Population Dynamics of 2 Preys 1 Predator
Models

In
the process of modeling a food net of one predator, two preys, the behavior of
the predator must be carefully considered.
For example, the predator may simply hunt for whatever prey it comes to
first, or it may be restricted in what prey it can hunt for by the time of day,
month, or year, or other environmental factors. These behavioral differences lead to distinct predation considerations
in mathematical models, leading to vastly different dynamical phenomena of the
system. The predation term models the
rate at which the predator consumes the prey according to Holling’s theory on species
predation.

These
models can be analyzed using techniques such as singular perturbation analysis,
and numerical computation. Our study
suggests that the ‘mixed’ predation model, having only one time unbiased predation
term, has relatively simple behavior and that it may be possible to group the
two preys together. This would reduce
the system to a traditional two-dimensional, one predator, one prey system.

The
other model, where the predator must schedule its hunt between two preys leads
to two ‘independent’ predation terms. Its behavior is much more complex. It can have equilibrium point, limit cycles
of distinct types, and chaotic dynamics.
Specifically, this model has a chaotic attractor containing a Shilnikov’s orbit as shown.

More work is needed on the independent predation model to
determine if other types of chaotic behaviors are possible and
their ecological implications.

(Preprint will be made available later)