Analysis Of A Nonautonomous Eco-epidemic Diffusive Model

Mathematics as a basic discipline has been widely infiltrated all areas of the natural sciences such as the discovering of many asteroids astronomical and the calculation of the track, they all relies heavily on mathematics; especially physics, quantum theory and relativity which are deeply proposed to lay a mark of mathematics.Mathematical Biology which has been one of the most well recognized subjects in modern applied mathematics is a frontier discipline overlapping life sciences and mathematics. It uses mathematical theory and computer technology to study quantities and types of spatial structure in nature in the life sciences, and to analyze inherent characteristics of complex biological systems, to find bio-informatics concealed in the data from a large number of biological experiments. Epidemic dynamics is an important component of biological mathematics. Based on the characteristics of population growth and the process of disease spreading, it uses to establish the appropriate mathematical models. Then through quantitative and qualitative analysis and numerical simulation, it uses to study the pathogenesis of infectious diseases, infectious pathways and characteristic of prevalence, which provides a theoretical basis for the people to prevent and control the disease.It is well known that Kermack and McKendrick constructed a mathematical model to study epidemiology in1927. From then on, more and more attention has been focused on epidemic models. But in the early model of infectious diseases, they just consider a single population. As we know, the actual situation is not so. In the natural environment, the population can not survive alone, they not only compete from other populations for the living space, food, but also suffer from the risk of being prey to other populations. So it is more biological significance to consider the interaction of populations when we study the dynamical behavior of infectious diseases model.Most previous work has considered the autonomous system, that is, the parameters were assumed to be constants, independent of time and position. However, non-autonomous phenomenon is so prevalent in the real life that many epidemiological problems can be modeled by non-autonomous systems of nonlinear differential equations. For example, many diseases show seasonal or regional behavior. So it is necessary to consider the non-autonomous eco-epidemiology model.Based on the above factors, in this paper we deal with the behavior of positive solutions to a nonautonomous reaction-diffusion system with homogeneous Neumann boundary conditions the model describes a two-species predator-prey system in which there is an infectious disease in prey.The first chapter, we briefly introduce the background and present the reaction-diffusion problem, and the main contents of this paper are also given.The second chapter is devoted to some notations, and some useful lemmas for the non-autonomous differential equation which will be useful in the sequel.The third chapter, we establish the sufficient conditions on the permanence of prey and the predator by using the comparison, and give the sufficient conditions on the spreading and vanishing of the disease. Also, by constructing a Lyapunov functional, we obtain the global attractivity of the corresponding autonomous model. Numerical simulations are also included in the final part to verify our analytical results.The fourth chapter deals with numerical simulations by using Matlab. The numerical results help one to better understand the theoretical conclusions obtained in former chapters.The fifth chapter, we first give a short summary, and then present some work to be considered.