| In recent years ,much inspiring progress about ecological systems has been made ,while the study of the stage-structured Lotka-Volterra system that is more correspond with nature phenomenon and more practical significance ,has not been studied very much .In Chapter I of this paper,we consider an autonomous predator prey Lotka-Volterra system in which individuals in the population may belong to one of two classes :the immatures and the matures ,the age to maturity is represented by a time delay . By using eigenvalue analysis ,we obtain some simple conditions for global asymptotic stability of the unique positive equilibrium point ,meanwhile the theorems extend the conrespond conclusions in which there have no stage structure . In Chapter II ,by using the comparing theorem and Razumikhin-type theorem and V-function method ,we consider a nonautonomous predator-prey system with stage-structure and time-delay . And we get the sufficient conditions for the uniform persistence and the solutions' global attractivity of this system . As a periodic system ,we obtain the existence and uniqueness of a positive periodic solution of this system . As an almost periodic system ,we prove the existence and the uniform asymptotic stability of the almost periodic solutions of this system.Chapter IThe Asymptotic Behavior of A Stage-structured Autonomous Predator-prey System with Time-delayIn this chapter ,we consider a stage-structured autonomous predator-prey system with time-delay in the following formWe denote predator species and prey species as species 1 and species 2 ,for system(1.2.1) ,xi(t) and yi(t) represent the densities of mature and immature of species i respectively ,here i = 1,2. Denote T;> 0 the length of time from the birth to maturity of species i . b,> 0 (i - 1,2) denote the birth rate of i-th immature population ;d,> 0 (i = 1,2) denote the death rate of i-th immature population ;Di > 0 (i = 1,2) denote the death rate of i-th mature population ;Suppose the preying quantity of the mature predator species preying on the mature prey in the unit time is proportional to the density of mature prey species in proportion to the constant 0 > 0,and we call it the preying rate in short . k > 0 is the preying effective coefficient and we call k the preying effective rate . For continuity of initial conditions ,we requireWe call the positive initial conditions of system (1.2.1) .For system,we are easy to get three nonnegative equilibria of it,as the followingsAnd if syst,ein(1.2.1) satisfy the conditionthen ,sy,stem(i.2.1) lias n unique positive equilibrium point ,hereWe get main conclusions about system(1.2.1) in the following.THEOREM 1.2.1 The solutions of system(1.2.1) with given initial conditions are positive and bounded for all t > 0.THEOREM 1.4.1 The nonnegative equilibria E,Ei of system(1.2.1) are unstable and under the condition (HI),E-2 is unstable.THEOREM 1.4.2 If the assumption (HI) andhold ,then the positive equilibrium E is globally asymptotically stable . THEOREM 1.5.1 If system(1.2.1) satisfiesthen the nonnegative equilibrium point E-2 is globally asymptotically stable .Chapter IIPeriodic Solutions and Almost Periodic Solutions ofA Noiiautonomous Predator-prey System with Stage-structure and Time-delayIn this chapter,we consider a stage-structured nonautonornous predator prey Lotka-Vblterra system with time-delay in the following formWhere Xi(t) and j i(t) represent the densities of mature and immature of species i at time t ,respectively ,here i = 1,2;Denote TJ > 0 the length of time from the birth to maturity of species i;bi(t) denote the bearing rate of mature i-th species ;di(t) denote the death rate of immature i-th species ;Dj(i) (i = 1,2) denote the death rate of mature i-ih species . Suppose the preying quantity of the mature predator species preying on the mature prey in the unit time is proportional to the density of mature prey species in proportion to the function 9(t) > 0,and we call it the preying rate in short . k(t) is the preying... |
