Chaotic Attractors and Physical Measures for Some Density-dependent Leslie Population Models

Following ecologists discoveries, mathematicians have begun studying extensions of the ubiquitous age structured Leslie population model to allow some survival probabilities and/or fertility rates depend on population densities. These nonlinear extensions commonly exhibit very complicated dynamics: through computer studies, some authors have discovered robust Hénon-like strange attractors in several families. Population researchers frequently wish to average a function over many generations and conclude that the average is independent of the initial population distribution. This type of “ergodicity” seems to be a fundamental tenet in population biology. In this manuscript we develop the first rigorous ergodic theoretic framework for density dependent Leslie population models. We study two generation models with Ricker and Hassell (recruitment type) fertility terms. We prove that for some parameter regions these models admit a chaotic (ergodic) attractor which supports a unique physical probability measure. This physical measure, having full Lebesgue measure basin, satisfies in the strongest possible sense the population biologist’s requirement for ergodicity in their population models. We use the celebrated work of Wang and Young [14], and our results are the first applications of their method to the biological sciences.