untitled

4 Both exploitation and interference competition (see Section 8.8.2) can occur

between species, although interference between species is relatively uncommon.

To obtain an understanding of what might be the expected outcome from a simple

and idealized interspecific competition we return to the logistic equation:

dN 1 /dt=rm1×N 1 ×(1 −N 1 /K 1 ) (9.1)

The term in parentheses (1 −N 1 /K 1 ) describes the impact of individuals upon other

individuals of the same species and on the population growth rate dN 1 /dt. We must

now add a term representing the impact of the second species N 2 on species 1. The

equation for the effect of species 2 on population growth of species 1 is:

dN 1 /dt=rm1×N 1 ×(1−N 1 /K 1 ) ×(a 12 ×N 2 /K 1 ) (9.2)

where rm1is the intrinsic rate of increase for species 1.

The ratioN 2 /K 1 represents the abundance of species 2 relative to the carrying capa-

city (K 1 ) of species 1. It is a measure of how much of the resource is used by species

2 that would have been used by species 1. Thecoefficient of competitiona 12 mea-

sures the competitive effect of species 2 on species 1. If we define the competitive

effect of one individual of species 1 upon the resource use of an individual of its own

population as unity, then the coefficient for the effect of other species is expected to

be less than unity. We expect this because individuals will compete more strongly

with those similar to themselves than with the dissimilar individuals of other species.

This does not always occur: when two species differ greatly in size an individual of

the larger species (l) may consume far more of a resource than one of the smaller

species (s) and in this case the aslcould be greater than unity. The converse effect

of species 1 on species 2 is denoted by the coefficienta 21 in the equation for the

other species:

dN 2 /dt=rm2×N 2 ×(1 −N 2 /K 2 ) ×(a 21 ×N 1 /K 2 ) (9.3)

These two equations (9.2, 9.3) are called the Lotka–Volterra equations, after the

two authors who produced them (Lotka 1925; Volterra 1926a). We can examine the

implications of the equations graphically by plotting the numbers of species 2

against those of species 1, as in Fig. 9.1a. First we plot the conditions for species 1

when dN 1 /dtis zero. There are the two extreme points when N 1 is atK 1 so that N 2

is zero, and when N 1 is zero because species 2 has taken all the resource. This lat-

ter point can be found from eqn. 9.2 by setting dN 1 /dtto zero and rearranging so

that it simplifies to:

N 1 =K 1 −a 12 ×N 2

If the resource is taken entirely by species 2, then:

N 1 =0, and N 2 =K 1 /a 12

Of course there can be any combination of N 1 and N 2 so that dN 1 /dtis zero; this is

seen from the diagonal line joining these two extreme points. To the left of this line

136 Chapter 9

9.2 Theoretical aspects of interspecific competition

9.2.1Graphical
models