Dynamics of Functional Differential Equations with Applications to Biology and Ecology

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Many biological and ecological processes involve time lags. For example, a predator needs time to convert its consumption of prey to its biomass; for many infectious diseases, time lags exist due to dispersal of populations and disease state changes. To model these time-lag involving processes, a natural choice is to use functional differential equations (i.e., differential equations subject to delays). In contrast to systems described by ordinary differential equations, the initial state of a functional differential equation is a function specified on an interval and the associated phase space is infinite dimensional. Since the resulting dynamical systems are infinite dimensional, the analysis of functional differential equations is extremely challenging, especially for equations with delay-dependent coefficients, nonlinearity and multiple state variables, which happen to be typical features of equations risen from biology and ecology. In this proposed research, I aim to develop new methodologies and techniques to depict the global dynamic structure of certain nonlinear systems of functional differential equations arising from important applications in biology and ecology. In particular, I plan to formulate biological and ecological models by mainly using functional differential equations and to develop new methodologies, techniques and algorithms to study their short-time (transient) and long-time (asymptotic) behaviors. The focus will be on the effects of time delays involved in the dispersal of populations, in intraguild predation and in induced resistance of plants in responding to insect herbivore's attack on the dynamics. Theoretical and applicable results to be established in this proposed research will increase our understanding of both short-time and long-time qualitative behaviors in biological and ecological models with time delay and nonlinearity, and thereby, provide useful and valuable guidance and suggestions on controlling the spread of infectious diseases, preventing the extinction of endangered species, and maintaining sustainable development of ecosystems. Anticipated newly developed theories, methodologies and techniques in this proposed research will highly enhance the advance of theoretical development in nonlinear functional differential equations and nonlinear dynamical systems. The established theoretical results will greatly improve our ability to deal with real world problems arising from biology and ecology and provide biologists and ecologists powerful tools to solve their specific problems that may be described by functional differential equations. This research will also provide excellent opportunities to train graduate students and postdoctoral researchers in dynamical systems and mathematical biology.