Mathematical Study On The Dynamical Behavior Of The Lotka-Volterra Competition System With Signal-dependent Diffusion

Under the Neumann boundary conditions,we consider two types of Lotka-Volterra competition systems with density-dependent dispersion in a 2-dimensional and 3-dimensional bounded domain.The presence of cross diffusion terms due to density-dependent diffu-sion,which results in the compare principle no longer applicable,is one of the main challenges in the analysis of two systems.In this paper,we use the weighted energy estimates and the damping effect of Lotka Volterra competition term to overcome the d-ificulty and establish the boundedness of solutions.Due to complexity of parameters and non-linear terms in the models,the analyses of the asymptotic behavior of solutions are much more technical.By constructing appropriate Lyapunov functionals,we study the large time behavior of solutions and explain the biological significance of results.More-over,our conclusion is different from " slower diffuser always wipes out faster diffuser".Precisely,1.For the Lotka-Volterra competition system with homogeneous resources,we use the weighted energy estimates to derive the lower order estimates of solutions solving the system,and utilize the Moser iteration method and parabolic regularity to show the global existence and uniform boundedness of solutions in a two-dimensional bounded domain.In three dimensional space,we prove that the global classical solutions exist with uniform-in time bound under some assumptions of diffusion functions.2.For the Lotka-Volterra competition system with dynamical resources,we make full use of the damping effect of the Lotka-Volterra competition term to overcome the difficulties caused by cross diffusion terms and obtain the L2-norm of the density of two species in a two-dimensional bounded domain.Furthermore,we establish the global existence and uniform boundedness of solutions to the system based on energy estimates and parabolic regularity.With some smallness conditions on diffusion,we prove that the system possesses global classical solutions in three dimensions.3.With the global existence and uniform boundedness of solutions,we discuss three scenarios in which two species eventually become coexistent or extinct in different parameter regimes by constructing appropriate Lyapunov functionals and using LaSalle's invariant principle.Our results show that if the resource has temporal dynamics,two species may coexist regardless of their diffusion coefficients and initial values.This is a very different scenario from "slower diffuser always wipes out faster diffuser" for given spatially heterogeneous resource.This paper is mainly divided into four chapters.In Chapter 1,we mainly introduce the background and research development related to the Lotka-Volterra competition system,and give a brief description of the main results of this paper.Chapter 2 is about the Lotka-Volterra competition system with homogeneous resources and signal-dependent diffusion.We study the global existence and large time behavior of solutions to the com-petition system.For the Lotka Volterra competitive system with dynamical resources,we prove the existence of the global solution of the system in Chapter 3.Moreover,we analyze the dynamic behavior of the system and explain the biological significance of the results.Chapter 4 summarizes the results of this dissertation and raises issues for further discussion.