Seminar in PSA 107

ABSRACT

Title: Discrete Models of Competition.

By: J. M. Cushing, Department of Mathematics & Program in Applied Mathematics, University of Arizona

In the 1960's P. H. Leslie and T. Park used a discrete time, stage-structured model in a famous experimental study that played a significant role in formulating the classical theory of competition. They used the model equations to explain the results of experiments (involving two species of beetles) which lent support to the assertion that two similar species cannot coexist on a single limiting resource. This notion is basic to the concept of ecological niche. However, Park's experimental results contained some anomalous outcomes that are not consistent with this fundamental principle. Although Leslie and Park puzzled over these anomalous results in several papers, they never arrived at an explanation. In the 1990's a more sophisticated model for the dynamics of the same species of beetles used in Park's experiments was developed for use in numerous laboratory studies in nonlinear population dynamics. I will discuss how this newer model, when extended to two competing species, offers a possible explanation for Park's data, including the anomalous results. Interestingly, the explanation stands in contradiction to classical competition theory and the notion of ecological niche. Mathematically, the explanation is based on the occurrence of multiple attractors (not all of which are equilibria) that arise through an unexpected sequence of bifurcations.