Asymptotic Property Research For Three Kinds Of Ecological Models

In this paper ,the asymptotic property of three ecological systems is studied by constructing Lyapunov function and Lyapunov functional ,establishing recursively sequences and employing differential inequalities etc.and using Lyapunov theorem, comparison theorem .Barbalat's lemma, the continuation theorem of coincidence degree theory etc.Here, the system's asymptotic property includes the global attractivity of the solutions,the global stability of the solutions, the uniform persistence ,the existence of the periodic solutions and the existence and uniqueness of the periodic solutions etc.In some specific ecological questions ,it is necessary to change the species equilibrium by mankind for the practical reasons. As we all know ,it is an effective method to introduce feedback controls into the model .In the first section of this paper,we study a class three-species predator-prey-competition system with feedback controls.The sufficient conditions of the globally asymptotical stable positive equilibrium are derived by Lyapuov function method .And the sufficient conditions of the globally asymptotical attracting positive equilibrium and the globally asymptotical stable positive equilibrium are respectively derived for the system with delays by establishing recursively sequences method and Lyapunov function method. The conclusion we obtained reveals the globally attractivity of the positive equilibrium is not affected by the delays.In real world ,a series of processes:species birth ,growth or death,predator-prey,competition or cooperation among species etc. are very complicated.The relations between them are not always linear but are all kinds of functional responsive functions Moreover ,the prey or inferior species exists by means of the shelter of the refuges .In the second section ,we study a class nonautonomous ratio-dependent three-species predator-prey system with refuges.Firstly.the sufficient conditions for uniform persistence and globally asymptotical stability are obtained by using differential inequalities and Lyapunov function method and Barbalat's lemma.Then the existence and uniqueness of the positive periodic solutions and positive almost periodic solutions for corresponding periodic system and almost periodic system are also discussed by using Brouwer fixed point theorem and constructing auxiliary system and Lyapounov function.In the classical competitive system ,it is assumed that each individual competitor admits the same ability to attack another competitor.However in natural world .almost all animals have a life history that takes them through two stages : immature and mature .And different stages have evident difference about their physiological characters(birthrate,deathrate, competition ability etc.