- The CAM-8 model illustrates the phenomenon of apparent
competition in an explicitly spatial model. But, the question still remains
whether the observed behavior is dependent upon the spatial structure of
the model, or whether it could also happen in a non-spatial version. Direct
comparison is virtually impossible, because even the local description is
64 dimensional. A more flexible approximation is a simple presence-absence
model, including direct competition of the prey species
- In the presence-absence model, each site can be occupied
by only a single individual; however, instead of interactions occurring
within a cell, they occur within a neighborhood in a manner similar to that
assumed by Durrett and Levin (1996). Thus, the cell size in this models
should be thought of as much smaller than that for the CAM-8. Births occur
onto neighboring empty sites, and predators and prey need only be in the
same neighborhood to affect each other's birth and death rates. In the basic
model, the interaction neighborhood is small, so that interactions are local;
this is compared with a global version, in which the interaction neighborhood
is the whole grid, and with a "mean-field" version of the global
model, which considers only expectations. The first two models are tools
for simulation; the mean field approximations is a system of difference
equations.
- All three models are fundamentally similar in their assumptions
and are based on the multi-state equations above. The death expressions
from these equations remain exactly the same, with the exception that numbers
of predators and prey are counted within the neighborhood. The birth probability,
however, now represents the probability of a birth onto an empty cell given
that the empty cell has P1 and P2 neighboring
prey of types 1 and 2 respectively, and Pd neighboring
predators. Transport involves random movement from an occupied to an unoccupied
neighboring cell.
- The probabilities of sites transitioning from one state
to another for the presence-absence model are described in Figure 2. The
mean field approximation (2) gives the expected values of the proportion
of patches occupied by prey 1 (P'1), prey 2 (P'2,),
predators (P'd) and nothing (E') respectively in
the new generation, given values in the preceding generations. The number
of patches is conserved, so that E=1-P1-P2-Ps
. b1 and b2 are the birth rates of the
prey, d is the death rate of prey, d1 and d2
are the additional death rates of prey due to predators, b
is the base birth rate of the predator, b1
and b2 are the additional birth rates
of the predator due to prey 1 and 2, d1
is the death rate of the predator, and d0
is the additional death rate of the predator due to lack of prey. Thus
 (2)
- A comparable spatial model can be constructed by implementing
the dynamics of equation 2 on a per site basis in the form of a cellular
automata (ca) model. Instead of Px representing the percent of
patches in the environment occupied by type x, it represents the
percent of patches in a given neighborhood that are of that type. Additionally,
P'x represents the probability of any given cell being
in any one of the three given states in the next time step. For the purpose
of this study, the spatial environment is a grid of 250x200 synchronously
updated cells, with periodic boundary conditions (i.e. a torus). The neighborhood
of interaction for the local dispersal case is a Moore neighborhood (box)
of range three, so that each individual interacts with many other individuals,
as in the multi-state model. There is also a transport phase, for the local
interaction case, in which each individual has a small probability (.001
for each species) of moving into an empty cell in its range three neighborhood.
For the global dispersal model, the neighborhood is the whole grid, and
transport is ignored since any movement would be irrelevant.
- I divide discussion into two main cases, as distinguished
by the dynamics of the mean field approximation which has several possible
equilibria. There is never an equilibrium with both prey in the absence
of the predator since the prey with the higher birth rate will always win.
For some ranges of parameters, the mean field model does not admit an equilibrium
with both prey species present if the predator persists. In this case, a
single boundary equilibrium will be globally attracting from any set of
initial conditions starting with all species present. Which of the five
possible boundary equilibria will prevail will depend upon exact parameters.
For other choices of parameters, a globally stable internal equilibrium
with all species present will exist; hence all boundary equilibria are unstable.
- As expected, the spatial model with global interactions
behaves similarly to the mean field approximation, except when extreme starting
conditions (very high or low densities) produce stochastic fluctuations.
Localized interactions, on the other hand, can lead to different results.
Parameters that permit the pest to persist with the predator for the mean
field case may permit persistence of the other prey and elimination of the
pest when interactions are local. Local interactions thus, through apparent
competition change the outcome of the competitive interaction of the two
prey species. When coexistence of all three species is possible in the mean
field equations, however, the effect of localization of interactions is
a quantitative rather than qualitative one. Apparent competition can lead
to a reduction in the pest species without reducing the other, but there
is no complete elimination of the pest. While the structure of this analysis
corresponds with mean field results, the closest comparisons are between
the local and global dispersal models, which both incorporate stochastic
effects: it is these comparisons that I will primarily address in the following
discussion.
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