Spatio-temporal dynamics of some reaction-diffusion population models in heterogeneous environments

Li, Zhimin (2024) Spatio-temporal dynamics of some reaction-diffusion population models in heterogeneous environments. Doctoral (PhD) thesis, Memorial University of Newfoundland.

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Abstract

Spatial and temporal evolutions are very important topics in epidemiology and ecology. This thesis is devoted to the study of global dynamics of some reaction-diffusion models incorporating environmental heterogeneities. As biological invasions significantly impact ecology and human society, how invasive species’ growth and spatial spread interact with the environment becomes a significant challenging problem. We start with an impulsive time-space periodic model to describe a single species with a birth pulse in the reproductive stage in Chapter 2. In-host viral infections commonly involve hepatitis B virus (HBV), hepatitis C virus (HCV), and human immunodeficiency virus (HIV). To explore the effects of the spread heterogeneity on the spread of within-host virus, we propose a time-delayed nonlocal reaction-diffusion model and obtain the threshold-type results in terms of the basic reproduction ratio in Chapter 3. In Chapter 4, we then explore the existence and nonexistence of traveling wave solutions for such a non-monotone system on an unbounded domain, and show that there is a minimum wave speed for traveling waves connecting the infection-free equilibrium and the endemic equilibrium. Mosquito-borne diseases are transmitted by the bite of infected mosquitoes, including Zika, West Nile, Chikungunya, dengue, and malaria. To investigate the effects of spatial and temporal heterogeneity on the spread of the Chikungunya virus, we develop a nonlocal periodic reaction-diffusion model of Chikungunya disease with periodic time delays in Chapter 5. We further establish two threshold-type results regarding the global dynamics of mosquito growth and disease transmission, respectively. At the end of this thesis, a brief summary and some future works are presented.

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