Dynamics Of Reaction Diffusion Models With Spatial Structure And Delay

The dynamical models in the form of reaction-diffusion equations have been extensively investigated in various natural sciences,such as ecology,epidemiology and biology.In order to reflect the real dynamical behaviors of models that de-pend on the past history of systems,it is reasonable to incorporate time delays into the systems.Especially in mathematical biology,many models of population dynamics can be described by the delayed reaction-diffusion equations.Moreover,in a reaction-diffusion model with time-delay effect,the individuals which were lo-cation x at previous times may not be at the same point in space presently.Thus,it is more reasonable to consider the diffusive type model with nonlocal delay.Be-sides,in the natural world,there are many species that go through several stages during their lifetime,such as a single-species growth model with stage structure consisting of immature and mature stages is developed using a discrete time delay.In this dissertation,we mainly investigate the existence and stability of the non-homongence steady state solution.Furthermore,we know that,in the fields of its nonlinear phenomenon,bifurcation is more important,which is related to the phe-nomenon that the entirely different results by the small changes of some objective conditions.The study of bifurcation problems in population models will help us to better understand the effect of parameters(e.g.,survival space and maturation period of species)on the population dynamics(e.g.,any sudden oscillations as parameters change),and hence,to control the species invasion and disease spread.This dissertation is organized as follows:Firstly,we investigate a generalized nonlocal delayed diffusive model under Dirichlet boundary conditions.By using Lyapunov-Schmidt reduction,we obtain the existence and multiplicity of steady state solutions.We investigate the stability of the spatially nonhomogeneous steady state solution,and describe the existence of Hopf bifurcation at the spatially nonhomogeneous steady state solution by an-alyzing the characteristic equation.Furthermore,we investigate the stability and bifurcation direction of Hopf bifurcating periodic orbits by using normal form the-orem and the center manifold theorem.Finally,we illustrate our general results by applications to models with homogeneous kernels and one-dimensional spatial domain.Secondly,we present a detailed analysis on the dynamics of a stage structure model with spatiotemporal delay and homogeneous Dirichlet boundary condition.The existence of steady state solution bifurcating from the trivial equilibrium is obtained by using Lyapunov-Schmidt reduction.The stability analysis of the pos-itive spatially nonhomogeneous steady state solution is investigated by a detailed analysis of the characteristic equation.Using the properties of the omega limit set,we obtain the global convergence of the solution with finite delay.Finally,we concerned with a Lotka-Volterra competition-diffusion model with stage structure and spatial heterogeneity.By analyzing the sign of the principal eigenvalue corresponding to each semi-trivial solution,we obtain the linear stability and global attractivity of the semi-trivial solution.In addition,an attracting region is obtained by means of the method of upper and lower solutions.