Aaron A. King, Ph.D.
Assistant Professor of Ecology & Evolutionary Biology and MathematicsUniversity of Michigan
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Weakly Dissipative Predator-Prey Systems
Bulletin of Mathematical Biology, 58:835-60, 1996.
Abstract
In the presence of seasonal forcing, predator-prey models with quadratic interaction terms and weak dissipation can exhibit infinite numbers of coexisting periodic attractors corresponding to cycles of
different magnitude and frequency. These motions are
best understood with reference to the conservative case, for which
the degree of dissipation is, by definition, zero. Here one
observes the now familiar mix of ``regular'' (neutrally-stable
orbits and tori) and chaotic motion typical of non-integrable
Hamiltonian systems. Perturbing away from the conservative limit,
the chaos becomes transitory. In addition, the invariant tori are
destroyed, and the neutrally stable periodic orbits become stable
limit cycles, the basins of attraction of which are intertwined in
a complicated fashion. As a result, stochastic perturbations can
bounce the system from one basin to another with consequent changes
in system behavior. Biologically, weak dissipation corresponds to
the case in which predators are able to regulate the density of
their prey well below carrying capacity. For reprints of this paper, contact me at aaron.king@umich.edu. An electronic reprint (PDF) is also available.