Analysis of a syrphid fly-aphid system with age structure

Predator-prey is one of the most prevalent interactions among species in the nature.Because their biological characteristics of each age are different,it is of very important theoretical significance and practical value to take into account the population models with age structure.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.In our model,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.First of all,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.Secondly,we get three equilibrium points of the system when we suppose the right-hand side is zero.We analyze these equilibrium points and the stability of the system based on the stability theory of differential equations.The results show that one equilibrium point E1(0,0,0) is unstable,and the boundary equilibrium point E2(K,0,0) is locally asymptotically stable under certain conditions.With time increasing at the boundary equilibrium point,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(x,y1,y2) 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(x,y1,y2). At this point we can also get the ratio of natural enemies by the pest,which can help control the pest population to reach the positive equilibrium point so as to prevent the outbreak of the pest population.Finally,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.Syrphidae insects include saprophagous and phytophagous.This paper only researches the predatory syrphid fly and aphids with predator-prey relationship.Because the Aphids are r-strategy users,the model in our research can be extended for application inthe syrphid fly-aphids system.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 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.