Analysis Of A Predator-Prey System With Age Structure

Predator-prey is one of the most prevalent interactions among species in the nature. For example, syrphid fly and aphid. Because the predator's biological characteristics of each age are different, it is of very important theoretical significance and practical value to take into account the predator-prey models with age structure. For example, their feeding habits, appetite, searching rate and detection rate are absolutely different. However, the existing predator prey models with age structure always assume that only mature predators catch prey. This is inconsistent with observed fact. In this paper we establish a syrphid fly-aphid model with age structure based on the biological characteristics of the predatory syrphid fly that only the immature predators can catch prey. We also explain the insect population dynamics by a mathematical analysis of the models, in order to provide a theoretical basis for Integrated Pest Management.Through the analyses of syrphid fly's biological characteristics that only the immature predators can catch prey. We establish a model with age structure that only the immature predators can catch prey. We also analyze the criterion for the stability and permanence of populations and the quantity change conditions and mechanism. Obtains the following conclusions:Firstly, a model with age structure that only the immature predators can catch prey was established.Secondly, according to the given initial conditions and characteristics of the system, we have proven that the solutions of the system are always positive and bounded.This is consistent with the ecological significance of the system.Thirdly, we get three equilibrium points of the system when we suppose the right-hand side is zero. The results show that one equilibrium point E1 is unstable, and the boundary equilibrium point E2 is locally asymptotically stable under certain conditions. With time increasing at the boundary equilibrium point, the aphid populations tend to survive and achieve the maximum capacity, while the syrphid fly species will get extinct, that would be the outbreak of the pest populations which should be prevented. The results also indicate that the positive equilibrium point E3 is locally asymptotically stable under certain conditions. With time increasing at this point, both the aphid and syrphid fly populations will tend to survive and approach a positive equilibrium E3. At this point we can also get the ratio of natural enemies by the pest, which is a very important parameter and can help control the pest population to reach the positive equilibrium point so as to prevent the outbreak of the pest population. So this ratio is a threshold value density to control harmful insect and the most important appraisal parameter in the effect of biological control.Last but not the least; we get the criterion for the permanence of populations using uniform persistence theory. Under the condition, both the predator and the prey populations will not be survive. Given certain corresponding parameters, we can draw some results of the system with numerical simulation analysis.To sum up, based on the biological characteristics of the predatory syrphid fly, in this article we establish a syrphid fly-aphid system with age structure, in which only the immature predators catch prey. Through mathematical analysis, we discuss the equilibrium points and stability of the system, and build a criterion for the permanence of populations. We can explain the insect population dynamics by these results. All of these results provide a foundation for Integrated Pest Management.