Asymptotical Behavior Of The Solution Of Nonautonomous Predator-prey Systems

This paper consists of four parts.Firstly, we consider a two-species model with nest parasitism, where both the two group aredevided to two stages. The conditions of permanence of it are gotten. Furthermore, existence andglobally asymptotical stability of periodic solution and al-most periodic solution have beenproved under the appropriate conditions.Secondly, we discuss an epidemic predator-prey system with time-delay and stage structure.By using some techniques of the differential inequalities, we obtained sufficient conditions forpermanence of the system. We obtained the existence and global attractivity of the positiveperiodic solution.Thirdy, we study a delayed predator-prey model with impulsive harvest and Holling IVfunctional response. The stability of the trivial equilibrium is analyzed by means of impulsiveFloquet theory. We show the existence of positive periodic solutions by using coincidence degreetheory. The system is then analyzed numerically, revealing that the presence of delays andimpulses may lead to chaotic solutions, quasi-periodic solutions,or multiple periodic solutions.Several simulations and examples are presented.Finally, a non-autonomous food-chain system with Beddington-DeAngelis functionalresponse and impulsive perturbation is considered. We obtained sufficient conditions forpermanence, ultimate boundedness, extinction of the system. Furthermore, it is shown that undersome assumptions, there exists a unique almost periodic solution which is globallyasymptotically stable..