Stability Analysis For Multi-species Strongly Coupled Population Dynamical Models

In mathematical ecology, the coexistence and the long time behavior of various species in an ecosystem are the chief contents of population dynamical models. With the rapid development of researches, more and more attention is recently concentrated on the strongly coupled reaction-diffusion equations with diffusion, self-diffusions and cross-diffusions. In this doctoral dissertation, we mainly discuss a representational class of strongly coupled nonlinear reaction-diffusion systems arising in population dynamics. By investigating the global behavior and the existence of non-constant positive steady states, we get a series of new and meaningful results. This thesis consists of four chapters.In Chapter 1, the background, historical development, advances in research and present situation and prospects of mathematical ecology and population dynamical models are introduced. The problems considered in this paper and the main ideas are presented.In Chapter 2, the preliminaries used in the main body such as some important lemmas and propositions are listed.In Chapter 3, we discuss the global solutions for a class of strongly coupled reaction-diffusion systems with nonlinear dissipative terms. Such system is n species SKT model including the effects of diffusions, self-diffusions as well as cross-diffusions. Based on the local existence of the nonnegative solutions from H. Amann, we consider the global existence and boundedness of the solutions by employing the standard technique of energy estimates and a few well-chosen Gagliardo-Nirenberg interpolation inequalities. If the diffusion matrix and competition matrix are all positive definite, the nonnegative solution for the system can exist globally and be uniformly bounded. Moreover, the non-existence of non-constant positive steady state is also obtained by constructing an appropriate Lyapunov function.In Chapter 4, the strongly coupled HP food chain model with three interacting species is investigated. When zero population flux across the boundary, by using the Leray-Schauder degree theory, we obtain the existence of non-constant positive steady states for this model. In addition, the stability of constant positive equilibrium point is discussed by elementary concept and methods in population dynamics. The results indicate that for strongly coupled system, when the effect of diffusion or self-diffusion of the second species is strong, there is no non-constant positive steady state for the system, whereas there is at least one non-constant positive steady state if cross-diffusion of the second species due to the first species or the third species due to the second species is large enough, and there is at least one non-constant positive steady state for weakly coupled system when diffusion of the third species is strong enough. Therefore, the emergence of stationary patterns is due to strong diffusion or cross-diffusions.In Chapter 5, we mainly discuss the golbal bifurcation of the steady state problem for the strongly coupled competitor-competitor-mutualist model under zero boundary conditions. The tools used in this chapter are the local and global bifurcation theorems of Rabinowitz. Regarding the intrinsic rate of natural increase of the first species as bifurcation parameter, we can obtain the positive solution branch of the steady state system which bifurcates from the semitrivial solution set the first component of which is zero by Rabinowitz's local bifurcation theorem. Further, Rabinowitz's global bifurcation theorem of positive solution shows that the local bifurcation branch is global. Therefore, the competitor-competitor-mutualist cross-diffusions system possess at least one coexistence if the intrinsic increasing rates of the three species fulfill proper conditions and the principal eigenvalue of an eigenvalue problem with nonlinear diffusion term is 0 and the algebra multiplicity of it is odd.