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Qualitative Apparent Competition:
Mean Field Single Prey-Predator Equilibria

In the multi-state model I showed a qualitative form of apparent competition; a complete elimination of one of the pest species in the presence of the predator and the alternative prey. A similar phenomenon occurs in the spatial presence-absence version with local interactions, but not with global interactions. Thus, clearly this form of apparent competition can be attributed to the local dispersal, at least for the range of parameters considered.

Since this case is restricted to the region of parameter space in which the prey species cannot coexist, five possible equilibria still exist for the mean field model. These equilibria can be solved analytically for the mean field case though the expressions are somewhat cumbersome.

For the mean field model, the pest only equilibrium is stable in the absence of the predator and the pest-predator equilibrium is stable to the introduction of the non-target prey. For the case of local interactions, the parameters (b1=.008, b2=.006, d=.004, d1=.08, d2=.04, b=.001, b1=.2, b2=.1, d1=.005, , and d0=.0001) approximate the behavior of case 2 from the CAM-8 model (i.e. cycling of predator and pest with 2 species present, and extinction of the pest when the non-target prey is added). These dynamics are illustrated in Figures 3a-c for the two prey, the pest and the predator and all three species respectively. The values of the equilibria for the corresponding mean field model are given in Table 1. While the values presented in this table are specific to the parameter set chosen, similar scenarios were produced for other sets of parameters.

Table 1:

Prey 1

Prey 2

Predator

0

0

0

.5

0

0

0

.333

0

.022

0

.043

0

.045

.038

The results from the local interaction model clearly indicate a qualitative form of apparent competition. In the absence of the predator, the pest eliminates the non-target species, whereas when the predator is present the non-target species persists and the pest does not. This result holds for a fairly wide range of initial conditions. Due to the stochastic nature of the model, extremely high initial densities (total grid occupancy of around 90% or higher) may cause the predator to go extinct which results in the pest displacing the non-target prey.

When interactions are global, the results (Figs 4a, b) look similar to Figures 3a,b. The pest still eliminates the non-target species and the pest-predator system persists going through damped oscillations. After a long period of time for the predator-pest system, the oscillations disappear. This is true for the whole environment in the local dispersal case as well, but the local oscillations appear to persist in the local dispersal case (Fig. 1).

When all three species are concurrently introduced in the global interaction case, the outcome depends on initial densities, as shown in Figure 5. When the species are introduced at low densities, the pest displaces the non-target species (Fig 4c). When initial densities are higher, however, stochastic fluctuations may cause the pest to be rapidly eliminated after a crash (Fig 4d). In the mean field model stochastic fluctuations are not present; hence the pest is never eliminated and consequently will always win.

It is unlikely that all three species would actually arrive in an environment at the same time. Therefore, rather than introducing all three species simultaneously, a more realistic situation is to introduce one of the prey species into an already established population of the predator and the other prey species. When this is done for the global interaction case the pest invades the non-target prey-predator system but the non-target prey does not invade the pest-predator system. When interactions are local, the situation is reversed: the pest does not invade the non-target prey-predator system, but the non-target prey invades the pest-predator system.

This example demonstrates that local interactions can induce apparent competition. In a broad sense, the apparent competition results from local densities of predator and prey significantly differing from overall densities when interactions are local. More specifically, it is caused by the aggregation of predators around the pest species, even when densities of pests are low. Figure 1 shows that predator densities are much higher in regions where densities of pests are also high, but some patches of pests are completely devoid of predators. This creates a situation in which the local predator-prey dynamics are much different than a mean field model would predict.

The predator fails to control the pest in the absence of the non-target prey because once the predators reach high densities, they rapidly consume the nearby pests. When the pests are nearly eliminated in a region, the predators in that region soon decline in density as well. The lower predator density allows the few pests that have escaped to increase in density after the predators have declined. Eventually these patches of pests are recolonized by predators from other regions, and the cycle begins again.

This escape mechanism does not work in the presence of the non-target species, which provides a corridor for the predators to reach all of the pests. The lower birth and attack rates of the predator in the presence of the non-target prey establishes a less volatile situation, in which the predators do not increase or decrease as rapidly as they do in the presence of the pest species. Consequently, after the predators quickly increase in density in regions of high pest density, and then crash after eliminating the pests, some of the predators persist at low densities with some of the remaining non-target prey. This allows the predators to eliminate the few remaining pests without going extinct in that region.

These dynamics do not normally occur when interactions are global. The pests can never escape the predators in such an environment. Therefore, pests don't reach densities high enough to produce the dramatic increase and decrease of predators that causes them to crash. Instead, the more moderate dynamics (compare Figures 3c and 4c) of the local dispersal favor the persistence of the pest species and the elimination of the non-target prey.

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