Global dynamics of two population models with spatial heterogeneity

Zhai, Tianren (2021) Global dynamics of two population models with spatial heterogeneity. Masters thesis, Memorial University of Newfoundland.

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Abstract

Mathematical models provide powerful tools to explain and predict population dynamics. A central problem is to study the long-term behavior of modeling systems. The patch models and reaction-diffusion models are widely applied to describe spatial heterogeneity and habitat connectivity. Basic reproduction number R₀ plays an important role in mathematical biology. In epidemiology, R₀ stands for the expected number of secondary cases produced in a completely susceptible population by a typical infective individual. The value of R₀ can determines the persistence or extinction of population. Nowadays, characterizing the basic reproduction number due to the effects of parameters becomes very significant for predicting and controlling disease transmission. This thesis consists of three chapters. In Chapter 1, we investigate the effect of spatial heterogeneity on the basic reproduction number for an SIS epidemic patch model, and compute R₀ numerically to show the influence of the spatial heterogeneity and movement. Chapter 2 is devoted to the study of the global dynamics of a reaction diffusion model arising from the dynamics of a kind of mosquitos named A. aegypti in Brazil. We first prove the global existence and boundedness of the solutions. Secondly, we establish the threshold type dynamics in terms of the basic reproduction ratio R₀. In Chapter 3, we briefly summarize the main results and present some future works.

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