Dynamics Of Reaction-diffusion Models With Chemotaxis Effect And Cross-diffusion

In the wake of applications of reaction diffusion system in ecology,researchers find that in addition to random dispersion,species could move upward along the re-source gradient,or be forced by external environment such as wind,water flow and so on,which usually can be described by an advection term.Based on the Lotka-Volterra system,in this dissertation we shall investigate several classes of reaction diffusion advection systems in detail by means of Leray-Schauder degree theory,maximum principle,Gagliardo-Nirenberg inequality,Lyapunov-Schmidt reduction,local bifurcation theory,and so on.The main content is organized as follows:Firstly,we focus on a stationary Leslie-Gower model with diffusion and ad-vection under homogeneous Neumann boundary conditions.Some existence con-ditions of nonconstant positive solutions are obtained by means of the Leray-Schauder degree theory.As diffusion and advection of one of the species both tend to infinity,we obtain a limiting system,which is a semi-linear elliptic equa-tion with nonlocal constraint.In the simplified 1D case,the global bifurcation structure of nonconstant solutions of the limiting system is classified.Secondly,we consider the following two-species chemotaxis system(?)under homogeneous Neumann boundary conditions.The parameters in the sys-tem are positive and the signal production function h is a prescribed C~1-regular function.The main objectives of this paper are two-fold:One is the existence and boundedness of global solutions,the other is the large time behavior of the global bounded solutions in two competition cases(i.e.,a weak competition case and a partially strong competition case).It is shown that the unique positive spatial-ly homogeneous equilibrium may be globally attractive in the weak competition case,while the constant stationary solution may be globally attractive and globally stable in the partially strong competition case.Finally,we consider a generalized predator-prey system with prey-taxis under Neumann boundary conditions,in which the predators can survive even in the absence of the prey species.It is proved that for arbitrary spatial dimension,the corresponding initial boundary value problem possesses a unique global bounded classical solution when the prey-taxis is restricted to a small range.Moreover,the local stabilities of constant steady states(including trivial,semi-trivial and positive constant steady states)are investigated.A further study on the co-existence steady state implies that the prey-taxis term suppresses the global asymptotical stability and influence the steady-state/Hopf bifurcations(if they exist).Analyses of steady-state bifurcation,Hopf bifurcation,and even Hopf/steady-state mode interaction are carried out in detail by means of Lyapunov-Schmidt procedure.In particular,we obtain stable or unstable steady states,time-periodic solutions,quasi-periodic solutions,and sphere-like surfaces of solutions.These results provide theoretical evidences to the complex spatio-temporal dynamics found by numerical simulation.