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Quantitative Apparent Competition:
Mean Field Three Species Equilibria

Quantitative effects of apparent competition, where the predator causes reductions in prey density rather than complete elimination, are more difficult to observe. As already discussed, this occurs in the region of parameter space in which all three species can coexist in the mean field case. I will demonstrate that under conditions of local interactions in the presence of the predator, the non-target prey causes reductions in pest density. Again, comparisons with comparable models employing global interactions will show that this form of apparent competition is a result of local interactions and spatial structure.

Under certain conditions for the mean field model, another equilibrium exists in which all three species coexist. When this equilibrium exists, the single prey-predator equilibria are not stable to the introduction of the other prey species. The four conditions for a three species equilibrium in the mean field model are:

(1) g>e

(2) c>d

(3)

(4)

A sample set of parameters satisfying these are (b1=.007, b2=.0055, d=.004, d1=.05, d2=.015, b=.001, b1=.2, b2=.06, d1=.01, , and d0=.0001), and will be the basis for simulation. The mean field equilibria for these parameters are shown in Table 2.

Table 2

Prey 1

Prey 2

Predator

0

0

0

.43

0

0

0

.27

0

.045

0

.047

0

.17

.026

.011

.13

.035

When no predators are present in the global and local interaction models, or when just the predator and a single prey are present, the dynamics and equilibria are similar to the case I described in the last section; either prey is capable of persisting without the other prey. For the situation in which all three species are introduced concurrently, however, all of the species persist in both the local and global interaction cases. The equilibrial densities are, however, different for these two modes of interaction (Fig 6).

The equilibrium values for the global interaction case (Fig 6a) are close to those predicted from the analytical calculation for the mean field model. For the local interaction case (Fig 6b), however, the equilibrium density of the non-target species is much closer to the mean field equilibrium value in the absence of the pest than it is to the analytically calculated three species equilibrium value. This implies that the non-target prey is experiencing an environment that is effectively free of the pest species.

I quantified the aggregation of the prey species as the proportional increase in probability above random chance that a neighboring cell will be occupied by one prey type given that the focal cell is occupied by the alternative prey type (Klopfer 1997). Specifically, let Px be the probability of a patch type x, Py be the probability of patch type y, and Pxy be the probability that a pair of adjacent cells is in the configuration (x,y); then the measure of aggregation is

R = (3)

This measure indicate that the two prey species have spatial distributions that are negatively related (-.35, where -1.0 is complete lack of overlap between the species and 0 is random mixing).

This negative relationship is caused by the great increase in predator density when predators are introduced to a patch of pests. This increase results from the high attack rates of the predator on the pests, and the births associated with these attacks. The high densities of predators also serve to reduce the number of non-target prey that are in the vicinity of the predators. Non-target prey that are spatially segregated from the pests experience lower predation pressure because nearby predators are lower in density due to the lower attack and birth rates of the predators when associated with only the non-target species. Consequently, the regions in which the non-target prey flourish are associated with a lack of pests. Since the pests are occupying only a small percentage of the patches (densities of .01-.02), and these are isolated from most of the non-target prey, the non-target species experiences an environment that is essentially free of the pest species.

In relation to apparent competition, the three species equilibrium case shows that while apparent competition is present when interactions are both global and local, it takes fundamentally different forms under these interaction modes. These differences are established by comparing densities of one prey species with and without the other prey in the presence of the predator. For the case of global interactions, the presence of the alternative prey type causes reductions in density of the prey for both prey species. This is a case of symmetric apparent competition. When interactions are local, however, this apparent competition is one sided and only the pest is reduced in density in the presence of the alternative prey type. The majority of the non-target prey are essentially isolated from the pest species and do not suffer a reduction in density. A few non-target prey remain among the pests and allow more complete control of the pests, thus reducing the pests' density. Again, local interactions have induced a change in apparent competition.

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