Studies On Patch Diffusion Ecological And Epidemic Dynamical Models

Population dynamic models and epidemic models are two classes of important biological dynamical models.They have been applied to study dynamic laws of species evolution or disease propagation in the spatial structure.Using ordinary differential equations and delay differential equations,especially monotone dynamical system theory,uniform persistence theory,matrix theory and graph theory,we,in this dissertation,study certain important biological models.They include a HTLV-I virus model with two time delays in homogeneous space,an epidemic model incorporating media impact in homogeneous space,a multi-patch predator-prey model with dispersal in heterogeneous space,a multi-patch predator-prey diffusion model with age structure in heterogeneous space and two special predator-prey diffusion models in a multi-patch environment.Firstly,we study a HTLV-I virus model with an intracellular delay and an immune delay.We prove stability of the infection-free equilibrium and the immune-free equilibrium using Lyapunov functionals.Our numerical simulations suggest that an increase of the intracellular delay may stabilize the long-term infection equilibrium such that the immune delay can destabilize it.If both delays increase,there exists a region where the long-term infection equilibrium is stable in this region,and is unstable outside the region.Secondly,we analyze an SEI epidemic model with nonlinear incidence rate,and derive dynamic behavior determined by the basic reproduction number;that is,the infectionfree equilibrium is globally asymptotically stable when the basic reproduction number is less than one,if the basic reproduction number is larger than one,there exist one or more endemic equilibria.The system occurs bifurcation with the increase of the basic reproduction number if there exist only one equilibrium.Thirdly,we study a predator-prey model with interdiffusion in a multi-patch environment.By using monotone dynamical system theory and uniform persistence theory,we determine thresholds for the dynamics based on the net reproduction number.More specifically,the predator-free equilibrium is globally attractive when the net reproduction number is less than one,and if the net reproduction number is greater than one,the system is persistent and has at least one coexistence equilibrium.Furthermore,if the prey population does not diffuse,we obtain the upper and lower bounds of the net reproduction number,and analyze the dependence of the net reproduction number on parameters.In addition,the monotonicity of the net reproduction number with respect to the mobility of the predator is obtained in a two-patch environment,and numerical simulations show that the net reproduction number becomes very complicated when the prey and the predators simultaneously diffuse.Fourthly,we consider a predator-prey diffusion model with age structure in a multipatch environment,and obtain thresholds for the dynamics based on the net reproduction number,and find that the net reproduction number is related the growth period of the juvenile predators.The predator-free equilibrium is globally attractive when the net reproduction number is less than one.If the net reproduction number is greater than one,the predator-free equilibrium is unstable,and the system is uniformly persistent and has at least one coexistence equilibrium.In addition,we formulate a the pest and natural enemy interaction model in a two-patch environment,where the natural enemies are linearly released.Using numerical simulations,we show that releasing natural enemies is not necessarily the best choice for pest control.Finally,we analyze the stability of two special predator-prey diffusion models in a multi-patch environment.If the net reproduction number is greater than one,we prove that the coexistence equilibrium is globally asymptotically stable by constructing suitable Lyapunov functions and using matrix tree theorem in graph theory.