| Predator-prey model is a mathematical model which describe the dynamic change in predator-prey system. In 1921 and 1923, Lotka an American ecologist and Voterra a Ital-ian mathematician came out with Lotka-Volterra system respectively. This model aroused people's wide concern. Many research men considered some important factors like stage structure, time delay, parental care and so on, and work out a lot of important conclusions which made a great step forward in the model. We know that classic Volterra predator-prey model is described by a differential equation with positive coefficients namedε1,ε2,γ1,γ2This model describes the interaction between predator and prey populations. We know that there are not only mature population, but also have immature population. They are same kind, but because of different age there are difference in consumption, food intake, food supply and so on. To consider in this way, Lotka-Volterra model is not very precise. In order to get a more accurate model, people considered many factors just like time delay and stage structure on this basis. Aiello, W. G. and Freedman, H. I. [2] studied a time-delay model of single-species growth with stage structureIn this paper, they discussed the problem of equilibrium, stability, oscillation and nonoscil- lation of solutions. They got many conclusions. The conclusions tells us that if immature population is not extinguished on t∈[-τ,0], and mature population is exist on t= 0, then the two populations will not extinguish on t> 0, and the system is stable, during time goes to infinity, it goes to a nontrivial equilibrium point. Wendi Wang and Lansun Chen[4]studied a predator-prey system with stage-structure for predator.They analyze permanence and stability of the model, and get relevant conclusions. Jing'an Cui and Yasuhiro Takeuchi[5] studied a predator-prey system with a stage structure for prey.They analyze the condition of permanence and the existence condition ofω-periodic solu-tion. Analysis showed that the system is permanence if and only if the growth by preda-tion is larger than death rate of predators. Zhihui Ma and Zizhen Li[6]pull the factors of stage structure and time delay into both predator population and prey population, and get a predator-prey system with stage structure and time delay.They analyze positivity and boundedness and permanence of the system. In the predator-prey system, functional response means during density of prey increase, the number of each predator consume prey is also increase. Based on experimental data, Holling proposed three types of prey dependent functional responses g(u)=k(t)u; P. Y.H.Pang,M.Wang[15] and M.U.Akhmet,M.Beklioglu[16] studied diffusive ratio-dependent predator-prey systems with Holling(Ⅱ) type of functional response.Xiuxiang Liu and Lihong Huang[7]considered a diffusive ratio-dependent predator-prey system.They analyze ultimately boundedness and permanence of the system. Wendi Wang,Yasuhiro Takeuchi[3] consider stage structure and parental care in the model, they divide the popula-tion into two stage, immature and mature. They assume that only the mature population have predatory ability, and the immature predator population get food from their parents. We get the modelAccording to the different inherent growth of prey, they analyze stability of model, and get important conclusion. If the growth of prey is of exponential type, the stability of the positive equilibrium is determined by the ratio of the death rate of immature predators to the death rate of mature predators. On the other hand, if the growth of prey is of Logistic growth type, predators will be extinguished when the birth coefficient of predators is less than the maturation coefficient. In the paper, they explained the difference generated by two different inherent growth. Based on the system with parental care, we considered space distribution and dispersal behavior[8], we got the reaction diffusion equation, using variable substitution In this system, we use U= (u, v1, v2) to denote its positive stationary solution. Discussing its dissipation and stability, we get some conclusionsTheorem There exist constants Ki, K2 which independent of initial value, such that for any positive solution of the system, we have the following calculationTheorem If a< b, then any positive solution of the system U satisfies (?)U= (a/b,0,0) on theΩ, if only u(x,0)≠0, then the stationary solution of the system (a/b,0,0) is globally asymptotically stable on the R+.Theorem If A satisfies and 4b+1≥m, U is locally uniformly asymptotically stable. Therefore, in some domains of U, there is no exist non-positive constant stable solution in the system.
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