Qualitative Research Of Some Biological Population Models

In this paper, we have systematically studied the dynamical behaviors of three kinds of biological population models with diffusion by mainly using the theories of nonlinear analysis and partial differential equations, especially those of reaction-diffusion equations and the corresponding elliptic equations. Some interesting results are obtained, which include existence, uniqueness, multiplic-ity of non-constant positive solutions, stability of positive steady states, global attractor, Turing instability, and so on. The main contents and results are as follows.In chapter1, the biological background of biological population models and some problems which we shall study were introduced. Then we give the main results in this paper.In chapter2, a diffusive one-prey and two-predators model with B-D func-tional response is studied. The main results may be divided three parts. In the first part, by using the fixed-point index theory, we get some sufficient conditions on existence of coexistence states. It turns out that the system has at least one coexistence state when the intrinsic growth rate r of prey is suitable high and the death rate c1or c2of predator remain in some suitable fields. In the second part, by using stability theory and topology degree theory, we show that the model has a unique positive steady solution which is non-degenerate and linear stable when predation rate a1and a2are small enough or when e1and e2are large enough. In the third part, by means of super-sub solutions method, we provide some suf-ficient conditions for the existence of positive global attractor and permanence of the time-dependent system, which show that the system will be permanent if the growth rate of prey r is suitable high and the death rate c1and c2of predators are both suitable low.In chapter3, we study the existence and multiplicity of positive solutions for a spatially heterogeneous cooperative system with cross-diffusion. The main results may be divided two parts. In the first part, we first get some necessary conditions on existence of positive solutions. The result shows that the nonexis-tence region of positive solutions in our paper is smaller than that obtained by Y-X Wang and W-T Li(J. Differential Equations251(2011)1670-1695) due to the different diffusion tragedy one of the cooperative species adopt. Then, regard-ing a as the bifurcation parameter, we obtain the global structure of the positive solutions bifurcating from one of the semi-trivial equilibria (a,0) and (0, b) by using local and global bifurcation theory. It turns out that, in either case of b>0or b<0, we can get a corresponding constant a*or a*such that there is at least one positive solutions of the system if a> a*or a> a*respectively. In the second part, the multiple existence of positive solutions to this system are determined by Lyapunov-Schmidt procedure and perturbation technique. The result shows that, if the parameter b>0is small enough, k is large enough and fΩ c(x)dxfΩ ρ(x)dx<fΩc(x)ρ(x)dx, then the system admits a roughly (?)-shaped smooth curve of coexistence solutions with respect to bifurcation parameter a, which implies that the system has at least one or two positive solutions when pa-rameter a remain in some suitable fields. If the parameter b<0is small enough, k is large enough, then the system also admits a roughly (?)-shaped smooth curve of coexistence solutions with respect to bifurcation parameter a when c(x) is large enough and d(x) is small enough.In chapter3, we study the effect of diffusion and cross-diffusion in a predator-prey model with a transmissible disease in the predator species. The main results may be divided two parts. In the first part, we first show that both self-diffusion and cross-diffusion can not cause Turing instability from the disease-free equilibria u0*. Then we find that the endemic equilibrium u*remains linearly stable for the reaction-diffusion system without cross-diffusion, while it becomes linearly unsta- ble when cross-diffusion also plays a role in the reaction-diffusion system. Hence we know that the instability is driven solely from the effect of cross-diffusion. In the second part, we further derive some results for the existence and non-existence of non-constant stationary solutions by using energy integral method and degree theory.