Mathematical Sciences: Stable and Chaotic Behavior of Deterministic Models of Interacting Populations with/withoutGenetic Variation

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The investigators study the dynamical behavior of systems of difference and differential equations that describe interacting populations with and without genetic variation. Special consideration will be given to the onset of chaos and of strange attractors, e.g., a period-doubling cascade to a strange attractor that is forced by changing a single intraspecific parameter. The proposers will attempt to reverse the onset of chaos by the periodic harvesting or seeding of a population. These studies will include numerical experiments and analytical arguments employing techniques such as stability theory, invariant set theory, bifurcation and perturbation theory. The development of an ecosystem, whether natural or managed, is affected by complex interactions among environmental and genetic components. The proposed research will study mathematical models that attempt to capture some of the essential features of these interactions. The objectives of the research include understanding: (i) how environmental and genetic parameters influence species coexistence and extinction; (ii) the onset of chaotic fluctuations in the densities of ecosystem populations; and (iii) how harvesting and seeding strategies affect ecosystem stability. Such knowledge may help in maintaining biodiversity in natural ecosystems, in the management of natural resources such as forests and lakes, and in the development of disease-resistant crops.