Qualitative Analysis Of Two Types Of Ecological Models

The dynamics of biological models has received intensive studies, and it has been an important aspect in the field of non-linear partial differential equations, and has gained many important and useful results. Depending on these methods and theories, the papers studies two kinds of biological models: one is a cooperative system with saturation, the other is a predator-prey system with non-monotonic functional response.The main contents and results in this paper are as follows:In section 1, we study a cooperative system (1).whereΩis a bounded domain in R~N(N≥1) with smooth boundary (?)Ω,△stands for the Laplace operator in R~N, u and v represent the densities of two species, a, b, c, d,γare positive constants,α,β, m, n each is more than the positive number 1,γmeasures the saturation level of the species.With the application of using Maximum principles,lower-upper solutions methodsand fixed points index of compact maps in cones, sufficient conditions of positive solutions for a cooperative system with saturation are obtained.In section 2, we study a predator-prey system (2).where u and v are the birth rate and death rate for the prey and predator respectively,a, c, d are positive constants,γis the half-saturation constant for the predator, b≥0, if and only if b > 0, it shows that predator has other sources of food. This chapter can be divided into two parts. First, we investigate the relationshipbetween equilibrium points and positive solutions over a large domain, and we obtain the uniqueness and the stability of positive solutions for the model with the assumption that the encounter rate with prey by a searching predator (the capturing rate)is sufficiently small. Second, the global structure of the coexistence solutions and their local stability are established by using bifurcation theory.