Coexistence And Periodicity Of A Two - Species Reciprocal Model

Biomathematics has become a current research focuses of modern applied mathematics. Biomathematics is an interdisciplinary subject which applies mathematics to study biological phenomena in various biologic areas. The general method is to establish mathematical models of the studied objects and analyze those models qualitatively and quantitatively. Earlier ecological models are described by differential equations, such as Logistic equation, Malthus equation, and Lotka-Volterra equation. Since that those problems have strong biological background, related Mathematical results can be used to explain some biological phenomena like co-existence, extinction and persistent existence. Nowadays, much attention has been paid to population biology. However, most models are expressed by differential equations with only consideration of time on population density. In fact, in certain areas, the population itself will diffuse and migrate from high density areas to low density areas in order to acquire enough food, and also the population will have cross diffusion characteristics to resist illness and the invasion of species. Therefore, it is more reasonable to use reaction-diffusion system to describe these models. By some transformations, these systems are divided into the weak-coupled and strong-coupled systems. And therefore, two problems have been discussed in the paper. One is the co-existence of strong-coupled elliptic problems with Dirichlet boundary conditions. The other is the periodicity of weak-coupled elliptic problems.This paper contains four sections. The first section is introduction of the related background, development and status. The second section deals with mutualistic models which describing by strong coupling elliptical problems with Dirichlet boundary conditions. The general elliptic problem is first introduced, and then sufficient condition for coexistence solutions of mutualistic models of strong-coupled elliptic problems has been given by using lower-upper solution method, monotone iterative method and Schauder fixed point theorem. Our results show that at least one positive solution exists for strong-coupled systems when cross-diffusion and interspecific interaction are relatively weak. The third section is devoted to the periodicity of weak-coupled elliptic system by applying lower-upper solution method and monotone iterative sequence. The fourth section presents direct visual graphs of the theoretical results by numerical simulations.