Analysis Of The Stage-Structure Model With Time-delay And Diffusion

In the study of the population dynamics,the population's persistence is a very interesting and important problem. In order to make the models more practical and accurate,more and more realistic factors have been considered,such as stage-structure (see [1-10]),diffusion (see [10-15,27-30]),time delay(see [14-18,31-40]),but the model with all the factors seemed rarely to be considered.Considering all these factors,the paper discusses two types of predator-prey models which are two-patch,two-stage and two-population dynamics. The model in Chapter 1 describes that each of the two patches has the immature and mature predator populations,and there is another prey population only in one patch;The model in Chapter 2 describes that there are immature and mature predator populations only in one patch,and the mature predator can disperse between the two patches,and there is a prey population on the patch which has no immature population. Because of the season's influence,it is reasonable to assume all the coefficients are periodic and almost periodic.Each chapter of the paper reads as follows. Firstly,the uniform persistence of the system is proved;Secondly,by using the independent subsystem method,the more brief sufficient conditions for the existence and global attractivity of the periodic solution and a mathematical example is obtained to show its practicality. Finally,the conditions of the existence of the positive almost periodic solution which is uniformly asymptotically stable are derived by the Razumikhin function method and a mathematicalexample is obtained.In the proof,by the subsystem which makes the discussion more convenient,we get more brief sufficient conditions. We also use the comparison theorem,Liapunov functional method and Razumikhin function method.In Chapter one,consider the following modelwhere represent the immature and mature predator populations densities in the patch i at time t respectively. z(t) represents the prey population density in the patch 1 at time t,and yi(t) in patch 1 is its predator at time t. The mature population yt(t) can disperse between the two patches at time t,Di(t) is the diffusion coefficient in the patch i at time t,where,t = 1,2.The models have their right background,for example,some kind of sea birds which live in two different islands,the adult birds can fly between the two islands,so the sufficient fish supply is one of most important driving forces. The initial conditions are as follows,For a bounded continuous function w(t) defined on For the continuity of the initial conditions,setThe main results are as follows,Theorem 1.3.1 If system(1.2.1) satisfies (1.2.3),then the solution of (1.2.1) is ultimately bounded.Theorem 1.3.2 If (H1),(H2)and (H3) hold,wherethen (1.2.1) is uniformly persistent.Let's consider the subsystem of (1.2.1)Theorem 1.4.1 If (H1) - (H6) hold,wherethen every positive solution of w-periodic system (1.4.1) is globally attractive in IntR .Theorem 1.4.2 If (H1) - (H6) hold,then u;-periodic system (1.4.1) has a unique positive w-periodic solution which is globally attractive.Theorem 1.4.3 If (H1) - (H6) hold,then w-periodic system (1.2.1) has a unique positive w-periodic solution which is globally attractive.For convenience,define the right part of (1.2.1) as F(t,xi,yi,z,xy,3 2)-Theorem 1.5.1 Assume system (1.4.1) satisfies (H1) - (H3) and (H7) - (H9),where,then equation (1.4.1) has an almost periodic solution p(t) which is uniformly asymptotically stable and mod(pi) C mod(Fi). Furthermore,if (1.4.1) is a w-periodic system in t,then (1.4.1) has a positive w-periodic solution.Theorem 1.5.2 Assume system (1.2.1) satisfies (H1) - (H3) and (H7) - (Hg),then equation (1.2.1) has an almost periodic solution p(t) which is uniformly asymptotically stable. Furthermore,if (1.2.1) is a w-periodic system in t.,then (1.2.1) has a w-periodic solution.In Chapter two,consider the following modelwhere xi(t),yi(t) represent the immature and mature predator population densities in the patch 1...