Qualitative Analysis Of Some Reaction-diffusion Systems

Understanding of mechanisms and patterns of spatial dispersal of interacting species is a central problem in biology,ecology and biochemical reactions.This dissertation con-cerns the qualitative properties of some reaction-diffusion systems under different back-grounds.Firstly,we introduce the research background and status of the problems and the main contents of this dissertation.Secondly,we investigate a diffusive competition model with homogeneous Dirichlet boundary conditions and functional response is in the form f（u,v）=1/（1+αu）（1+βv）.The trivial and semi-trivial solutions include（0,0）,（θ_a,0）and（0,θ_c）.The positive coexistence states are more partical interest.For arbitrarily fixed parameters,we obtain the existence results of positive solution of propose model.For any parameters a,c>λ₁,we can choose α or β suitably large such that the two species competition model has at least one coexistence state which is linear stable.When a<λ₁（bθ_c/1+βθ_c）and α>>1,or c<λ₁（dθ_a/1+αθq）and β>>1,we obtain at least two coexistence states of the system.Moreover,we investigate the bifurcation occours when a coexistence state emanates from semi-trivial solutions.In the third part,we investigate predator-prey interaction with Holling type-Ⅱ func-tional responses.In this part,we assumed a no-flux boundary condition for both species to predator and prey species live in a closed ecosystem;the boundary of the protection zone does not affect the dispersal of prey,but it works as a barrier to block the predator from entering.We give the a priori estimates and obtain the nonexistence of non-constant positive solution as the diffusion coefficients are large enough.Moreover,we demonstrate the existence and stability of non-constant steady-state solutions branching from constant semi-trivial solutions.The fourth part of this dissertation explores the indirect effects of predators and prey.Without being directly killed by predators,prey reproduction success is largely reduced by avoidance behaviors.We propose a model that incorporates the impact of fear effect in prey reproduction and study dynamical properties of time-dependent solutions and the stationary patterns induced by diffusion（Turning patterns）.In the fifth part,this chapter aims to provide another insight into the minimal Keller-Segel model with logistic damping.We apply a delicate method to show the known con-clusion:The weak chemotactic effect can ensure the global existence and boundedness of the solutions of the minimal Keller-Segel model with logistic growth in any dimension-al cases.Moreover,we obtain the explicit uniform in time upper bound for the global solution.Finally,we deal with one kind of Belousov-Zhabotinskii reaction model.Linear stability discusses for the spatially homogeneous problem firstly.Then we focus on the stationary problem with diffusion.Non-existence and existence of non-constant positive equilibrium solutions obtained by using implicit function theorem and Leray-Schauder degree theory,respectively.