Stability,Controllability And Optimal Harvesting Of A Nonlinear Discrete Hierarchical Population Model

Biomathematics is an interdisciplinary subject between mathematics and biology.It is concerned with the evolution and adjustment of populations by mathematical modeling.With the rapid development of industrialization,people gradually realize the importance of protecting the ecological environment.In order to protect the bio-diversity,maintain the ecological balance and make rational use of resources,scholars at home and abroad have established a large number of mathematical models.These models can be roughly divided into two categories: continuous models and discrete models.For some populations,it is convenient to use discrete models to describe the evolution of the species.In this dissertation,a class of non-linear discrete hierarchical population model is established,the body consists of Chapters 2 and 3.In the second chapter,the evolutionary behaviors of the non-linear discrete hierarchical population are studied.Firstly,we formulate the model and analyze the basic properties of the model solutions,such as non-negativeness,existence and uniqueness of positive equilibria,estimation of the positions of positive equilibria,definition of population regeneration number R~0.Then we investigate the stability of the steady states.It is concluded that when R~0≤ 1,the model has no positive equilibrium state,and when R~0> 1,the model has only one positive equilibrium state.The asymptotic stability of trivial and positive equilibria is studied by using primitive matrixes.The global stability of zero equilibrium and positive equilibrium is discussed by means of Lyapunov functions,and the criterion is provided.In addition,the absence of 2-periodic solutions to the model is analyzed,and a set of conditions are obtained.Finally,numerical simulations are carried out to verify the conclusions.Chapter 3 mainly discusses the controllability and optimal harvesting of the population system.Firstly,the upper controllability and the lower controllability of the population system are studied by using the comparison principle,and the stabilization of the system is obtained.The conditions for the exact controllability of the system are presented by using the fixed point principle.Secondly,the optimal harvesting problem of the population is considered,the existence of the optimal policies is analyzed,and the necessary optimality conditions are derived by the discrete maximum principle.Formulas ofoptimal harvesting in three special cases are given.In order to achieve maximum benefits for sustainable population development,the optimal harvesting when the population is in equilibrium is analyzed.Finally,some numerical simulations are carried out to show the effects of individual values and rank-changing rates on the optimal harvesting.