Study On Dynamic Behavior Of Population Models With Road Diffusion And Nonlocal Interaction

In recent years,the problem of biological invasion has attracted much attention This is mainly because the invasion of alien organisms not only affect the balance of the local ecosystem;and what's more serious is that invasive species may bring infectious diseases and endanger human health.Therefore,it is necessary to theoretically analyze the dynamic mechanism of alien species spreading along the road.In this paper,we establish a mathematical model based on the reaction-diffusion equation,consider nonlocal diffusion and road diffusion,and make analyses by using the relevant theories and methods of partial differential equation.In Chapter 1,we introduce the background knowledge of biological invasion and the research status and development of this subject,and describe two kinds of models studied in this paper.The first model is the mosquito dynamics model with River-Field reaction-diffusion equations with nonlocal term,and the second is the two species Lotka-Volterra competition dynamics model with "road" diffusion and nonlocal effect.In Chapter 2,we mainly study the first kind of single species dynamic model By establishing a new comparison principle and constructing the upper and lower solutions,we prove the existence and uniqueness of the solution.We give a sufficient condition for the global asymptotic stability of the positive equilibrium solution.Finally,we illustrate the dynamic behavior of the initial diffusion by numerical simulation.In Chapter 3,we mainly study the second kind of two species dynamic model First,we define a new set of upper and lower solutions and reconstruct them.Then,we obtain the comparison principle of the equation with convolution term by using the counter evidence and the strong maximum principle.Then,we construct the monotone iterative sequence by using the comparison principle,and introduce the Green function to express the solution to prove the existence and uniqueness of the solution and give its upper bound.Furthermore,we discuss the effects of three competition relations:weak-strong competition,strong-weak competition and weak-weak competition on the dynamic behavior.Finally,the asymptotic diffusion velocity is given to better understand the influence of road diffusion and nonlocal effect on the propagation mechanism of populationIn the conclusion part,we summarize the research results of this paper,and point out that there are still some problems to be further studied.