Population Dynamics of 2 Preys 1 Predator Models

Students: Brian Bockelman, Elizabeth Green, Leslie Lippitt, and Jason Sherman,

Advisors: Bo Deng, and Wendy Hines

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)