Growth Regional Logistic Equation Of Diffusion On The Asymptotic Behavior Of Solution

Biomathematics has been one of the most well-recognized subjects in modern applied mathematics. Especiallly, mathematics is playing a more important role in ecology. Ecology produces interesting problems, mathematics provides models and ways to understand them, and ecology returns to test the mathematical models. The function of mathematical ecology is to exploit the natural relationship between ecology and mathematics and is to help predict and interpret what we observe.So far large numbers of ecology mathematical models have been constructed, most of them can be described mathematically by nonlinear parabolic and elliptic partial differential equations. The diffusive Logistic equation is a classical scalar reaction-diffusion equation. It also forms the nucleus of more complex multi-speices models in ecology. This equation and its generalization have been extremely studied and considerable results have been obtained. It should be noted that in all these researches the domains are fixed. A natural question arises that what is the effect on behavior of solution when the domain grows. In recent years growing domains have been an interesting topic for both mathematics and biology. It comes from the research of Turing pattern, which the effects on pattern formation and choice are considered when the organism is growing. In this thesis we try to introduce the growing domain to the population ecology, and study the asymptotic behavior of solution to the diffusive Logistic equation on growing domain.In Chapter1, the background and history about the related work are first introduced and the major work of this thesis is presented.Chapter2is devoted to summarize and prove some classical results of diffusive Logistic equation with Dirichlet boundary condition on fixed domain.In Chapter3, a general reaction-diffusion equation with domain growth is developed in n-dimensional Euclidean space. When the domain growth is isotropic, a diffusive Logistic equation on one dimensional growing domain is presented. In fact, in this case, it is transformed into a new reaction-diffusion equation on a fixed domain.Chapter4deals with the asymptotic analysis of solution to the diffusive Logistic equation with Dirichlet boundary condition on an isotropic growing domain. Two cases of domain growth. i.e. the Logistic growth and the infinite growth, are considered. In the case of Logistic growth, the definition of upper and lower solution and compare principle are first given. Then the global asymptotic behaviors of the new equation are obtained by the method of upper and lower solutions and the classical results. Our results show that the growth of domain takes a positive effect on the asymptotic stability of positive steady-state solution while it takes a negative effect on the asymptotic stability of the trivial solution. In the case of infinite growth, the similar methods are applied to study the asymptotic behavior of solution to the new problem. Our results show that when the domain grows slowly, the species successfully spreads to the whole habitat and stabilizes at a positive steady state, while it dies out in the long run if the domain grows fast.In Chapter5, some numerical simulations about two cases of growing domain are performed to justify the analytical results by software Matlab7.0.We summarize our work in Chapter6and propose some problems which need to make further consideration for future research.