Dynamics Analysis Of Several Classes Of Temporally Discrete Fixed And Free Boundary Problems

Discrete time biological modeling is a natural choice in many cases,and the dynamics of discrete models are often richer.A number of time-discrete spatially continuous diffusion equations provide insights into population ecology,and integrodifference equations are an important class of these equations,but their application is limited by their inability to account for boundaries.Another class of time-discrete spatial diffusion equations has a good performance,which not only can accurately describe the diffusion phenomenon,but also can be transformed into equivalent integrodifference equations,obtaining rich analytical methods.In recent years,the free boundary problem has received much attention because it can reflect the spatial and temporal evolution of individuals more realistically,but its current research is mainly based on continuous time equations.In this paper,we consider the free boundary problem based on a single discrete equation and a discrete competing system,and give some conclusions on spreading-vanishing dichotomy and the existence uniqueness of the semi-wave solution,which provide new ideas for the related research.In addition,current work on integrodifference equations focuses on traveling wave solution theory.In this paper,we discuss the global dynamics of an integrodifference competition model over a bounded domain and apply it to the ecology of Wolbachia population invasions to help in decision making.The paper consists of five chapters,as follows:Chapter 1 introduces the research background and progress of this thesis,as well as the main contents.In Chapter 2,a temporally discrete Stefan-type free boundary problem with a single relatively general form equation is formulated with the aim of giving a basic theoretical framework for the long-time behavior of the associated problem.To this end,the regularity theory and global dynamics of the problem on a fixed bounded domain are first obtained by discussion.On this basis,we consider the free boundary problem,giving well-posedness of the solutions and a modified comparison principle.Then the spreading-vanishing dichotomy result is proved by some estimation and upper and lower solution methods,and some sufficient conditions for judging the dichotomy are given.Finally,we prove the uniqueness and monotonicity of the solution of the corresponding semi-wave problem when spreading occurs,which provides an important foundation for the subsequent study of spreading speed.In Chapter 3,we introduce Stefan’s condition on a discrete-time diffusive competitive system to formulate a free boundary problem in which a new population invades and competes with a native species.Extending the approach of Chapter 2 to the system,we prove well-posedness of the solutions and a modified comparison principle of this competitive free boundary problem,and give the long time behavior of the solutions.The results show that when the invasive species is at a competitive disadvantage,it must fail to invade.When the invasive species is the superior competitor,the dichotomous conclusion indicates that it may succeed in spreading or fail to invade due to factors such as insufficient expansion capacity,which fits the actual situation.Chapter 4 analyzes the invasion dynamics of Wolbachia populations from a competitive point of view.Based on a competitive difference system with CI effect,a Laplace kernel function is introduced to build a competitive integrodifference system and analyzed to give the conditions for successful invasion of Wolbachia infected populations.The average dispersal success approximation is applied to make the equations precisely analyzable.Finally,the influence of spatial factors on Wolbachia invasion is illustrated by comparing the dynamics of non-spatial models.The final chapter summarizes the entire text and provides an outlook for future work.