting the models (equations 3.5 and 3.6) described
the results are shown as plots of k-values against log,, above to these data by statistical means, we can see
density in Fig. 3.12 (experimental data) and Fig. 3. l 3 that they do indeed give a very close fit to the observed
(field data). As originallynoted by Hassell (1975),there results. It appears (again) that our basic difference
is a tendency, particularly under laboratory condi- equation model is very satisfactory in describing
tions, for the model to be very satisfactory at high and reality. Since, however,we can interpret the constants
low densities, but incapable of describing the sudden in the models in biologic$ terms we can make further
transition from a shallow to a steep slope. This may statements about the Agrostemma population. The
be a result of the data reflecting two superimposed data suggest that the maximum population size that
density-dependent processes (one weak and one can be supported (if the experiment is repeated again
strong),while the model treats them as a single process under identical conditions) is 10 869 seed-bearing
(Stubbs,1977).Nevertheless, the model's performance plants m-2 ( = m-' = U(9.2 X lW5), and that the maxi-
overall is impressive: whether the plot is more or less mum seed yield of an isolated plant, h, is 3685 seeds.
straight or curved,its fit to the data is very satisfactory. Also, to grow to maximum size, a plant required
325 cm2(a = 0.0325)of space. The value of b (1.15)in
A final example of the utility of the difference equation 3.6 was significantly greater than unity,
equation model is shown in Fig. 3.14. The data come indicating overcompensation at high density. This is
from a pot experiment in which Agrostemma githago, evident when the components of seed yield are
corncockle, was sown over a wide range of densities. examined (Fig. 3.14~).The effects of density did not
We can see that there were two components of fall equally on all parts of the plant. Most density stress
density-dependent regulation: plant momlity and was absorbed by the number of capsules per plant
plasticity. Self-thinning was noticeable above sowing which declined at a constant rate with density as did
densities of 1000 seeds m-2 (Fig. 3.14a), whilst the seed weight. The number of seeds per capsule,

CHAPTER 3: MODELS OF SINGLE-SPECIES POPULATIONS 65

Fig. 3.13 Field data on intraspecific competition and their One of the earliest uses of this approach was in
description based on equation 3 and 4. (a) The limpet fisheries research in the 1950s in investigations of
Patella cochlear: a = 4.13 X 10-5, b = 51.49 (Branch, 1975). sustainable fishing of natural stocks.
(b)The Colorado beetle Leptinotarsa decemlineata:
a = 3.97 X 10-6, b = 30.95 (Harcourt, 1971).(c)Larch tortrix If the procedure is carried out to encompass a
Zeiraphera diniana: a = 1.8 X lO-', b = 0.11 (Auer, 1968). number of generations, then we might envisage a
(After Hassell, 1975.) curve of the form illustrated in Fig. 3.15. The curve
has been termed a recruitment or reproduction curve.
however, fell sharply at high density reflecting the To investigate the behaviour of the population we
overcompensatory density response. The difference examine the changes in population size by comparing
equation model, despite its simplicity, does indeed
appear to encapsulate the essential characteristics of ,the reproduction curve with a one to one reference
the behaviour of single-species populations.
line (N,+ = N,), plotting the data on log scales. Any
3.4.3 'Cobwebbing'-a more general approach population lying on this reference line will be at
equilibrium.
Whilst the behaviour of populations described by
models such as equation 3.4 can be examined mathe- Four theoretical recruitment curves are illustrated
matically, there will of course be situations where we in Fig. 3.15 and we can predict the size of the popula-
tion over succeeding generations by iteration using
,cannot find appropriate functions relating N, + to N, a method known as cobwebbing the reproduction
curve(Hoppensteadt,1982).In the first(Fig. 3.l5a)the
and even when we can, they may not be tractable
analytically. In such cases, however, we can examine ,recruitment curve parallels the reference line and for
the behaviour of the population by geometric iteration,
bearing in mind that this method will have its limita- a chosen starting size (N,) we may evaluate N, + on
tions (see below). In the simplest case for discrete the reproduction curve and reflect this value back to
generations, the approach involves plotting the size of the horizontal axis through the one to one line. Inspec-
the population after a generation of growth against its tion of Fig. 3.15a shows that the population exhibits
former (or starting)size. In section 1.2 we introduced exponential growth increasing in size by a factor of
this idea of examining the dynamics of a population 10 each generation.
graphically by plotting its size over successive genera-
tions, each generation against the next (see Fig. 1.3). When growth-rate is density-dependent, reproduc-
tion curves will not parallel the reference line but will
intersect it at a locus which by definition is an
equilibrium population size N,. Of particular interest,
however, is the slope of the reproduction curve at the

66 PART 1: SINGLE-SPECIES POPULATIONS

Fig. 3.14 Density-dependent regulation in Agrostemma sating), then as Fig. 3.15d indicates population sizes
githago:(a)survival to reproduction;(b)seed production;
(c)yield components.(From Watkinson, 1981.) may fluctuate around N,and in a chaotic manner-a

approach to the point of intersection.Where this slope pattern we have already observed in the circum-
stances for the specific case of equation 3.4, and to
is in the range of zero to + 1 (indicating either which we will return in section 6.12.

undercompensatingdensity-dependence (slope > 0) or Figure 3.16 provides selected reproduction curves
exactly compensating density-dependence (slope = 0)) that have been observed in natural or laboratory
we can see that population shows a monotonic ap- populations and we will return to use them in
proach to the equilibrium point. Where the population explaining mechanisms of population regulation, par-
approaches the equilibrium point from below, the ticularly in plants. Before we leave this topic, we must
incremental gain diminishes with each generation, voice a cautionary word of warning. Whilst intuitive
and when it approaches from above, the incremental and simple, graphical analysis by cobwebbing is lim-
decline diminishes with each generation (Fig. 3.15b). ited since it will not necessarily expose the full range of
Population perturbations away from this density in dynamical behaviour that a population may exhibit
either direction will result in a smooth return to it. (see Yodzis, 1989).For example, in particular circum-
stances, the outcome will depend on the population
Reproduction curves, however, may exhibit a size chosen to start the exploration.
'hump' such that population growth rate is maximal
at some intermediate density between N , and very low 3.5 Incorporationof age-specific
densities. This is indicative of overcompensating fecundity and mortality
density-dependence. Two theoretical examples are
shown in Fig. 3.l5c and 3.l5d, and by cobwebbing we We have seen from the life-table data for annual
can illustrate two further forms of behaviour. Where meadow grass and red deer (Chapter 2) that mortality
the slope of the reproduction curve is negative but less and fecundity are often age-specific. The models we
than - 1 in the region of N, (mildly overcompensat- have developed so far, however, do not include these
ing), damped oscillations towards a stable equilibrium important features, and to this extent they are defi-
population size will occur (Fig. 3.154. If, however, the cient. It may seem a formidable task to attempt to
slope is greater than - 1(more strongly overcompen- model a population of overlapping generations in

CHAPTER 3: MODELS OF SINGLE-SPECIES POPULATIONS 67

Fig. 3.15 Theoretical recruitment (or reproduction) curves. these terms, but by the application of appropriate
In each case (a-d) the size of the population after a mathematical techniques it can be achieved fairly
generation of growth is plotted against initial size. Dotted easily,

lines illustrate the process of cobwebbing. (a) Exponential The best starting point is the diagrammaticlife table
growth; (b)monotonic approach to equilbrium N,; for overlapping generations (see Fig. 1.g), expanded
(c) damped oscillations towards equilibrium; (d) chaotic
into a more complete form (Fig.3.17). Here we sup-
behaviour (see text for details).

