Bang-Bang Control Of The Catastrophe Modle For Pest Populations

In the "Renewable Resource Management",the author Clark has mentioned three classes of models,namely,the pure compensation model,the depensation model and the critical depensation model.In fact,the latter two models are a good description for the catastrophe behavior of the populations and a powerful tool to reveal the outbreak of pest populations.In this paper,considering the prevention cost of pest populations,the controllers for the depensation model and the critical depensation model are designed to eliminate the outbreak of the pest populations and minimize the cost using the optimal control principle model.The main cottents are as follows:Part I.Firstly,based on the depensation model,the corresponding cusp catastrophe model of the pest populations are established.Considering the control cost of the pest populations,the objective functional is established by using the economic principles,thus the optimal control model is obtained.Using the optimal control principle,bang-bang controller is designed to eliminate the outbreak of pest population and to minimize the cost.Finally,the validity of the control is verified by some numerical simulations.Part II.Firstly,based on the critical depensation model,the corresponding cusp catastrophe model of the pest populations are established.Secondly,considering the economic threshold of the pest populations,the objective functional is established,further the optimal control model is obtained.Thirdly,using the optimal control principle,the bang-bang controller is designed so that the pest populations do not affect the normal growth of the crop.Finally,the conclusion is verified by numerical simulations.In order to show the ecological mechanism of the outbreak of the pest populations,the bang-bang control principle is applied to the depensation model and the critical depensation model,respectively.In addition to the application of bang-bang control principle in pure compensation system.The whole system of bang-bang control of three models which are mentioned by Clark is formed.The conclusion provides the theoretical basis for the quick response of pest populations control.