The Effects Of Carrying Capacity And Intrinsic Growth Rate On Species Dynamics Behaviour

In recent decades,it has been an emerging trend and also a significant tool in the study of population dynamics to use partial differential equation(PDE)models,especially reaction-diffusion equations and systems.By developing a series of PDE models,studying and comparing their qualitative properties and long time behaviors,we can get a deeper understandings of some fundamental questions in ecology and the underlying mechanism.For instance,in a diffusive logistic model in a spatially heterogeneous environment,the joint effects of carrying capacity–K(x),intrinsic growth rate–r(x) and the random dispersal of the species could produce some interesting,and sometimes,astonishing phenomena.Thus,the main goal of this thesis is to study the qualitative properties of logistic equation and the long time behaviors of Lotka-Volterra competition diffusion systems.Generally speaking,the total biomass of a population is an important indicator to measure its long-term viability and competitiveness.Therefore,we first consider the properties of equilibrium solutions of the logistic model under different relations between r(x) and K(x).Our result shows that the total population at equilibrium is strictly smaller than the total carrying capacity when r(x) is a constant.However,when r(x) is proportional to K(x),it is proved by Lou[53]that the total population is always strictly greater than the total carrying capacity,which is in drastic contrast to the case when r(x) is a constant.Therefore,the relations between r(x) and K(x) have significant impacts on the qualitative properties of the logistic model in heterogeneous environments.Furthermore,we consider the dynamics of the corresponding Lotka-Volterra competition-diffusion systems in the case r(x)?const.Assuming that the resource distributions of the two species are different,and the competition abilities and the amount of total resources are the same,we show that the species with heterogenous distribution of resources no longer has a competition advantage in terms of the two species' diffusion rates.This result is also quite different from the case r(x)?K(x).Finally,we consider the global dynamics of the general Lotka-Volterra competitiondiffusion systems in heterogenous environments,and obtain its global dynamics under a general criteria.This thesis consists of the following six chapters.In Chapter 1,we briefly introduce the background and some current results on the study of the logistic model and the Lotka-Volterra competition-diffusion systems.In Chapter 2,we establish some preliminary results which will be used in the remainder of this thesis,including Banach ordered space,theory of monotone dynamical systems,and the theory of principal eigenvalue of elliptic eigenvalue problems.In Chapter 3,we first introduce some properties of the steady-state solution of the logistic model.Then we consider qualitative properties of the steady state solution when r(x) is a constant.We prove that the total population at equilibrium is strictly smaller than the total carrying capacity.In Chapter 4,we study the dynamics of the Lotka-Volterra competitiondiffusion model when the intrinsic growth rates of both species equal 1.Inspired by the works of He & Ni [33,34] which focus on the case r(x)?K(x),we also assume that the two species have the identical competition abilities and the same amount of total resources,but the resource distributions of the two species are different.We investigate the dynamics of the Lotka-Volterra competitiondiffusion systems under two different cases:(i)the distribution of resources is heterogeneous for one species but homogeneous for the other species;(ii)the distributions of resources for both species are heterogeneous.Our results indicate that the species with heterogenous distribution of resources no longer have a competition advantage in terms of their diffusion rates,which is quite different from the case r(x)?K(x).In Chapter 5,we consider a more general Lotka-Volterra competition-diffusion systems in heterogeneous environments,and establish its global dynamics under a general criteria in terms of their competition coefficients.Furthermore,when the ratios of the intrinsic growth rate to the carrying capacity for each species are proportional,this criterion reduces to what He & Ni have obtained in [32].We also study the detailed dynamics in terms of the two dispersal rates of the two species in this special case.In Chapter 6,we summarize the main results of this thesis and propose some interesting problems to study in the future.The mains tools we used in this thesis are theory of monotone dynamical systems,linearization theory and the theory of principal eigenvalue of elliptic eigenvalue problems.We analyze the linear stability properties of the two semitrivial steady states of the Lotka-Volterra competition-diffusion systems.When we establish the global dynamics of the more general Lotka-Volterra competitiondiffusion systems in heterogeneous environments,our main idea is to prove that any coexistence steady statess of the model,if it exists,is linearly stable.Therefore by the monotone flow theory,if one of these two semitrivial steady states of the model is linearly stable,it is globally asymptotically stable;on the other hand,if both semitrivial steady states of the systems are linearly unstable,there exists a coexistence steady state which must be unique and hence is globally asymptotically stable.