68 PART l : SINGLE-SPECIES POPULATIONS

Fig. 3.17 The diagrammaticlife table for a population with
overlapping generations:a, numbers in diierent age
groups; B, age-specific fecundities; p, age-specific
survivorships.

pose that a population can be conveniently divided
into four age groups: a,, a,, a, and a,; a, representing
the youngest adults and a, the oldest. In a single time
step, t, to t,, individualsfrom group a,, a, and a,, pass
to the next respective age group; each age group
contributes new individuals to a, (through birth); and
the individuals in a , die. This clearly rests on the
assumptionthat the population consistsof discrete age
groupings and has discrete survivorship and birth
statistics, in contrast to the reality of a continuously
ageing population.

We can write a series of algebraic equations to
express the changes that are occurring in Fig. 3.17.
These are:

Fig. 3.16 Representative reproduction curves for (a) Vulpia
fasiculata (Watkinson & Harper, 1978);(b)Senecio vulgaris
(Watson, 1987);(c) Caflosobruchusmaculatus (Utida, 1967)-
population fluctuations of these data are shown in
Fig. 3.11~.

C H A P T E R 3: MODELS OF SINGLE-SPECIES POPULATIONS 69

where the numbers in the age group are subscripted t, 3.5.1 The matrix model
or t, to identify the time period to which they refer.
There are four equations because there are four age The equations 3.10-3.13are called 'linear recurrence
groups, and they specifically state how the numbers in equations', but it is clear that in this form our model is
age groups are determined over the time-step t, to t,. rather cumbersome, consisting as it does of as many
equations as there are age groups. One way of
Suppose that our population at t, consists of 2000 expressing our model in a much more compact form is
individuals distributed among the age groups as by the use of matrix algebra. In essence, matrix
follows: algebra is a tool designed to manipulate and store
large sets of data. We will certainly not need to cover
Suppose, further, that the age-specific birth- and all aspects of matrix algebra to understand how our
survival-rates are as foIlows: model is constructed in matrix terms, but it is vital
that the basic concepts are clearly understood.
Then, for instance, no offspring are born to adults
aged a,, whilst adults in age group a, are the most A matrix is simply a group, or table, or array of
fecund. Also, only 10%~of the individuals aged a,
survive the next time-step and become incorporated :;a]numbers. For instance, the numbers for B and p
into a,, whereas none of the individuals in a, survive
(p= 0).The numbers at time t, will be: envisaged above could be set out as the matrix:

10 0.0

and we signify that it is a matrix by surrounding the
numbers by square brackets. Conventionally,matrices
are symbolized by a letter in bold face, say X.

The number of rows and columns that comprise a
matrix may vary, and we may have matrices consist-
ing of only a single column. We can, in fact, write the
initial age structure of our previous population in this
way:

During this time-step our population has increased by and symbolize it as tlA.Our age distribution at t,, tZA,
765 individuals and our age distribution has changed: can obviously then be mitten as

from a, = 1750to 2500 The two matrices ,lA and are technically called
a, = 100 175 column vectors, indicating that our matrices are in fact
a,= 100 60 just one column of figures.
a,= 50 30
We have seen how our population changes from ,lA
It is important to realize that equations state that
individuals reproduce before they die. Taking age group
a,, for instance, the figure 100 was used in
equation 3.10 to compute the number of births
(100X 15 = l5OO),but then used in equation 3.l3 to
compute the number of a, survivors into age group a,
(100X 0.3= 30).

70 PART 1: SINGLE-SPECIES POPULATIONS

to through the medium of the recurrence equa- groups, it is:
tions. Now, to complete our matrix model, we have to
constructa suitable matrix to enable tlAto become t2A
by mutiplication. The construction of this matrix is
determined by the rules of matrix multiplication, as
we shall learn below, and in consequence it appears
as:

Note that matrix T is square, and that we have entered which may, alternatively, be vvritten:
into it all our age-specific survival and birth statistics
in particular positions, writing all the other numbers T X ,lA = t2A.
(or matrix elements as they are called) as zero.
T is called the transition matrix, which, when post-
The multiplication of our initial, age-distributed multiplied by the vector of ages at t,, gives the age
population occurs according to the rules of matrix distribution at t,. (In matrix algebra, in contrast to
algebra, as follows. Take, in turn, each row of elements conventional algebra, T X A is diierent from A X T (cf.
in T. Each individual element in the row is multiplied X X b = b X X),and we distinguish between the two by
with the corresponding element in the first with referring to post- and pre-multiplication.)
the first, the second with the second, and so on. These
pairwise multiplications are then summed, and the By now it must be clear that our matrix model
sum entered as the appropriate element in a new allows us to condense the complexities of age-specific
column vector. Thus, the first row of the square schedules into a simply written but explicit form. It
matrix leads to the first element of the new column might also seem that to use the model requires
vector. the second leads to the second, and so on. This endlessly repeated multiplications and additions. This
may be illustrated as: disadvantage has been removed, however, by the
widespread use of computers,which are ideally suited
5 15 10- (not to say designed) to perform such iterative proce-
00 dures at speed. Our model is, therefore, of great value.
0
The specific rules arid procedures of matrix algebra
0 0.3 0 are numerous. We have only dealt here with those
which are necessary in this particular context. If
further explanations are needed, reference may be
made to one of the several books on matrix algebra for
biologists (e.g. Searle, 1966).

We can observe now that the positions of the zeros in 3.5.2 Using the model

the matrix T are critical, because they reduce terns in If we now wish to compute the changing size of our
each row of the multiplication to zero, giving rise in t2A population, on the assumption that the birth and
to the age structure that we have already seen. mortality statistics are constant from one time to the
next, we can write:
This matrix model was first introduced to popula-
tion biology by P.H. Leslie in 1945 and is often known T X tlA= t2A; T X t2A= t3A; T X t3A= tqA'
as the Leslie matrix model. In general form, for n age
and so on. This is a process of iterative pre-
multiplication of the successive age groups by the

CHAPTER 3 : MODELS OF SINGLE-SPECIES POPULATIONS 71

transition matrix, giving:

and

TX

Over these three time-steps the age distribution is
changing and the population is increasing in size; the
numbers of individuals in all age-groups are oscillat-
ing. If we were to continue the iteration, however,
these oscillations would disappear (Fig. 3.18), and
after 17 and 18 multiplications the vectors would be

respectively. Note that the population has grown
enormously, but that the ratios of ,JI,:t,,a, are all
equal (to two significant figures at least), i.e.

