| The prohlc-m of persistence of the biological population is very important with regard to biomath-ematics and other related courses. There have been a fair amount of previous works on the standard Lotka - Volterra type prey-predator model (see [1]-[3]),in which the mean predation rate was also assumed to be dependent on the density of the prey. But recently more and more papers have been proved that,the mean predation rate should be dependent on the theory of " ratio-dependent " by the proofs of the biology and experiments (see [4]-[7]). Nowadays it has been provided that the predator hunting for the preys according to the different rates of different preys by [8]. Because in many environments,especially the predators have to search for food or maybe the prey is so big or strong that the predators must hunt for the prey with the help of the other predators,then they must share the food with each other. These facts have been proved by the [5,C,9,10). In order to make the biological model more practical,many biomathematicians and biologists have been working on the stage-structured models (see[11-19,31-40j).On the other hand,more and more people have begun to find out that the time-delay to biologicalsense can't be ignored and the too longer time-delay would destabilize the stability of the solutions of the model (see [20,21]). Recently many papers have been shown the influence of time-delay on the asymptotic properties of the solutions (see [21]-[24]).In [8],it is provided that the functional response is dependent on the different rates of different preys. Such model is as followswhere xj,x denotes the density of the two competitive populations A'i,A'2 respectively;13 denotes the density of the predator population X3. a1,a2 denotes the rate of AS hunting fot the populations X1,X2,f is the functional response;.In [25],a new prey-predator modle with stage-structured has been provided in which the predator might reduce the death rate by feeding on the preys. Such model is as followswhere Xi(t),xm(t) denotes the density of the immature and the mature population predator A' at time t respectively;y(t) denotes the density of the prey population Y at time t. The mature predator of X can reduce tins death rate;by hunting and feeding on the prey }'.In this paper,( ) combined with the factor of () is considered. It show the influence of the stage structure on the uniform persistence and the existence and uniqueness of the positive positive periodic and almost periodic solution and their corresponding stability. In Chapter one,model (1.1.1) is considered ;In Chapter two,model (2.1.1) is considered,then mathematical examples are given to show their practical respectively;In Chapter three,through comparison of the conditions of (1.1.1) and (2.1.1),which gurantee the same results,many interesting results are obtained. Such as,in spite of the consideration of stage-structure,the conditions,which can gurantee the uniform persistence and the properties of the same results of the positive periodic and almost periodic solution,are much simpler. So considering the stage-structure is very essential. The paper lias obvious biological background,for example,Chinese mantid preys on the bees and butterflies in the forest and field. But the immature Chinese mantid,i. e. its spawn,can not prey on them. Butterflies feed on the flowers and bees also need them,so they affect and compete with each other. From the right model,we can see that our theoretical results accord with the biological facts.The model in Chapter 1 is defined as followsLet J(t) is a strictly positive bounded continous function,set Note x(t) = (xi(t),xa(t), The initial conditions are as follows,The main results are as follows,Lemma 1.2.1 Assume that every solution of system (1.1.1) satisfies the initial condition (1.1.2),then is the positive invariant set,of the system (1.1.1).Theorem 1.2.2 Assume that the system (1.1.1) with the initial condition (1.1.2) satisfies the following conditions,then system (1.1.1) is uniformly persis...
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