Quantitative Analysis Of Discrete Size-Structured Population Models With Slow Growth

The study of the development and evolution of biological populations needs to be based on certain conditions,then we establish the corresponding models to do analysis and control.The population models could be continuous or discrete,in which discrete models are more close to the natural and the real situations.More than one hundred years ago,a mathematician,Verhulst,established a Logistic model to describe the growth of human population.During the first World War,the decline in fishing capacity resulted in a reduction in non carnivorous fish and the increase of the carnivorous fish amount at the Finme port in Italy.Volterra established a mathematical model with the interaction between small and big fishes to explain the phenomenon.In 1945,Leslie proposed a very important discrete population model,which promoted the study of the discrete biological population models.On the other hand,the investigation of the control of the biological dynamic systems not only has important theoretical value,but also is closely related to the bio-diversity protection,the optimization of the ecological environments and the utilization of the renewable resources.In this dissertation,we study the discrete population models.Firstly,we establish the population model.Then we use the matrix theory and numerical analysis method to analyze the model,discuss the existence condition of the equilibrium,the stability conditions of the equilibrium,and carry out numerical analysis using the MATLAB software(figures can intuitively display the evolution trend of the population).Finally,we explain the ecological significance of the results based on the results.In the second chapter,we studies the size-structured model with resource dependence.Firstly,the existence condition of the non-negative solutions of the model is analyzed.Then the equilibrium of the model is examined,including zero and non-zero equilibria,as well as the conditions for their stability or instability.The stability conditions is obtained by the analysis of characteristic value of the Jacobian matrix corresponding to the equilibrium state.Finally,the model was simulated by MATLAB,and the effect of the slow growth rate on population is observed.It is found that slow growth can promote the stability of the population.The third chapter is devoted to optimal harvesting problem of the size-structured population model with two stages.Firstly,the necessary conditions for optimal policy are given by citing a result in the literature.Secondly,considering the effect of harvest strength and slow growth rate on the optimal biomass harvest,the model is simulated by MATLAB.We observed that the slowgrowth rate has a promotion for the optimal harvest.Finally,the conclusion is proved by a series of computations.