Howard Weiss
Department of Mathematics
The Pennsylvania State University
Abstract:
We study the dynamics of an overcompensatory Leslie population model where the fertility rates decay exponentially with population size. This nonlinear and noninvertible model has been highlighted by Caswell in his treatise on population models. Through careful numerical studies, we find a plethora of remarkably complicated dynamical behaviors, many of which have not been previously observed in population (or even biological) models, and which may give rise to new paradigms in population biology and demography.
We study the two and three dimensional models and find a large variety of complicated behaviors: all codimension one local bifurcations, period doubling cascades, attracting closed curves which bifurcate into strange attractors, various routes to chaos, multiple co-existing strange attractors with large basins (which cause an intrinsic lack of "ergodicity"), crises -- which can cause a discontinuous large population swing, creation and destruction of periodic orbits, merging of attractors, phase locking, and transient chaos. We show that some of the more exotic bifurcations arise from homoclinic tangencies.