Qualitative Analysis To Predator-prey Models Of Ecosystems

This paper is a survey on the studies of predator-prey systems . In this paper, our main interest is to study a class of predator-prey problem. We focus on the stability of the positive uilibrium point and the limit cycles, the existence, uniqueness and the global asymptotic stability of positive periodic solutions and positive almost periodic solutions, and specific applications of predator-prey systems of ecosystems. The paper is divided into five chapters. The main contents are as follows:In chapter one, we introduce a survey to the development of biological mathematics, particularly the biological species ecology major impact on biological mathematics. We also summarize main results in the dissertation .In chapter two, we discuss models interacting between the predator species and the prey species. First, we study simple cases between two species of predator and prey and establish a mathematical model of the system describedas followswhere a₁,a₂,b₁,b₂ are constant, b₂ > 0. That is the classic of Lotka-Volterra model of the system. By using qualitative analysis methods for analysising andimproving the models, and considers predator-prey model with two species, weobtain modified Lotka-Volterra modelsdxxoi ix Cly,dyStability of the positive equilibrium point to system (2.2) is discussed with the use of V-functional, under certain conditions we obtain the sufficient conditions or sufficient and necessary conditions on the stability of the systems (2.2). For the sake of convenience, we now state the lemmaLemma 2.1: Suppose that (2.2) has positive equilibrium point M(x\, x^)-Then stable if and only to if of the point M is locally asymptotical :Lemma 2.2: Suppose that (2.2) holds positive equilibrium point M(x\, x^) Then stable if of the point M is global :(1) point M is locally asymptotically stable;(2) Both species are density dependents.Subsequently, we discuss the system of harvest rate or invest rate, and study the impacts to the global stability of predator-prey system of the positive equilibrium point. We consider a class of modelsx = x(b - anx - any),(2.3)andI t = T.(h — 0. Then the system (3.1) has a unique stable limitcycle, where aj = — > 0, ai = —, 03 = —r- < 0, and a,\ — a > 0, a2 = aw-b, a p p03 = — bu < 0, (3 = ea — ud.Similarly, we consider predator-prey systems with Holling's type III functional response,x = ax — bx2 — 0);bi(t), di(t), ei(t), ri(t), (i = 1,2) are strictly positive.Suppose ku > kl > au > a!,pi > ^ > 0, (i = 1,2), where— hunu-----^-,0 < xm 0.1 xmBy using the theory of system permanence and the theory of system ultimately bounded, we obtainTheorem 4.1: Suppose contition (Hi) holds. Then The System (4.1) is sustainable, f2 = {(x,yi,y2)\xm