This means that we have reached a situation in which Fig. 3.18 The behaviour of a matrix population model:
the age distribution is proportionally constant or stable oscillationsin an age-distributed population as a stable age
as the population increases. In other words, a, :a, : distribution is approached. See text for details.
a, :a,, is the same in all subsequent generations. In
fact, this particular stable age distribution is an in fact, the net reproductive-rate or finite rate of
attribute of the transition matrix, and would have increase per unit of time, R. Repeated pre-
been reached irrespective of the initial column vector; multiplication is, therefore, the mathematically sim-
and in general, populations, when subjected to re- plest means of determining R from an age-specific
peated pre-multiplication by the same transition ma- schedule of births and deaths.
trix, achieve the stable age distribution characteristic
of that matrix. We should not, however, simply dismiss the matrix
model as a device for only handling sets of recurrence
Moreover, as Fig. 3.18 shows, the ratios from one equations in an iterative manner. A transition matrix,
generation to the next (1.18 in this case) are also stating age-specific data as it does, is a precise
constant: the population (as well as each of the age- description of the population statistics and can be
classes within it) is increasing at a constant rate. This subjected to algebraic analysis. First, we can find R,
rate is also a reflection of the transition matrix, and is, by calculating 'the dominant latent root' of the

72 PART l : SINGLE-SPECIES POPULATIONS

transition matrix; and second, we can conduct sensi- we can assume instead that, in general terms, for r
tivity analyses to investigatethe relative importance of time periods:
the different transitions in determining population
growth-rate. Both techniques require a knowledge of Furthermore, we can incorporate the idea that the
matrix algebra which is beyond the scope of this book birth and death statistics are density-dependent by
(but which are clearly discussed in Caswell, 1989). A varying the fecundity and survival elements in the
further usage of transition matrices comes in harvest- transition matrix in relation to population size. To
ing theory (Chapter S), which allows us to predict the achieve this, we need simply derive the relevant
level of 'cropping' that can occur without driving a elements of the successive transition matrices from
population to extinction. All that need be appreciated equations which relate fecundity and survivorship to
at this stage is that matrix modelling can lead to the population size, which is itself the sum of the
sophisticated analyses of populations and their behav- elements of the most recent age vector.
iour.
3.5.3 A working example: Poa annua
Our computations of R and the stable age distribu-
tion rest on the assumption that the elements of the Both of these additions to the model are well illus-
transition matrix are constant over time and independent trated in practice by the work of Law (1975) on the
of population density. In these respects our matrix annual meadow grass Poa annua: a particularly suc-
model, in its present form, is very much divorced from cessful colonizer of open habitats, which responds
the biological reality of natural populations. We can strongly to intraspecific competition. In modelling
incorporate temporal changes in fecundity and sur- populations of this species,Law envisaged that the life
vival in a very straightforward manner, by simply cycle included four ages of plants besides s'eeds
changing our transition matrix with every time-step. (Fig. 3.19), and that the span of each age group was
Thus, whereas we have assumed that: approximately 8 weeks. The transition matrix appro-
priate to this life cycle is given in Table 3.1 and,
so that although most of the elements are familiar to us, the
incorporation of a seed bank into the model requires

Fig. 3.19 The life cycle for annual meadow grass Poa surviving to become old adults. Fecundities (0or >O): B,,
annua. (After Law, 1975.) Proportions (ranging from 0 to 1): B, and B,, seed produced by young, medium and old
p, , seeds surviving;g,, seeds germinating; p,, seedlings adults, respectively. All events occur over the time period t
to t + 1.
surviving to become young adults; p,, young adults
surviving to become medium adults; p,, medium adults

CHAPTER 3: MODELS OF SINGLE-SPECIES POPULATIONS 73

Table 3.1 A matrix model of the life cycle outlined in Table 3.2 A transition matrix, P, for a Poa annua
Fig. 3.19. population. (After Law, 1975).

that the first element of the matrix (p,) is not a specific assertion that the survivorships of individuals
fecundity but the probability of a seed surviving in the in these age-classes are not subject to density-
bank if it does not germinate. dependence. The maximum seed production of young
and old adults at very low population density is 100
Law was able to develop curves of both seedling seeds per plant, but, with increasing density, this
survivorship to young adults, and age-specific seed decreases in a negative exponential fashion. Similarly,
output per plant in relation to overall population the maximum seed production of medium adults is
density. These are shown in Fig. 3.20 together with 200 seeds per plant, which also decreases with
the equations that describe these relationships. The density. This reflects P. annua's schedule of fecundity
transition matrix which incorporates these density- in which seed output peaks in the medium adult age-
dependent functions of survival and reproduction class (see Fig. 1.13).
takes the form shown in Table 3.2. Between succes-
sive time intervals, 0.2 of the seeds in the banks The matrix model,
remain dormant and survive, while 0.05 of them
germinate. The proportion of individuals that survive P X , p = ,r+lA
from the seedling age-class, po(N), is initially 0.75 at
very low population density, but declines according to can now be used to simulate the behaviour of the
the function shown in Fig. 3.20. The proportion of population, which is depicted in Fig. 3.21. We can see
individualsthat survive from the young adult age-class that each group of individuals undergoes oscillations
onward, however, is fixed at 0.75. This represents the in numbers, but when 18 time periods have elapsed
the size of each has stabilized and population increase

Fig. 3.20 (a) The proportion of seedlings of Poa annua production per plant for young (B,) medium (B,) and old
surviving to become young adults as a function of (B,) adults of Poa annua as a function of population density:
B,(N) = B,(N) = 100 exp ( - 0.0001 N); B,(N) = 200 exp
population density: p,(N) = 0.75-0.25 exp (0.00005 N) if ( - 0.0001 N). (From Law, 1975.)
Nc27726; p,(N) = 0 if N>27726. (b) Age-specific seed

74 PART l : SINGLE-SPECIES POPULATIONS

Fig. 3.21 Application of matrix

model: simulation of the density-
regulated changes in density of five
age-classes in a population of Poa
annua. (From Law, 1975.)

has been halted by intraspecific competition. (Note can derive a figure for R as previously described.
that the age-structure stabilization after 18 time Moreover, by consideringa variety of transition matri-
periods both here and in section 3.5.2 is purely ces, based on a range of densities, we can examine the
coincidental.) density-dependent effects on the actual reproductive-
rate. This is illustrated in Fig. 3.22.
Seed germination and seed production in P. annua
vary considerably with the time of year, and to model Fig, 3-22 Net reproductive-rate of Pea annua growing at a
this realistically it would be necessary to introduce a range of population densities. (Data from Law, 1975.)
different transition matrix for each month of the year.
The number of seeds per individual plant in each age
class would then be a function of population density
and time during the season. Although we do not need
to examine the workings of this sophistication, we
should realize that the facility of changing elements in
successive transition matrices is essential for biological
reality.

