Global dynamics of two population models with spatial heterogeneity

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

Selected topics on reaction-diffusion-advection models from spatial ecology

We discuss the effects of movement and spatial heterogeneity on population dynamics via reaction-diffusion-advection models, focusing on the persistence, competition, and evolution of organisms in spatially heterogeneous environments. Topics include Lokta-Volterra competition models, river models, evolution of biased movement, phytoplankton growth, and spatial spread of epidemic disease. Open problems and conjectures are presented

Spread of infectious diseases: Effects of the treatment of population

In a metapopulation network, infectious diseases spread widely because of the travel of individuals. In the present study, we consider a modified metapopulation Susceptible-Infected-Removed (SIR) model with a latent period, which we call the SHIR model. In the SHIR model, an infectious period is divided into two stages. In the first stage, which corresponds to the latent period, infectious individuals can travel. However, in the second stage, the same individuals cannot travel since they are seriously ill. Final size distributions of the metapopulation SIR and SHIR models are simulated with two different methods and compared. In Monte Carlo simulations, in which the population is treated as an integer, the distributions show similar behavior. However, in reaction-diffusion systems, in which the population is treated as a real number, the final size distribution of the SHIR model has a discontinuous jump, and that of the SIR model shows a continuous transition. The discontinuous jump is found to be an artifact that occurs owing to an inappropriate termination condition.Comment: 6 pages, 4 figure