The Survival Analysis For The Predator-prey Diffusive System With The Functional Response In A Polluted Environment

By applying integral in mean, comparison theorem and construction of auxiliary function, this paper studys how diffusion affect the predator-prey system with functional response in polluted patch. This paper is divided into two parts, the main content can be described as follows:In the first part, this article studies survival analysis of autonomous predator-prey systems with the Holling II functional response in polluted patch. Under the premise of that the toxicant input has a limit, firstly, we prove that the certain populations of the system are bounded above by using integral in mean. Secondly, we prove the certain populations of the system are bounded below and get sufficient conditions of permanent living of populations by applying comparison theorem. finally, applying constructing a suitable auxiliary function, some sufficient conditions which the population will go to extinction are obtained. when the toxicant input is bounded, boundedness of toxin density in population in patch one and in the environment of patch one are proved, the populations are bounded above and limit inferior of the populations are bounded below by comparison theorem and constructing a suitable auxiliary function, some sufficient conditions which the population will be uniformly persistent or go to extinction are obtained.In the second part, this paper studies survival analysis of nonautonomous predator-prey systems with the Michaelis-menten functional response in polluted patch. Take time varying coefficients of system are strictly bounded into account, by scaling on system, we translate nonautonomous system into autonomous system. when the toxicant input in the polluted patch has a limit, some sufficient conditions which the population will be permanent or go to extinction are obtained by comparison theorem and constructing a suitable auxiliary function. when the toxicant input in the polluted patch is bounded, some sufficient conditions which the population will be uniformly persistent or go to extinction are obtained by comparison theorem and constructing a suitable auxiliary function.When the pollution and diffusion existing at the same time, the research on the predator-prey systems with the functional response are studied. It was discovered that we can achieve the purpose of saving endangered species at last by controlling the diffusivity between the two patches, it has both important and realistic significance.