Finally, it is interesting to note that the net
reproductive-rate, R, can still be estimated, even from
this more sophisticated form of the model. R is, by
definition, the rate of increase in the absence of
competition. If we, therefore, consider the transition
matrix P with its elements unaffected by density, we

Part 2

This page intentionally left blank

Chapter 4

4.1 The nature of interspecific host, do not usually cause its immediate death. All
interactions
aspects of the ' + - ' interaction will be considered in
In Chapters 2 and 3 we examined the consequences of detail in Chapter 5.
individuals of the same species competing with one
another. Natural communities, however, are usually We can only make passing reference to the ' + O',
assemblages of species, and in many habitats the ' - 0' and ' + + ' interactions. This is not to belie their
'neighbours' of any one individual may well be of a
different species to the individual itself. Yet, although biological significance,but more to indicate the lack of
the process of interaction between species might occur consideration they have received in terms of popula-
in a variety of ways (fighting in animals, shading in
plants, etc.), there are only three basic efects. One tion analysis. Commensalism ( + 0) is the state in
species may cause (or be the cause of) increases in the which prerequisite conditions for the existence of one
survival, growth or fecundity of another species, or it species are maintained or provided by a second, but in
may cause decreases, or may have no effect at all. which there is no associated adverse effect for the
Placing these effects in symbolicform in Table 4.1, we second species. Saprophytism between fungi and
can see that there are six possible outcomes, in five of higher plants might enter this category, as would
which at least one species is affected. 'parasites' that have no measurable effect on their
host. The reverse of commensalism is amensalism
The most obvious example of the ' + - ' type of ( - 0), an often-cited example of which is allelopathy
between plants: toxic metabolite production by one
interaction is a predator-prey relationship in which species causing growth reduction in another. Allelo-
one species is eaten by the other; but there are other pathy is difficult to investigatein controlled laboratory
situations in which one species provides a food source experiments, and its role in the field is uncertain (see
or growth requirement for a second species at the first Harper, 1977).
species' expense. Herbivores 'prey' on vegetation but
often do not kill entire plants by grazing; and para- Mutualism ( + + ) completes this group of three
sites, while reducing the vitality or fecundity of their
'minor' categories, and although it represents a
Table 4.1 The effects of species 1on survival, growth or uniquely fascinating topic in evolutionaryterms, it has
fecundity in species 2. The effects of species 2 on survival, received relatively little attention from population
growth or fecundity in species 1. ecologists. Nevertheless, a single example (Janzen,
1966) will illustrate the essential continuity between
mutualism and other interactions. The bullhorn aca-
cia of Central America (Acacia cornigera)gets its name
from the pairs of large, swollen, hollow spines it bears
on its trunk. The spines have a patch of thin tissue on
one side, and small, aggressive, stinging ants (Pseudo-
mymexfermginea)perforate this tissue and nest inside
the spine. The ants also feed on nectar produced at the
base of the bullhorn acacia leaves, and on protein-rich
(Beltian) bodies, produced at the leaf-tips, and they
are, therefore, able to complete their whole life cycle
on A. cornigera. In addition, however, the ants attack
any insects that attempt to eat the acacia leaves, and

78 PART 2: INTERSPECIFIC INTERACTIONS

Table 4.2 Effects of the ant Pseudomyrmex fermginea on the will act through some combination of fecundity and
bullhorn acacia Acacia comigera. (After Janzen, 1966.) survivorship; that the interaction will be essentially
reciprocal; that competition will be for a resource
they cut the shoots of any other plants which come which is in limited supply; and that the effects will be
into contact with the acacia and may shade it. As density-dependent.
Table 4.2 shows, therefore, the ants, apart from ob-
taining food and a protected place to live from the We can, however, expect important differences
acacia, also cause measurable improvements in the between the precise nature of interspecific competi-
fecundity and the survivorship of the acacia itself by tion in animals and plants. All animals obtain their
protecting it from predators and competitors. Fecun- food from the growth, reproduction and by-products
dity and survivorship are clearly the common cur- of other living organisms, and they commonly com-
rency linking mutualism with every other ecological pete for this food. Conversely, the factors required for
interaction. plant growth, and for which they compete, are neither
self-sustaining nor the products of reproductive pro-
4.2 lnterspecific competition cesses. Moreover, because plants are sessile organ-
The final interaction in Table 4.1 is interspecific com- isms, they will, once rooted and fixed in position,
petition, and the ' - - ' symbolism stresses its essential interfere mainly with their neighbours' growth and
aspect: that the two species cause demonstrable reduc- reproduction (also true of sessile animals, of course).
tions in each other's survival, growth or fecundity. Conversely, amongst most animals there is rarely a
Nevertheless, having stressed this, it must also be continuous struggle between two, or even a few,
emphasized that whenever two species compete there individuals. Moreover, whereas two plants in close
will be some circumstances in which one species will proximity to one another may immediately suggest the
be very much more affected than the other. In such possibility of competition for a limited resource (per-
'one-sided' cases of competition it may even be haps soil nitrogen, or light in a canopy of leaves),
impossible to discern any measurable detrimental many animal species may never even encounter their
effects on the stronger competitor. These cases will competitors (because of differences in foraging strat-
appear to be amensal. (The relationship between egy, feeding times and so on). In assessing the nature
amensalism and interspecific competition is a subtle of competitive interactions, therefore, particularly
one which we shall consider in more detail in section amongst animals, very detailed observation and exact-
4.11.) ing experimentation are required. Nevertheless, de-
spite these difficulties and differences, there is, as we
As with intraspecific competition, we can expect shall see, a coherent view of interspecific competition
that the detrimental effect of interspecific competition which applies to both animals and plants.

4.3 A field example: granivorous ants

Desert environments are of interest to physiologists
because of the opportunities they present for studying
animals and plants adapted to the extremes of water
shortage; but because they are rather simple environ-
ments, there are also many instances in which ecolog-
ical studies in deserts have been very instructive. One
example is the work of Davidson (1977a,b, 1978)and
Brown and Davidson (1977) on interspecific competi-
tion in the seed-eating ants and rodents living in the
deserts of the south-western USA. In examining this

CHAPTER 4: INTERSPECIFIC COMPETITION 79

example, we shall take the approach which we try to taking seeds from foraging ants and rodents, Brown
follow throughout this chapter. First, we shall discover and Davidson were also able to show that the two
whether or not interspecific competition occurs, then guilds overlapped considerably in the sizes of seeds
we shall examine the form it takes and its conse- they ate (Fig. 4.2) suggesting not only that the ants
quences. and rodents are limited by their food, but also that the
two guilds compete with one another for this limiting
Seeds play a major role in desert ecology, since they resource.
constitute a dormant, resistant stage in the life his-
tories of plants, allowing them to survive the long, These are only suggestions, however: plausible
unfavourable intervals between short periods of deductions from field correlations. Realizing this,
growth. But these seeds are also a food source for Brown and Davidson performed an experiment in
several, distantly related taxa (including ants and which four types of 36-m diameter plots were estab-
rodents), which feed as specialized granivores. It is lished in relatively level, homogeneous desert scrub.
well established that in arid regions mean annual In two plots, rodents were excluded by trapping
precipitation is a good measure of productivity; and residents and fencing to preclude immigration; in
this productivity will determine the size of the seed another two plots, ants were removed by repeated
resource availableto these granivorous animals. Thus, insecticide applications; in two further plots, both
the graphs in Fig. 4.1 (Brown & Davidson, 1977) of rodents and ants were removed and excluded; and,
numbers of common ant and rodent species against finally, two plots were reserved as unmanipulated
mean annual precipitation, indicate that species controls. The results are shown in Table 4.3, and
number is correlated with the size of the seed re- constitute positive evidence that the two guilds com-
source. This suggests that for both granivorous guilds pete interspecifically with one another. When either
- ants and rodents - the size of the food resource rodents or ants were removed, there was a statistically
limits the number of common species, and probably
also the total number of individuals(a 'guild' is defined
as a group of species exploiting the same resource in a
similar fashion; Root, 1967). Moreover, by actually

