Dynamics And Controlanalysis Of A Class Of Resource Allocation Models

Nowadays,all kinds of renewable resources are under extreme pressure worldwide and new efforts are to design better regulations in terms of economic or control instruments and measures of stocks and catches.The maintenance of species diversity depends on the sum of each species responses to the environment and on the interactions among them.Little is known about the overall distribution of the proportion of consumers and resources in real ecosystems which is systematically changing between habitats or extensive taxonomic groups together with resource harvesting activity.This dissertation mainly studies the effects of trade-off specificity and harvesting on a consumer-resource model and modeling and optimization analysis of heterogeneous renewable resource harvesting with extraction budget.This dissertation is divided into four chapters:In Chapter 1,we introduce the background and biological significance of the resource allocation models and the theoretical knowledge of ordinary differential equations.In Chapter 2,we research the relationship between resources and consumers together with harvesting by using a modified Rosenzweig-MacArthur model,in which the proportion of each genotype available for consumption consists of two components: an intrinsic part and a combination from all genotypes present in the resource.The trade-off of these components is characterized by a specificity parameter.From the viewpoint of population dynamics,we observe that the ability of the mutant to invade the resident resource strongly depends on the values of this parameter.Based on the sufficient conditions for mutant invasion by using the theory of ordinary differential equations.Finally,we illustrate the possible evolutionary outcomes via numerical simulations.In Chapter 3,we introduce the ODE model of concern,stating the relevant economical and biological assumptions together with the definitions of the various parameters employed.We shall propose two possible harvest scenarios and thenanalytically investigate the corresponding stability.Then,we state the control problem and introduce the objective functional to be maximized,the Hamiltonian method being then applied to find the sufficient conditions and,respectively,the necessary conditions for the existence of the optimal control,and the singular solution of this problem is thereby obtained.Finally,we complement the previous theoretical analysis with several numerical simulations and give a discussion of the main results and findings.In Chapter 4,we summarize the dissertation according to the research contents.