Mathematical Sciences: Global Qualitative Analysis of Ecological Models with Delays and Diffusions

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9306239 Kuang The investigator studies the global qualitative behavior of solutions of some ecological models with delays or diffusions. The principal themes of this work are the fundamental problems of finding various criteria for the uniform persistence of these considered systems, the global stability of steady states, the existence of periodic solutions and spatial patterns, and the existence of chaotic behaviors. These topics indeed cover most aspects of the so-called global qualitative analysis of nonlinear systems. In particular, the investigator studies some mathematical models of populations dispersing among patches in heterogeneous environments, where various kind of interactions are considered. In addition to performing the traditional local stability or bifurcation analysis, he considers global stability, bifurcation, and chaos aspects of the nonlinear systems. Time delays and space variations constitute integral components of most ecological, biological, physiological, chemical, physical, economic and atmospheric processes. Almost all these real-life processes are nonlinear in nature. They are normally too complicated to simulate in a lab, or too difficult to understand from experimental work alone. In order to better manage or control such processes, it is necessary to study them mathematically. Most of the previous approaches to these applied mathematical problems have been local (pretending the system behaves in a small region as if it were linear rather than nonlinear) and thus do not paint the whole picture of the real systems. This project tries to gain a full view of the real picture by focusing on the nonlinear effects of the dynamics of the considered systems. This work offers a better understanding of the dynamic interplay of the effects of time delays and diffusions. In particular, the final results can be useful to ecologists, engineers, and system scientists. ***