- 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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