Dynamics Of Stochastic Dispersal Equations And Nonlocal Dispersal Equations

In the study of mathematical biology,population dynamics has become an important branch,of which the study of population models with either stochastic diffusion or nonlocal diffusion has attracted much attention and has been investigated by many mathematicians and biologists.In addition,in order to reflect the real dynamical behaviors of models that relay on the past history of systems,it is reasonable to bring time delays into the systems.In particular,in mathematical biology,there are a lot of population dynamics models described by reactiondiffusion equations with time delays.Due to the differences in the energy during the transmission and transformation of individual biological,the investigation of the dynamic properties of predator-prey systems with different functional response functions and diffusion possesses a strong theoretical and practical significance,such as the existence and stability of the steady state solutions,and delay-induced Hopf bifurcation phenomena.In nonlinear sciences,the study of bifurcation problems becomes more and more important,since it can reflect the fact that the subtle changes of one or some factors in the real world can cause a huge impact on the change of material.In the study of population dynamics,bifurcation problem can help us understand that the change of some of the parameters(such as living space and the maturation period)causes the change of the population dynamics(such as the stability or oscillation of stationary states).The regulation of these data can make the species develop towards the expected direction.This dissertation is devoted to the study of the existence,stability and bifurcation of the spatially nonhomogeneous steady state solutions.The main content of this dissertation is organized as follows:Firstly,we investigate a generalized delayed diffusive model subject to Dirichlet boundary conditions.By applying Lyapunov–Schmidt reduction,we obtain the existence and multiplicity of steady state solutions.We also 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 analyzing the characteristic equation.Moreover,we investigate the stability and bifurcation direction of Hopf bifurcating periodic orbits by applying the normal form theorem and the center manifold theorem.Finally,we illustrate our general results by applications to the Nicholson blowflies equation in one-dimensional spatial domain.In particular,we obtain some other results of the Nicholson blowflies equation by our detailed analysis.Secondly,we present a detailed analysis on the dynamics of the 2-dimensional delayed diffusive Lotka-Volterra equation with general functional response function subject to a homogeneous Dirichlet boundary condition.By applying Lyapunov–Schmidt reduction,we obtain the existence of the nonhomogeneous steady states under different parameters conditions.In addition,we also derive the existence and the stability of the spatially nonhomogeneous steady state solutions bifurcating from these nonhomogeneous steady state solutions,and describe the existence of Hopf bifurcation at these spatially nonhomogeneous steady state solution.Finally,we illustrate our general results by applications to models with a single delay and one-dimensional spatial domain with different functional response functions.Finally,we are concerned with a nonlocal dispersal logistic model with nonlocal reaction term.By using the method of upper and lower solutions,fixed point theory,Lyapunov–Schmidt reduction and bifurcation theory,we derive the existence,multiplicity and stability of the steady states under the different parameter conditions.