Fig. 4.1 Patterns of species diversity of seed-eating rodents Fig. 4.2 The foods of ants and rodents overlap: sizes of
(A) and ants (0)inhabiting sandy soils in a geographic native seeds harvested by coexisting ants and rodents near
gradients of precipitation and productivity. (After Brown & Portal, Arizona. (After Brown & Davidson, 1977.)
Davidson, 1977.)

80 PART 2: INTERSPECIFIC INTERACTIONS

Table 4.3 Competition affects competitors and the resource
competed for. Responses of ants, rodents and seed density
to ant and rodent removal. (After Brown & Davidson,
1977.)

significant increase in the numbers of the other guild; Fig. 4.3 The relationship between worker body length and
the reciprocally depressive effect of interspecific com- seed size index for experiments with eight species of seed-
petition was clearly shown. Moreover, when rodents eating ants near Rodeo, New Mexico. Species designations
were removed, the ants ate as many seeds as the are as follows: N.c., Novomessor cockerelli; Ph.d., Pheidole
rodents and ants had previously eaten between them, desertorum; Ph.m., Pheidole militicidu; Ph.s., Pheidole sitarches;
as did the rodents when the ants were removed; only Ph.x., Pheidole xerophila; Pd., Pogonomyrmex desertorum; Rr.,
when both were removed did the amount of resource Pogonomyrmex rugosus S.X., Solenopsis xyloni. All species
increase. In other words, under normal circumstances except Pheidole militicidu coexist at Rodeo. (After Davidson,
both guilds eat less and achieve lower levels of abund- 1977a.)
ance than they would do if the other guild was absent.
This clearly indicates that the rodents and ants, al- species tends to specialize in seeds of a particular size
though they coexist in the same habitat, compete inter- depending on its own size.
specificallywith one another. It also suggests strongly
that the resource for which they compete is seed. The second facet studied by Davidson was the
species' foraging strategy (Davidson, 1977b),of which
Davidson (1977a, b) went on from this to examine there were essentially two types: 'group' and 'individ-
the various species of ants more closely. She was ual'. Workers of group-foraging species tend to move
particularly interested in two facets of the ants' together in well-defined columns, so that, at any one
feeding ecology. The first of these was the relationship time, most of the searchingand feeding take place in a
between a species' worker body length and the size of restricted portion of the area surrounding the nest. By
the seeds which the species harvested. Some of her contrast, in colonies of individual foragers, workers
data are illustrated in Fig. 4.3 (Davidson, 1977a), search for and collect seeds independently of one
which represents the results of an experiment in another, and, as a result, all of the area surrounding
which eight species of granivorous ants were pre- the colony is continuously and simultaneously
sented with an artificially produced range of seeds and searched. From a series of observations and experi-
seed fragments of various sizes. Workers and the seeds ments, Davidson was able to show that group foraging
they were carrying were then sample and measured. was more efficient than individual foraging when seed
The points in Fig. 4.3 refer to mean values, and densities were high and when the distribution of seeds
therefore fail to illustrate the fact that species overlap was clumped; but the relative efficiencies were re-
considerably in the sizes of seed that they take. versed at low seed densities and when the seeds were
Nevertheless,it is clear from Fig. 4.3 that seed size and more evenly distributed. The situation with regard to
body size are strongly correlated, and that each foraging is therefore directly comparable with that
regarding size. Group and individual foragers show a

CHAPTER 4: INTERSPECIFIC COMPETITION 81

considerable potential for overlap in the seeds which at a site they differ in foraging strategy, and when
they harvest, but each specializes in a particular species of similar foraging strategy coexist at a site
arrangement of the resource. In fact, the specialization they differ in size. The only apparent exception, in
resulting from foraging strategy tends to be a temporal fact, is the coexistence of Pogonomyrmex desertorum
one. Group foragers have marked peaks of activity and P. maricopa at site A, and of these the latter occurs
coinciding with periods of high seed density, and pass only rarely. Certainly this should not prevent us from
less favourable periods in a 'resting' state. Individual drawing the general conclusion that when several
foragers, although very active at high seed densities, species of granivorous ant coexist at a single site, each
retain intermediate levels of activity even during the specializes in a different way in its utilization of the
less favourable periods. food resource.

Bearing these observations on size and foraging Conversely, when the species compositions of dif-
strategy in mind, we can turn now to some of ferent sites are compared, it is apparent that the
Davidson's (1977a)results concerning the occurrence species similar in both size and foraging strategy act as
of various ant species at a range of sites. These are 'ecological replacements' for one another. Thus, of the
illustrated in Fig. 4.4. It will be convenient,initially, to two group foragers exceeding 9 mm in length, P.
restrict our discussion to the ant species with mean barbatus and P. rugosus, only one ever inhabits a site;
worker sizes exceeding 3 mm. If we do this, there are and of the three individual foragers between 6 and
several important conclusions we can draw. Almost 7.6 mm in length, P. californicus. P. desertorum and
without exception, when species of similar size coexist P. maricopa, there is never more than one that is

Site
A
B
C

D

E

012345678910
Size (mm)

Fig. 4.4 Mean worker body lengths of seed-eating ants at P.C., Pogonomyrmex calijornicus; P. m., Pogonomyrmex
five sites: A, Rodeo 'A', New Mexico; B, Rodeo 'B', New maricopa: P.p., Pogonomyrmex pima; V.p., Veromessor
Mexico; C, Casa Grande, Arizona; D, Ajo, Arizona; E, Gila pergandei. G, group forager; I, individual forager; species
Bend, Arizona. Species designations as for Fig. 4.3 plus designated by open circles occur only rarely. (After
Ph.g., Pheidole gilvescens; P.b., Pogonomyrmex barbatus:
Davidson, 1977a.)

82 PART 2: INTERSPECIFIC INTERACTIONS

common. Given the numbers of species and sites, this Fig. 4.5 The relationship between the within-colony
is very unlikely to be mere coincidence. Overall, of variation (CV) in mandible length for
therefore, it appears that although amonsgt the guild
of granivorous ants there is overlap in resource ants in the community. (After Davidson, 1978.)
utilization and interspecific competition for food, there
is coexistence only between species that differ in size, though they utilize the resource in the same way. The
or foraging strategy,or both. Species that do not differ second explanation is that the Pheidole species do
in at least one of these respects are apparently unable utilize the seed resource in different ways, but at
to coexist.

