Several Types Of Population-optimized Harvest Control Problems With Pulse Effects

Biological resources are a kind of renewable resources.In recent years,the is-sue of the development and utilization of renewable resources has become a common concern of many scholars.The renewable resources should be moderately exploited,that is,human beings should pursue the maximum sustainable yield or the best economic benefit under the premise of ensuring the sustainable development of bio-logical resources.In this paper,by applying the maximum principle of impulsive differential e-quation and the basic theory of population dynamics,we consider two classes of optimal control problem of single population systems with impulsive harvest and a class of inshore-offshore fishery system in a polluted environment.These results are significant to the theoretical research and practical applications for the optimal management of renewable resources.The main research contents and results in this thesis are as follows:(1)We study a class of optimal harvesting problem for a general non-autonomous Gompertz model with impulsive harvests,and the species is harvested at fixed mo?ments.By choosing the harvesting efforts as control variables,we study the max-imum harvesting problem in given time range.Firstly,the existence of optimal control strategy is proved.Further,the singularity harvesting strategy is obtained by using the maximum principle of impulsive differential system.It can be seen that the singularity harvesting strategy is the optimal harvest strategy when the conditions required by the singular harvesting strategy are satisfied.Secondly,we consider the control problems for the situations in which the singular controls are blocked at some harvesting moments.Therefore we first establish an optimization principle:the optimal path lies as close as possible to the singular path.And based on this optimization principle,the optimal harvest strategies in all blocked situations are obtained.Finally,the correctness of the obtained theoretical results is verified by numerical simulation.(2)We study a class of optimal harvesting problem for a non-autonomous Giplin-Ayala model with impulsive harvests,and the species is harvested at fixed moments.We choose the harvesting effort as control variable and the maximum harvest yield in given time range as control objective.By using the analogous opti-mal principle as(1),we study and obtain the optimal harvest strategy in arbitrary parameter situation for the harvesting system.(3)Considering the effect of toxicant on the population growth,the dynamic behavior and optimal control problem of a class of inshore-offshore fishery system with periodic impulsive diffusion and continuous harvesting is studied.Firstly,the sufficient conditions for existence and stability of positive periodic solutions are ob-tained by using impulsive differential equation theory.Moreover,under the condition that the system is global asymptotic stability,a related optimal control problem is investigated by applying the maximum principle of impulsive differential systems.The purpose is to control the harvesting effort as to maximize the profit which is the difference between economic revenue and cost.The exact expression of opti-mal harvesting polices and achievable conditions are given explicitly.Finally,the correctness of the obtained theoretical results is verified by numerical simulation.