A Qualitative Study Of The Solution Of A Class Of Competitive Model Problems With Free Boundaries

In the natural ecological environment,there are predation,competition and other mutual constraints.When some invasive species invade,they will compete with native species,thus disrupting the food chain and affecting the balance of biological populations.In addition,the birth and death rates of species may vary depending on the seasonal changes and the geographical environment,so that the scale and rate of invasion of invasive species may also vary.These phenomena have attracted extensive attention of many scholars to establish different mathematical models to analyze the spread of exotic species.In this thesis,the Lotka-Volterra competition model with Robin boundary conditions and free boundary conditions is used to study the competitive relationship between native and invasive species.Firstly,this thesis introduces the theoretical background of the Lotka-Volterra competition model,the current state of research and the preparatory knowledge needed for this thesis.Secondly,this thesis investigates the Lotka-Volterra competition model in the homogeneous time-space environment,mainly considering the situation where the native species is the inferior competitor and the invasive species is the superior competitor.Under this condition,the existence of the global solution of the Lotka-Volterra competition model is proved to be unique,and the conclusion of spreading-vanishing dichotomy is given: either spreading occurs,i.e.,both the native species and the invasive species can spread successfully,and their upper and lower bounds can be obtained at this time;or vanishing occurs,i.e.,the native species cannot spread successfully and become extinct as time tends to infinity,The invasive species will have a dichotomous result,either extinction or convergence to a positive solution of the stationary problem.In addition,the criteria for spreading and vanishing are given.Finally,the thesis discusses the Lotka-Volterra competition model in the time-periodic environment.In this model,the variable intrinsic growth rates of these two species change signs.Under this assumption,the uniqueness of the solution to time-periodic initial boundary value problem is proved and a spreading-vanishing dichotomy conclusion is given: either spreading occurs,i.e.,both the native species and the invasive species can successfully spread out,at which point their upper and lower bounds are obtained;or vanishing occurs,i.e.,the native species cannot successfully spread out and becomes extinct as time tends to infinity,while the invasive species will happen a dichotomous result,either extinction or convergence to a positive solution of the time-periodic initial boundary value problem.In addition,the criteria for spreading-vanishing,and the asymptotic spreading speed at the free boundary is estimated.