Some Types Of Ecological Models With Free Boundary

Using the initial boundary value problem of fixed boundary to study biological invasion often has great defects,and the free boundary problem widely appears in various fields of natural science and engineering technology.This paper first introduces the complete theory and research method of free boundary problem by studying two predator-prey models;Then these theories and research methods are applied to simulate the mosquito swarm carrying W olbachia spreading in a given area.By an-alyzing the solutions of the corresponding model,appropriate release strategies are determined to provide theoretical support for workers in practical departments.In Chapter 2,we consider a Leslie-Gower predator-prey model in one-dimensional environment.We study the asymptotic behavior of two species evolving in a domain with a free boundary.Sufficient conditions for spreading success and spreading failure are obtained.We also derive sharp criteria for spreading and vanishing of the two species.Finally,when spreading is successful,we show that the spreading speed is between the minimal speed of traveling wavefront solutions for the predator-prey model on the whole real line（without a free boundary）and an elliptic problem that follows from the original model.In Chapter 3,we study a predator-prey system with free boundary in a one-dimensional environment.The predator v is the invader which exists initially in a subinterval[0,s₀]of [0,L]and has the Leslie-Gower terms that measure the loss in the predator population due to rarity of the prey.The prey u（the native species）is initially distributed over the whole region[0,L].Our primary goal is to understand how the success or failure of the predator’s invasion is affected by the initial datum v₀.We derive a spreading-vanishing dichotomy and give sharp criteria for spreading and vanishing in this model.Scientists have been seeking ways for many years to use Wolbachia to eliminate the mosquitoes that spread human diseases.Could Wolbachia be the determining factor in controlling the mosquito-borne infectious diseases?To answer this question mathematically,In chapter 4,we develop a reaction-diffusion model with free boundary in one-dimensional environment.We divide the mosquito population into two groups:one is the uninfected mosquito population that grows in the whole region while the other is the mosquito population infected with Wolbachia that occupies a finite small region and invades the environment with a spreading front governed by a free boundary satisfying the well-known one-phase Stefan condition.For the resulting free boundary problem,we establish criteria for spreading and vanishing.Our results provide useful insights on designing feasible mosquito releasing strategy to invade the whole mosquito population with Wolbachia infection and thus eventually eradicate the mosquito-borne diseases.Releasing mosquitoes with Wolbachia into the wild mosquito population is becoming the very promising strategy to control mosquito-borne infections.To investigate the effects of wind and critical patch size on the Wolbachia establishment in the wild mosquito population,in Chapter 5,we propose a diffusion-reaction-advection system in a heterogeneous environment.By studying the related eigenvalue problems,we derive various conditions under which Wolbachia can fully establish in the entire wild mosquito population.Our findings may provide some useful insights on designing practical releasing strategies to control the mosquito population.The summary of this dissertation and the outlook for the further research work are stated in Chapter 6.