Competition is one of three major kinds of interactions that can occur between the populations of two species of organism. In the feedback structure created by competition between two species, each population can have a negative effect on its own rate of increase, as represented by arrows from each population to itself, and can also have a negative effect on the rate of increase of the other species, as represented by the arrows bridging the two populations. The self-regulating feedback could be due to intraspecific competition for food or space, or to the influence of natural enemies. The interspecific effect could be due to the removal of food or space required by the other species, interference with mating or other essential processes, or direct predation on the other species. Notice that this structure has two negative feedback loops representing the two separate intraspecific interactions, and one positive feedback loop, representing the interspecific interaction; i.e., the loop is positive because the product of two negative causal links gives rise to a positive loop.
Let us return to the logistic R-function and write it in the following way
where C11 refers to the marginal impact of species 1 on its own kind, N1 to the density of species 1, and the time symbolism (t-1) is omitted for convenience. It is easy to extend this equation to two competing species
where C12 is the marginal impact on species 1 of species 2. The equation for species 2 is identical so we need not repeat it here. This equation states that the rate of change of species 1 is equal to its maximum A1 minus the intraspecific effect C11 * N1 minus the interspecific effect C12 * N2. The zero growth isocline for species 1 can now be determined by setting R1 to zero and solving for one species or the other,
or, assuming that K1 = A1/C11, where K1 is the equilibrium density of species 1 in the absence of species 2 (or its carrying capacity), then
This isocline is a straight line with an intercept at K1 on the N1-axis And at K1 * C11/C12 on the N2-axis. Notice that the slope of the line, C12/C11, measures the effect of the competitor relative to the species in question; i.e., when C12 = C11 then the two species are competitive homologues. Of course, the isocline for the other species is identical.
When the two isoclines are superimposed in the phase space of
the two species we obtain a graph which has equilibrium points
at K1 and K2, and may also have a community equilibrium
where the two isoclines cross (see figure).
NOTE the two-species trajectory, the gray line starting at the
magenta spot and converging to the equilibrium point.
The community equilibrium is stable, and the two species coexist, if and only if
or the combined effect of intraspecific competition (-feedback) is stronger than the combined effect of interspecific competition (+feedback).
On the other hand, if this condition is not met, or the combined effect of interspecific competition is stronger than the combined effect of intraspecific competition, then the system is unstable (because +feedback dominates) and coexistence is impossible. Under this condition:
The equation can be generalized to include non-linear intraspecific competition by adding a coefficient of curvature Q
where i identifies the object species and j its competitor. The zero growth isocline from this equation has a concave shape when Q < 1, convexity when Q < 1 and will be linear when Q = 1. Of course, we could also place a curvature coefficient on the interspecific effect but, in order to keep down the number of parameters, PAS models do not currently include such effects.