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K 2 , which is the maximum number of species 2 that the environment can hold, is

less than that necessary to drive down species 1. Therefore species 2 always loses

when the two species occur together, as can be seen by the resultant arrows and by

the fact that the species 1 isocline is always outside that of species 2.

The above outcome is not the only possible solution, for this depends on the rel-

ative positions of the two isoclines that are shown in Figs 9.2b – d. Figure 9.2b is the

converse to that of Fig. 9.2a so that species 2 always wins. In Fig. 9.2c we see that

K 2 >K 1 /a 12 and K 1 >K 2 /a 21 so that, depending on the exact combination of the two

population sizes, either can win. Where the two isoclines cross there is an equilib-

rium point but this is unstable in the sense that any slight change in the populations

will cause the system to move to either K 1 or K 2 and the extinction of one of the

species. In nature we would never see such an equilibrium.

Figure 9.2d also shows the two isoclines crossing, but in this case K 2 <K 1 /a 12 and

K 1 <K 2 /a 21 (i.e. individuals of the same species affect each other more than do indi-

viduals of the other species, and neither is capable of excluding the other). This also

means that intraspecific competition is always greater than interspecific competition.

Hence, whatever the combination of the two populations, the arrows show that the

system moves to the equilibrium point, which is therefore stable. This situation can

occur only if there is some form of separation in the resources that they use, which

we call niche partitioning (see Section 9.6).

1 We can see from the figures that the outcome of competition depends upon the

carrying capacities (K 1 and K 2 ) and the competition coefficients (a 12 and a 21 ) accord-

ing to the Lotka–Volterra model. The intrinsic rate of increase has no influence on

which species will be the eventual winner.

2 Coexistence occurs when intraspecific competition within both species is greater

than interspecific competition between them.

3 These equations can be expanded to include the effects of several species on species

1 by summing the a×Nterms. This assumes that each species acts independently

on species 1.

4 There are several other assumptions underpinning the logistic equation, for

example constant environmental conditions leading to constantrand K, and no

lags in competing species’ responses to each other. Furthermore, the competition

coefficients are constant: the intensity of competition does not change with size, age,

or density of the competing species.

These assumptions mean that the Lotka–Volterra equations, like the logistic one, are

simplistic and idealized. It is unlikely that the assumptions hold, although they may

be approximated in some cases. The real value of these models is that they show

how it is possible for coexistence to occur in the presence of competition, and that

exclusion is not necessarily predetermined but may depend on the relative densities

of the competing species.

Much of the work in ecology has assumed that competition has occurred and is

necessary for the coexistence of species, and competition is one of the major

assumptions in Darwin’s theory of natural selection. Nevertheless, it is necessary to

demonstrate that interspecific competition does actually take place. One of the most

direct approaches is to carry out a removal experimentwhereby one of the species

is removed, or reduced in number, and the responses of the other species are then

138 Chapter 9

9.2.2Implications
and assumptions

9.3 Experimental demonstrations of competition

9.3.1Perturbation
experiments