Dynamics Of Reaction-Diffusion Models With Spatial Structure And Heterogeneity

Reaction-diffusion models play a crucial role in the development of epidemiology and biological population dynamics.With the deepening of research,scholars have found that the migration of species and the spread of microorganisms in space may not only be random diffusion,but also directional movement due to external environmental forces such as water currents,ocean currents,quicksand and wind direction.In addition,species and microorganisms may also have a certain ability to discriminate some external signals,such as spouse,food,smells or some chemicals,and thus produce chemotactic movement.For this reason,researchers add advection term or chemotaxis term to the classical reaction-diffusion equation to improve the practical significance of the mathematical model.In this paper,the local and global dynamics of the reaction-diffusion model with different spatial structures(i.e.,spatial heterogeneity,time delay effect,advection effect and chemotaxis effect)are investigated by using the implicit function theorem,Lyapunov-Schmidt reduction method,central manifold theorem,normal form theory,local bifurcation theory,Banach’s fixed point theorem,parabolic regularity theory,a priori estimation,Moser iteration,Lyapunov functional method and La Salle’s invariance principle.The specific research content is divided into the following parts:Firstly,an SIS epidemic system with time delay and external recruitment rate in heterogeneous environment is investigated.The principal eigenvalues of the weighted eigenvalue problem is explored to obtain the stability of the diseasefree equilibrium and the influence of environmental heterogeneity on the stability region.It is observed that the time delay has no essential effect on the stability of disease-free equilibrium.When the disease-free equilibrium point is unstable,the existence interval and stability conclusion of endemic equilibrium are obtained,and the influence of environmental heterogeneity and delay τ is described as well.Regarding the time delay τ as a bifurcation parameter,we obtain the condition of Hopf bifurcation near the endemic equilibrium and obtain the bifurcation direction and stability of the bifurcating periodic solution by applying the central manifold theorem and normal form theory.Secondly,we consider a classical Lotka-Volterra type competition-diffusion system with advection terms under the homogeneous Dirichlet boundary condition.Notice that an elliptic operator with advection term is not self-adjoint,which causes some trouble in the spatial decomposition,explicit expressions of steady-state solutions and some deductive processes related to infinitesimal generators.Moreover,unlike other work,the advection rate here depends on the spatial position,which increases some difficulties in the investigation of the principal eigenvalue.We classify the intrinsic growth rate parameters of the two competing species as exhaustively as possible,and then obtain the existence and multiplicity of the spatially nonhomogeneous steady-state solutions by using implicit function theorem and Lyapunov-Schmidt reduction method.The asymptotic behavior and stability of the spatially nonhomogeneous steady-state solutions are obtained by analyzing the spectral distribution of the infinitesimal generators of the linearized system at the steady-state solutions.It is observed that the Hopf bifurcation does not occur.In particular,two specific examples are proved to verify the validity of our theoretical analysis.Finally,we consider a Lotka-Volterra type two-species competition-chemotaxis system affected by different phenomena under Neumann boundary conditions.The local existence of the classical solution is established by means of the Banach’s fixed-point theorem and the parabolic regularity theory,and the regularity is improved in three steps to obtain the global boundedness of the classical solution by using a priori estimation and the Moser iteration method.Then by virtue of the Lyapunov functional method and La Salle’s invariant principle,the global stability and asymptotic behavior of the classical solution are investigated in the weak competition situation and the asymmetric competition situation,respectively.Furthermore,we not only obtain the exponential convergence of the classical solution in the case of weak competition,but also the exponential/algebraic convergence in the case of asymmetric competition.It is observed that the existence of LotkaVolterra kinetics makes the classical solution not blow-up no matter how strong the chemotaxis is.Moreover,different from the usual hypothesis in literature,we no longer need all the chemotaxis intensities to be so small in order to ensure the global stability of the classical solutions.