Asymptotic behavior and traveling waves for some population models

Xu, Dashun (2004) Asymptotic behavior and traveling waves for some population models. Doctoral (PhD) thesis, Memorial University of Newfoundland.

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Abstract

Since the 1970s, more and more mathematicians have been trying to propose reasonable models for the growth of species in all kinds of environments and for the spread of epidemic diseases, and to understand the long-term behavior of their modelling systems. This thesis, consisting of five chapters, mainly deals with the dynamics of population and epidemic models represented by some time-delayed ordinary and partial differential equations, and reaction-diffusion systems. -- In Chapter 1, we present some basic concepts and theorems, which involve the theories of monotone dynamics, uniform persistence, essential spectrum of linear operators, asymptotic speeds of spread and minimal traveling wave speed. -- Based on some specific competitive models, we formulate in Chapter 2 a class of asymptotically periodic delay differential equations, which models multi-species competition, and investigate the global dynamics of the model. More precisely, we established the sufficient conditions for competitive coexistence, exclusion and uniform persistence via theories of competitive systems on Banach spaces, uniform persistence, periodic and asymptotically periodic semiflows. -- Chapter 3 focuses on a nonlocal reaction-diffusion equation modelling the growth of a single species. For this model, we obtain a threshold dynamics and the global attractivity of a positive steady state. We also discuss the effects of spatial dispersal and maturation period on the evolutionary behavior in two specific cases. Our numerical investigation seems to suggest that the model admits a unique positive steady state even without monotonicity conditions. -- In Chapter 4, we consider an epidemic model represented by a reaction-diffusion equation coupled with an ordinary differential equation, which is proposed by Capasso et al. Here, the existence, uniqueness (up to translation) and global exponential stability with phase shift of bistable traveling waves are studied by phase plane techniques, monotone semiflow approaches and a detailed spectrum analysis. -- In Chapter 5, the asymptotic speeds of spread for solutions and traveling wave solutions to the integral version of the epidemic model in Chapter 4 are investigated. Our results show that the minimal wave speed for monotone traveling waves coincides with the asymptotic speed of spread for solutions with initial functions having compact supports. Some numerical simulations are also provided.

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