The data for ant species less than 3 mm in length,
however, do not conform to this pattern. It could be
claimed that Pheidole sitarches and Ph. gilvescens are
ecological replacement for one another, but, overall, it

any differentiation in size or foraging strategy. There
are three possible explanations. The first is that the
mode of coexistence of the larger species, described
above, is not a general phenomenon; these small
species compete for a resource and coexist, even

Fig. 4.6 Some frequency
distributions of mandible size classes

for Veromessor pergandei, with the
mean mandible lengths of

competitors most similar in size

indicated by arrows. (After
Davidson, 1978.)

C H A P T E R 4: INTERSPECIFIC COMPETITION 83

present the basis for the differentiation-some third tion, that size itself is strongly influenced by those
facet of their feeding ecology-is unknown to us. species which are most similar to V. pergandei in
Finally, the third explanation is that these species are mandible length. The effect is particularly noticeable
not limited and do not compete for the seed resource at Ajo, where Pogonomyrrnex pima (mean mandible
because they are limited in some other way; perhaps length, 0.64 mm) coexists with V. pergandei (0.81 mm).
by some other resource, or by a predator that keeps Note that, because of the nature of the data, we
their densities so low that there is no competition. cannot conclude with absolute certainty that V. per-
gandei competes with the other ant species for food;
The most important aspect of these three alternative but we can say that, at each site, a form of V, pergandei
explanations is that they illustrate some very basic has evolved which competes less with, the other
methodological problems in the study of interspecific species than it might have done. Thus, this single,
competition. The second and third explanations are elegant example illustrates four important points.
based on our ignorance concerning the ants' ecology; 1 Species from distantly related taxa can compete
but since we are always likely to be ignorant of a with one another for a limited resource.
species' ecology to some extent, these explanations 2 Competition need not lead to exclusion: competitors
can never be discounted. The first explanation is, can still coexist.
therefore, left as a last resort: to be used only when we 3 Coexisting species tend to differ in at least one
are confident of our own infallibility! Furthermore, respect in the way they utilize the limited resource,
these second and third explanations are essentially and species which utilize the resource in the same
dependent on there being some 'other' important way tend to exclude one another from a site.
differences between the Pheidole species. Yet the mere 4 The precise nature of a species,and thus the precise
discovery of differences between the species cannot, in way in which it utilizes the resource, can itself
itself, support either of the explanations. The differ- respond to the species' competitive milieu.
ences must also be shown to reflect differential utili-
zation of a resource for which the species do compete. 4.4 Competition between plant species:
Coexistence requires differences; but there are likely experimental approaches
to be many differences that have nothing to do with
coexistence. There are, therefore, some very real Experimental approaches to interspecific competition
difficulties in studying interspecific competition (to in plants have followed one or other of two broad
which we shall return in section 4.11). Yet, as we shall pathways. On the one hand, for a particular controlled
see below, concerted effort often confirms that when suite of resources, investigators have altered the
competing species coexist they do so by differential number of individual plants present in a two-species
utilization of the resource for which they are compet- mixture, varying either the density or the relative
ing. frequency of species, or both. On the other hand,
deliberate manipulation of particular resources has
Finally, Davidson (1978) looked closer still at just been made to assess the responses of competing spe-
one of these species (Verornessor pergandei) that differs cies. Plants raised from seed or from ramets in a pot are
in size, and size variation,from site to site. Some of her easy to manipulate experimentally and it is not sur-
results are illustrated in Figs 4.5 and 4.6. In this case prising that changes to density in two-species mixtures
Davidson used mandible length as a measure of size, have provided the main thrust, at least historically.
and, as Fig. 4.5 shows, the variability of this measure-
ment decreased significantly as the diversity of poten- 4.4.1 Manipulating density
tial competitors at a site increased. In other words, V.
pergandei is apparently more of a size specialist at those Deliberate manipulation of the number of plants in a
sites in which interspecific competition is most likely, pot allows the density and proportion of species in
and less of a specialist where it is least likely. This is
also apparent from Fig. 4.6, which suggests, in addi-

84 PART 2 : INTERSPECIFIC INTERACTIONS

mixture to be changed at will, and, since the plants are Fig. 4.7 The range of experimental designs used to assess
together in the same pot, there will, at suitable plant competition. represents a combination of planting
densities, necessarily be root competition for finite densities of the two species. (See text for details.)
(nutrient and water) resources in the soil, and shading
amongst adjacent leaves resulting in competition for (horizontal and vertical combinations Fig. 4.7~).We
incident radiation within the canopy. However, al- will call this a full additive design.
though the execution of a competition experiment
may appear straightforward,its design is not. Consider Simple additive experiments have been used by
a pot containing a mixture of plants: 100 of species A agronomists to demonstrate the outcome of competi-
and 50 of species B. Addition of 50 plants of species B tion between crops (the target species)and weeds and
to this pot has two immediate effects. First, it changes by plant ecologists investigating the effects of neigh-
the overall density, increasing it by a third; but it also bours on the yield of a single target plant (Goldberg,
alters the proportion of species B to A in the mixture 1987). Figure 4.8 illustrates the results of a typical
from 113to 112.In other words, additive expriments like additive experiment, in this case wheat experiencing
this confound two important variables which should, competition from the weed Bromus sterilis. Bromus
ideally, be clearly separated. One solution to this seeds were established over a wide density range (from
problem in design is deliberately to maintain the 0 to over 100 plants) into pots containing a common
overall plant density constant, but to vary the propor- density of wheat. The design therefore follows
tions of the mixture by substitution.The inception and
development of the design and analysis of these
substitutive competition experiments has largely been
the work of de Wit and his colleagues in the Nether-
lands (de Wit, 1960). The basis of the experimental
design is the 'replacement series' in which seeds of
two species are sown to constant overall density, but
the proportions of both species are varied in the
mixture from 0 to 10O01o.At a density of 200 plants we
may have 100A and 100B, 50 A and 150B, 0 A and
200 B and so on, representing a set of mixtures all at
constant density. Such a series may be repeated at
different densities if required.

Figure 4.7 summarizes the types of experimental
designs that have been used in investigating competi-
tion between two species. In the simplest case of an
additive design (Fig. 4.7a) species Y is added over a
range of densities to a constant density of a target
species (X), such that with each addition the overall
number of plants per pot increases. Case 4.7b illus-
trates the substitutive design in which the overall
density remains constant and the ratio of species X to
species Y is varied. In the third design (Fig. 4 . 7 ~a)ll
possible combinations of density and frequency are
considered such that at any one density there is
substitutive series (diagonal combinations, Fig. 4 . 7 ~ )
and there are also addition series for both species

CHAPTER 4: INTERSPECIFIC COMPETITION 85

Fig. 4.8 The response of wheat to
competition from Bromus sterilis in

an additive series experiment. (After
Firbank et al., 1990.)

