Traveling Wave Solutions Of Predator And Competition Models And Their Propagation Speed

In this dissertation,three diffusion-reaction models are established,which are Lotka–Volterra type two predators-one prey model,predator-prey model with nonlocal delayed effects and two competing species model under a changing climate.We investigate the existence and nonexistence of traveling wave solutions,determine the minimal speed.These results can provide a theoretical strategy for biological control.The main results in this thesis are as follows:In Chapter 1,we describe the background of models,mathematical methods and the history and present conditions of the subject both at home and abroad.In Chapter 2,based on Competition Exclusion Principle,we consider a model with one basal resource and two species of predators feeding by the same resource.Unlike the most other traveling wave solutions connecting zero equilibrium and positive equilibrium,or two equilibria that admit a partial ordering,the three non-trivial boundary equilibria do not have an ordering in common sense.So either the method of upper–lower solutions or monotone iteration is invalid.Using an improved high-dimensional shooting method,the Wazewski'principle,Lyapunov function and LaSalle invariant principle,we establish the conditions for the existence of traveling wave solutions(meaning successful invasion),and determine the minimal wave speed.In Chapter 3,a predator-prey model with general functional response and nonlocal delay effects is proposed.As is well known,the time delay and spatial diffusion is ubiquitous in nature.In recent years,many investigators began to the combined effects of both time delay and spatial diffusion on the dynamical behaviors of differential equations,where the nonlinearities involving a spatial averaging the whole of the infinite spatial domain and the whole of the previous time.The study leads to a new class of infinite dimensional dynamical system: reaction-diffusion equations with nonlocal delays.In contrast with(delayed)reaction-diffusion equations,these systems can reflect the reality more accurately.On the other hand,the time delay and spatial non-locality also bring many mathematical difficulties and complicated dynamics.Therefore,it is interesting and valuable both in theory and practice to study such kinds of the systems.For a predator-prey system with general functional response and nonlocal delay effects,it is difficult to compute the characteristic roots and to construct a proper Lyapunov function due to the nonlocal delay term and the general functional response.The existence of and persistence of the weak traveling wave solutions is established based on the methods of upper-lower solutions as follows:(1)First,using the minimal positive characteristic roots of linearized system to construct a pair of appropriate upper-lower solutions,a closed and convex invariant set is thus obtained;(2)Transforming system into an integral operator equation,checking the invariance of the integral operator equation;(3)Using Schauder fixed point theorem to get the fixed point of the operator equation,that is the traveling wave solution of the original system;(4)Applying negative onesided Laplace transform to prove the nonexistence of the traveling wave solution,and minimal wave speed is derived by combining(3).Finally,the persistence of the weak traveling wave solution is obtained by using the persistence theory of Hale and Waltmann.The highlight of this chapter is synthesizing nonlocal delay effects and general functional response,the model is more closer to the reality,and the results are more general.In Chapter 4,we considers effects of a climate-induced range shift on outcomes of two competitive species,which is modeled by a reaction-diffusion system with the increasing growth rates of species along a shifting habitat gradient.Analytical conditions are established for the coexistence or competitive exclusion of two-competitors under the climate change,which present the control strategies to maintain the persistence of species.Global climate change is altering environmental suitability across species' s ranges,and an increasing appreciation that ecosystem response to climate change are complex and widespread has promoted a focus on understanding and predicting biological impacts(either positive or negative effects).Some of the most negative impacts are the loss of diversity-including genetic,species,and functional-that accompany extinctions.Whether populations and species will persist at the local and global scale,respectively,depends on their abilities to endure future climate shifts.That is,persistence then requires that a species keep pace with the movement of its habitat band.Modeling and exploring the issue of species spread in the context of climate change can benefit in theoretically and empirically.And also,an understanding of spreading speeds can provide insight into invasion processes.By applying analytical approaches,we have determined the critical invasion speed for each species,and analyze the issues of coexistence,competition exclusion,and extinction.Finally,we provide a number of simulations to illustrate the results.Innovations of this thesis are as follows:1.The first innovation consists of two aspects: first,the model we consider involves three species.Then it is more difficult to construct a variant of Wazewski set,because five equations would be involved and the geometry in R5 is more complicated than in R3.As far as we know,our paper seems to be the first try.Second,we will identify conditions under which the model has a traveling wave solution,which connects the equilibrium where prey and the first predator(less fit species)are present(i.e.,E1)to another equilibrium at which the first predator extinct while the prey and the second predator(superior species)coexist(i.e.,E2).2.The second innovation in chapter 3 is to incorporate spatial effects by deriving,in one spatial dimensional,a stage-structured diffusive predator-prey model in which the delay term is rigorously derived by solving a von Foerster equation.The before kind of models either simply added a diffusion term,or simply included a Holling I type functional response.So,our model is more reasonable and more general.3.The third innovation is the novelty of the model.The study of invasion dynamics in the context of climate change is in its infancy.Spatial heterogeneity has also been considered for models in time almost periodic and space periodic media.But in our model,the population growth rate is defined as r(x-ct),it involves not only spatial location and time,but also the speed c at which the edge of the habitat suitable for species growth is shifting.