Global dynamics of some population models with spatial dispersal

Zhang, Yuxiang (2012) Global dynamics of some population models with spatial dispersal. Doctoral (PhD) thesis, Memorial University of Newfoundland.

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Abstract

Spatial evolution is a very important phenomenon in ecology and epidemiology. In mathematics, integro-difference/differential equations or reaction-diffusion equations are often used to describe different spatial spread/invasion phenomena. In this thesis, we investigate the global dynamic of some integro-difference and reaction-diffusion population models with spatial dispersal and temporal heterogeneities. -- In Chapter 1, we present some basic terminologies and theorems which are used in this thesis. They are involved in monotone dynamics, spreading speeds and traveling waves, basic reproduction ratio, and chain transitive sets. -- Chapter 2 is devoted to investigate the spatial dynamics for a class of discrete-time recursion systems, which describe the spatial propagation of two competitive invaders. The existence and global stability of bistable traveling waves are established for such systems under appropriate conditions. -- In Chapter 3, we study spreading speeds and traveling waves for a class of reaction-diffusion equations with distributed delay. Such an equation describes growth and diffusion in a population where the individuals enter a quiescent phase exponentially and stay quiescent for some arbitrary time that is given by a probability density function. The existence of the spreading speed and its coincidence with the minimum wave speed of monostable traveling waves are established via the finite-delay approximation approach. We also prove the existence of bistable traveling waves in the case where the associated reaction system admits a bistable structure. Moreover, the global stability and uniqueness of the bistable waves are obtained in the case where the density function has zero tail. -- In Chapter 4, we investigate a periodic reaction-diffusion competition model, which describes the propagation of two competitive species in bad and good seasons. The existence and global stability of time-periodic bistable traveling waves are established for such a system under appropriate conditions. -- In order to study the evolution dynamics of the Lyme disease in a periodic environment, in Chapter 5, we propose a reaction-diffusion Lyme disease model with seasonality. In the case of a bounded habitat, we obtain a threshold result on the global stability of either disease-free or endemic periodic solution. In the case of an unbounded habitat, we establish the existence of the disease spreading speed and its coincidence with the minimal wave speed for time-periodic traveling wave solutions. We also estimate parameter values based on some published data, and use them to study the Lyme disease transmission in Port Dove, Ontario. -- In Chapter 6, we present a brief summary of this thesis and some future works.

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