Fig. 4.7a in which species X is the wheat and species Y 128 and 256 plants per pot-and five frequencies-0,
is the Bromus. Seeds of both species germinated largely 12.5, 50, 87.5 and 100%of the total sown. After 29
simultaneously, no plants died during the experiment weeks of growth in a greenhouse,the yield of spikelets
and the yield of the wheat was measured shortly per pot was assessed for each species. To measure the
before the plants flowered. We can see (Fig. 4.8) that intensity of intraspec$c competition across the range of
the yield of the wheat diminishes with increasing densities in the mixtures, both species were also
density of Bromus. Such a yield relationship may be grown in monoculture (pure stand). Figure 4.9 shows
described as hyperbolic, the yield of the target species that the yield per pot in monoculture finally became
(wheat)declining towards an asymptote as the density independent of initial sowing density. This is the
of the added species (Bromus)increases. background of intraspecificcompetition against which

This form of relationship in simple additive experi- Fig, 4.9 Intraspecific competition in Avena jatua and A.
ments has been widely observed for many plant barbata. (From Marshal1 & Jain, 1969.)
species (Silvertown & Lovett Doust, 1993). Whilst
being useful in practice for empirically ranking the
damage done by weed species to a crop it is not
possible to disentangle the competitive effect of the
weed on crop yield, from the overall effect of increas-
ing total plant density in the mixture. Substitutive
designs have therefore been employed to investigate
competitive effects at constant total density and an
early but exemplary approach of how this may be
done comes from work on wild oats.

In the annual grasslands of California two wild oat
species, Avena fatua and A. barbata, occur naturally
together, and Marshall and Jain (1969) undertook an
experimental analysis of the competitive interaction
between them, in order to elucidate the extent and
pattern of their cohabitation. The two oats were
grown together from seed at four densities-32, 64,

86 P A R T 2: INTERSPECIFIC INTERACTIONS

we must assess the reaction of each species grown in responses of A. fatua in mixture and pure stand are
mixture. almost identical, suggesting that this species is not
responding to the presence of A. barbata in mixture.
Figure 4.10 shows the replacement diagrams for Conversely,for A. barbata the yield is depressed so that
each of the four densities, with yields per pot plotted at the equiproportional mixture it is reduced to only
against the proportion in mixture. The dotted lines in 44%. The interaction appears to be amensal.
these diagrams are the appropriate yield responses
from Fig. 4.9 of each species grown on its own; solid Examining the higher planting densities, however,
lines show the responses of each in mixture. Compar- (Fig.4.10b-d) reveals a truly competitive interaction
ison of these lines for each species thus allows us to between the species. The monocultureyield responses
gauge the effect of interspecific competition. At the are all convex, indicating yield limitation by intraspe-
lowest total density (32 plants per pot), the yield cific competition. Yet in mixture the yield responses of

Fig. 4.10 De Wit replacement
diagrams for Avena fatua and A.
barbata when grown in mixture at
four overall densities. Dotted lines
are the yield expectations in

monoculture, solid lines the yield
in mixture. (After Marshal1 & Jain,

1969.)

CHAPTER 4: INTERSPECIFIC COMPETITION 87

A. barbata are all concave, and with increasing total 9-cm diameter pot, together with monoculture plant-
plant density they become substantially depressed ings over the same range. After fruiting was complete,
below the expected yield in pure stand. The same is they measured seed yield and examined graphically
also true for A. fatua, since if we examine this species' the response surface arising from competition in
response in mixture over the four densities, spikelet relating to the sown density combinations (Fig. 4.11).
yield departs more and more from the monoculture Comparison of the two surfaces reveals that in general
expectation. There is, therefore, mutual depression Phleum was more sensitive to increases in density than
resulting from interspecific competition. However, the Vulpia-yields decline more sharply for Phleum, top left
shape of A. fatua's response curve changes from to bottom right across the mixtures. More importantly
convex (Fig. 4.lOa), through linear (Fig 4.10c), to al- the data suggest that there is an intergrading of the
most concave (Fig. 4.10d). So, whilst suffering inter- effects of inter- and intraspecific competition across
specific competition, A. fatua is performing relatively the total plant density range. When present in mixture
better than A. barbata in mixture: it has a competitive at low density, both species were sensitive to an
advantage over A. barbata; and for both species the increase in the density of the other, a response which
intensity of interspecific competition increases with diminished with increasing density of the mixture. At
overall density. the highest sowing density, each species (200 plants
per pot), responded little to the presence of the
Interpretation of replacement diagrams, then, in- competitor over the entire density range. We may
volves two consideration~: infer from this that the most important competitive
1 an assessment of what a species might do when interactions at high density tend to be intraspecific
growing on its own at densities equivalent to the ones.
densities it experiences in the replacement series; and
2 a measurement of the departure from this response All of the above experimental approaches provide
when grown in mixture. methods that may illustrate the existence of competi-
tion between pairs of species but not its long-term
A response in mixture that is concave necessarily outcome over generations, a point we will consider
means that a species is suffering interspecific compe- more fully below. Earlier editions of this and other
tition, since the yield-density response in monoculture textbooks often refer to a further form of analysis of
can only be linear or convex (intraspecificallylimited). data collected on a substitutive series design and the
However, a convex response does not mean that a use of relative yields to suggest whether or not pro-
species is not experiencing interspecific competition. longed competition between two plant species will
We can only assess this with the additional knowledge result in competitive exclusion or coexistence. The
of the equivalent pure stand response. Our important relative yield of a species in a given mixture is the ratio
conclusions from this experiment are (i) that interspe- of its yield in mixture to its yield in monoculture in the
cific competition between two plant species may affect replacement series. Calculating yield in this way
the performance of both, but to different extents, i.e. it removes any absolute yield differences that may exist
is asymmetric; and (ii)that the intensity of competition between species and refers both yields to the same
is dependent on the density at which the interaction scale. The relative yield total (RYT) is then the sum of
the two relative yields in a mixture. The original
takes place. proponents of this approach argued that if the RYT of
Marshal1 and Jain's experimental design represents a particular mixture was unity then both species are
competing for the same resources for growth in that
a set of replacement series (four diagonals, see the relative gain of one species (in numbers and
Fig. 4.7b). However, to fully explore the outcome of resource acquisition) is exactly balanced by the rela-
two-species competition we must use a complete tive loss of the other. RYT values greater than unity
additive design (four diagonals, Fig. 4 . 7 ~ ) .Law and therefore signify differing resource demands-one
Watkinson(1987)grew the sand dune annuals Phleum
awenarium and Vulpia fasiculata in eight density com-
binations over a density range of 1-200 plants per

88 PART 2: INTERSPECIFIC INTERACTIONS

XI XI
Phleum
Phleum Densities shown are per 9 cm diameter pot

Fig. 4.11 The response surface measured as as seed yeild Glycine javanica) typical of Australian pastures. The
per plant of (a)Phleum arenarium, and (b)Vulpia fasiculata grass Panicum can only acquire nitrogen from the soil
arising from sowing in monoculture and mixture over a while Glycine acquires part of its nitrogen by nitrogen
range of densities and frequencies. (After Law & Watkinson, fixation from the air through its mycorrhizal associa-
1987.) tion with Rhizobium. Both species were grown in a
species' gain is accompanied by a comparatively substitutive series with and without an inoculation of
minor loss of the other. Figure 4.12 illustrates data Rhizobium. The RYTs for both dry matter yield and
analysed in this way. de Wit et al. (1966)experimented nitrogen content of the plant were not significantly
on a grass legume mixture (Panicum maximum and different from 1 in all mixtures in the absence of
Rhizobium. In contrast, when inoculated with Rhizo-
Fig. 4.12 Relative yield totals in the analysis of interspecific bium in all mixtures investigated RYTs exceeded 1.
competition between Panicum maximum (P)and Glycine
javanica (G) in the presence and absence of Rhizobium. The clear inference from this experiment is that
(From de Wit et al., 1966.) when both grass and legume compete for the same
nitrogen source (soil nitrogen),the legume will experi-
ence such intense competition from the grass that it

will ultimately disappear from the mixture; but if the
legume is able to utilize a different source, namely
nitrogen from the air fixed by Rhizobium, then the two
species will coexist.

