Global dynamics of some spatially heterogeneous population models

Zhang, Fang (2006) Global dynamics of some spatially heterogeneous population models. Doctoral (PhD) thesis, Memorial University of Newfoundland.

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Abstract

The conjoining of nonlinear dynamics and biology has brought about significant advances in both areas, with biology promoting developments in the theory of dynamical systems and nonlinear dynamics providing a tool for understanding biological phenomena. Since the 1970's, various differential equations models have been proposed to study the evolutionary (long term) behavior of interacting species, the transmission of infectious diseases, biological invasions and disease spread. The purpose of this PhD thesis research project is to investigate the global dynamics and traveling waves in some spatially heterogeneous population models. -- In chapter 1, we present some elementary concepts and theorems based on the theories of uniform persistence and coexistence state, chain transitive sets, monotone dynamics, spreading speeds and traveling waves. -- In chapter 2, we study the global dynamics of a non-autonomous predator-prey system with dispersion. We establish sufficient conditions for uniform persistence and global extinction, the existence, uniqueness, and global stability of the positive periodic solutions. After that, we lift these results to asymptotically periodic systems. -- It has been observed that population dispersal affects the spread of many infectious diseases. An epidemic model in a patchy environment with periodic coefficients is investigated in chapter 3. Motivated by the works of Wang and Zhao [51], we present a disease transmission model with population dispersal among n patches, and we assumed that these coefficients are periodic with a common period due to the seasonal effects. We focus mainly on establishing a threshold between the extinction and the uniform persistence of the disease, and the conditions under which the positive periodic solution is globally asymptotically stable. -- In the book [41], L. Rass and J. Radcliffe raised an open problem on the spreading speed and traveling waves for an epidemic model on the integer lattice Z. We address this problem in chapter 4 by appealing to the theory of spreading speeds and traveling waves for monotone semiflows [34]. More precisely, we establish the existence of asymptotic speeds of spread, and show that this spreading speed coincides with the minimal wave speed for monotone traveling waves. -- Chapter 5 is devoted to the investigation of the asymptotic behavior for a reaction diffusion model with a quiescent stage, which was proposed by Hadeler and Lewis [18]. By appealing to the theory of spreading speeds and traveling waves for monotone semiflows, we establish the existence of asymptotic speed of spread and show that it coincides with the minimal wave speed for monotone traveling waves. By the theory of monotone dynamical systems and the persistence theory, we prove a threshold type result on the global stability of either the zero solution or a unique positive steady state in the case where the spatial domain is bounded. -- To illustrate the obtained mathematical results, we also provide numerical simulations in chapters 2-5. -- At last, we summarize the results we have obtained in the thesis, and also point out some problems for future research in chapter 6.

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