The Optimal Control Problem Of State-dependent Impulse Model Under Random Influence

How to reasonably develop and utilize natural resources is a crucial issue in the development of human society.As an important part of natural resources,fishery resources have been widely concerned by many scholars in recent years.Although a fish population under exploitation by a fishery could be influenced by many factors,man's action is considered to be the main controlling agent.The core issue of this paper is to investigate how to take measures to guide human activities,so that we could get the optimal catch of fish stocks.Assuming that an economic fish lives in a constant environment with a limited food supply and meets logistic growth,we are concerned about what the capture strategy is optimal.If the fishing activity is too sparse or the fishing intensity is too small,we can not achieve the purpose of exploiting resources.On the other hand,intensive fishing activity may pose a threat to resources,which can not main-tain the long-term development of resources.Therefore,sustainable exploitation of renewable resources is crucial for fishery resource management.In fact,there are different interpretation of optimization,such as the largest amount of fish,or the maximum profits.This paper studies how to maximize fish populations at an end of a harvesting season.Many scholars have established deterministic impulsive models to investigate this problem and proposed meaningful solutions.In reality,however,the growth of population will be influenced by many random disturbances,such as weather changes,predators,natural illness and death,etc.Typically,these random factors can not be ignored.In order to satisfy different practical needs,it is necessary to establish a stochastic model to reconsider this issue.Suppose that the intrinsic growth rate of the population is affected by envi-ronmental noise,we obtain a state-dependent random impulsive model.We first consider the case of performing one pulse harvest in a harvesting season.According to the theory of the stochastic differential equation and the maximum principle,the optimal threshold for one-pulse harvesting model is obtained,that is,we should perform fishing activity once the fish population reaches the threshold so that in the subsequent growth,the expectation of the population will reach its maximum.Then this is expanded into a two-pulse constant harvesting model,followed by a more general case of a multi-pulse harvesting system.Investigations of the effects of different parameters reveal that theoretical predictions from the new stochastic model accord with those from the deterministic case.At the same time,we estimate the time when the fish population density will attain these thresholds which requires some more detailed analyses compared with the estimation of Malthusian case.We have also tested our conclusions by numerical simulation,the predictions of which are in good agreement with numerical experiments.Our method is a tentative study of fishery resources which can be extended to other renewable resources such as agricultural systems with pests controlled accord-ing to the principles of integrated pest management(IPM).In this investigation we will use our models determine the optimal time within a planting season of crops that minimizes a pest' s density at the end of the season.The proposed method in this case is more accurate and specific than the analysis of deterministic model.All of the theoretical predictions axe verified by numerical simulations.Another interesting question is to discuss the influence of random disturbance on multi-population impulsive systems.In the third chapter,the tw?-dimensional impulsive differential equation under random influence is established.The stability of the zero solutions of subsystems are give.Then,we propose the comparison theorem of stochastic impulsive equation and analyze the boundary conditions of population.Our work can be regarded as a meaningful attempt of multi-population impulsive systems.