Connolly (1986)has pointed out that the interpreta-
tion of RYTs is based on restrictive assumptions. The
first is that the yields of the individual species in the
monocultures(the 'ends' of the replacement series)are
the same as those that would be achieved if each
species was grown alone at the constituent densities
used over the series. In other words, each species is

CHAPTER 4: INTERSPECIFIC COMPETITION 89

sown in mixture and monoculture at a density that Fig. 4.13 (a)The effect on resource availability caused by
gives constant final yield (Chapter 2). Second, RYTs plants, and (b)the response to resource availability by
greater than 1 indicate potential coexistence in mix- plants. (From Goldberg, 1990.)
ture only at the particular proportion and density
investigated. Because of these restrictions, Connolly and coexistence may be determined by resource
proposed an index based on a concept of relative depletion and species' response to it has come in large
resource acquisition by species in mixture. He argued part from the work of Tilman (1976, 1982) using
that for a species A grown at a density (Nu,,) in planktonic algal species. Silicate and phosphate are
association with species B (total mixture density, essential compounds in the aqueous environment of
Asterionella formosa and Cyclotella meneghiniana, as
Nu,, + N,,,) there will be a density Nuin monoculture they are needed for the development of skeletal
structure. The growth rate of populations of these
which will give the same per plant yield ( Y ) as that single-celled organisms may be expected to be gov-
observed in mixture. The reciprocal of this density erned by the supply of these compounds in the
(l/Na),the area occupied by an individual plant, is a aqueous environment. From experimental studies,
measure of the resources needed to achieve Y. The Tilman (1976) showed that the actual growth-rate of
area required to produce the equivalent yield of Nu,, each species was determined by the concentration of
individuals with yield Y is then Na,,/Na. By similar whichever compound was the more limiting and
argument, the corresponding area for species B is moreover there were absolute limits below which
Nb,,/Nb, where terms are equivalently defined. The growth could not occur. He was thus able to quantify
relative resource total (RRT)is then the sum of these two the ranges of silicate and phosphate concentration at
ratios. If the RRT is greater than 1, the mixture of A which both species would exhibit positive population
and B returns a higher yield than the monocultures growth-rate, when these compounds were supplied at
and thus this index measures the extent to which a constant rate. Figure 4.14a illustrates the funda-
species utilize different resources. It should be noted, mental niches of each species with respect to silicate
however, that if RRT is greater than 1, this does not and phosphate. Neither species can inhabit environ-
imply that the mixture shows superior yielding to both ments characterized as region 1as the supply of both
the monocultures (Connolly, 1986). Further discus- resources is just too sparse. In region 2, however,
sion of the strengths and weaknesses of the designs silicate levels are above the threshold for net positive
available for investigating plant competition is given growth of Cyclotellabut not for Asterionella. In region 3
by Firbank and Watkinson (1990) and Snaydon the converse is the case, but here it is phosphate that
(1991). is the limiting resource. In the concentration ranges
shown by region 4, populations of both species were
4.4.2 Manipulating resources not limited by either resource and competition can be

The alternative approach to understanding interspe-
cific competition has developed from the recognition
that the methods above describe outcomes of compe-
tition but shed little light on the resources which may
be limiting and hence on underlying mechanisms.
Goldberg (1990)has drawn attention to the distinction
between the effects of competition and the responses to
competition. Competing plants will result in a lower-
ing of resource availability but the significance of this
reduction depends on the fitness of individuals at the
lowered resource level(s)(Fig. 4.13).

The development of our ideas of how competition

90 PART 2: INTERSPECIFIC INTERACTIONS

Fig. 4.14 Competition between two algae, Asterionella Consider point X (Fig. 4.14a), which is the first combi-
formosa and Cyclotella meneghiniana. (a)The fundamental nation of resources at which both species may poten-
niche of each species is defined. (b)Predicted outcome of tially coexist. They will do so only if the ratio of
competition between the species. Symbols indicate the consumption-rate to supply-rate is exactly balanced
winners and losers in competition experiments in aqueous and sufficient for each species with respect to each
environments of varying silicate/phosphate concentration. resource (i.e. not in region 2 or 4). If consumption of a
The predicted outcomes of competition according to the resource by one species is not matched by supply and
supply of phosphate and silicate are indicated by alternative the resulting concentration falls below the threshold
shading. Data points represent the results of experimental needed by the other species then competitive exclu-
tests of these predictions. In all but two experiments sion will occur. Thus consumption of phosphate by
predictions were confirmed. (O), Asterionella wins; (0) Asterionella in excess of supply rate by the environ-
Cyclotella wins; (+), coexistence of both species. (From ment will reduce phosphate concentrationto a level at
Tilman, 1977,1982.) which Cyclotella cannot persist (region 3 , below point
X). Similarly, over consumption of silicate by Cyclotella
expected to occur between them. Tilman postulated will lead to exclusion of Asterionella.
that the outcome of competition in region 4 was
determined by the relative species' consumption-rates Extension of this argument (see Tilman, 1980;
of the two compounds in relation to the supply-rates. Begon et al., 1990) leads to the following conclusion.
Competitive exclusion will always occur unless each
species consumes its more limiting resource at a rate
in excess of the supply; or, in other words, coexistence
can only arise because of concurrent disproportionate
exploitation of the more limiting resource by each
species. In Fig. 4.14b, the hatched region illustrates
the combinations of silicate and phosphate concentra-
tion in which such disproportionate exploitation was
predicted to occur, together with experimental tests of
these predictions (Tilrnan, 1982). We can see that
pleasingly in all but two circumstances the experi-
ments confirmed the predictions.

In conclusion, these experiments on plants confirm
the findings and interpretations of the studies on ants.
Two species may compete with one another and
coexist, even though the competition has a detrimen-
tal effect on both species. Moreover, the conclusion
has also been reinforced that for two species to coexist
they must in some way avoid making identical de-
mands for limited environmental resources.

4.5 The ecological niche

It is necessary to digress slightlyat this point. The term
'ecological niche' has been in the ecological vocabu-
lary for over 70 years, but for more than half of this
time its meaning was rather vague (see Vandermeer,
1972 for a historical review). Here, we shall concern