Behaviors Of Solutions To A Parabolic System Describing Two Populations In A Cooperating Model With Nonlocal Sources

Mathematical Ecology has been one of the most well recognized subjects in modern applied mathematics. 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.This dissertation is devoted to nonlinear partial differential systems established in ecology. A lot of effort has been devoted to the study of the two—species Lotka-Volterra models. The properties of solutions to the corresponding partial differential equations are well-known. But few concerns non-local problems. A two-species Lotka-Volterra model with non-local source of weakly coupled reaction-diffusion systems is studied in this dissertation. We will consider the influence of non-local source on the properties of solutions and show how the non-local source affects the permanence and blowup of the species.In part 1, the background and history about the related work are given.In Part 2, we give some preliminary materials that are necessary for our study presented in this dissertation.In parts 3 and 4, the global existence and blowup results of solutions are given using upper and lower solutions, comparison principle and related results and techniques from ordinary differential equations. It is shown that global solutions exist if the intra-specific competitions are strong, whereas blowup solutions may exist if the intra-specific competitions are weak.In part 5, we first give the relationship between the maximums of components of the solution near the blowup time. Then the upper bound estimates for blowup rates are given by using integral representation formula.