Stability Analysis And Optimal Harvesting Problems For Discrete Multi-Stage Size-Structured Population Models

The first glory of mathematical biology was created by Lotka,Volterra,Kolmogorov et al in 1920 s and 1930 s.The emergence of a large number of practical problems and the establishment of modern differential equation theory promoted the development of mathematical biology,especially mathematical ecology in 1970 s.From 1980 s,related research work in China increased gradually.So far,there has been a lot of work comparable with their international counterparts’.The research on biological population has been attracting the attention of many scholars,not only because it is a need of ecological development,but also is the high economic and environmental value.We mainly establish the population model to research its evolution and apply mathematical methods to the investigation process.Structured models can be roughly divided into two kinds: one is age-structured and the other is size-structured.The former has been researched extensively,while the latter is still one of focuses in recent years.Body size is denoted by individual physiological or statistical indexes,including length,weight,surface area and so on.In a sense,an age-structured model is a special case of size-structured model,and the latter is more close to ecological reality.The population models can be mainly divided into continuous and discrete ones.The methods for presenting them are differential equations and difference equations,respectively.A difference equation may be regarded as discretization of a differential equation.Thus,both methods are complementary to each other.In this dissertation,we mainly study some discrete size-structured models,considering cross-group and slow growth,and analyze the stability,controllability,stabilizability and optimal harvesting problems of linear and nonlinear models.The main researches are as follows:In the second chapter,under the constrains of ecological balance,based on the linear and nonlinear models respectively.For the linear model,we consider cross-group and slow growth,divide the individuals in the population into 3 groups according to the individual size from small to large,and look for the optimal harvesting strategies which make the maximum economic value.The results show that the two-stage strategy is optimal,i.e.the first group aren’t harvested;a fraction of the second group and all of the third group are harvested.Then the controllability of the system is studied.We control the number of the larvae by inflow or catching,and derive the conditions on which the system is completely controllable.Finally we consider a nonlinear model by introducing a reproduction rate with crowding effects,and discuss the existence and stability of equilibrium the model.Also,we estimate the optimal harvesting rate by numerical simulation,and find that the second group is harvested hardly,all of the third group are harvested.In the third chapter,we consider the stability of a general nonlinear model in details.The individuals are divided into n groups and some conditions for the stability of the steady states are obtained by making use of the disc theorem and matrix theory.Numerical experiments with Matlab are used to verify the theoretic results.Then the system is linearized around the zero equilibrium and stabilizability is discussed.When the population is divided into 3 groups,we observe the number of individuals in these groups,by changing the parameters c and the reproductive rate of the second group,with numerical simulation method.The results show that c is very sensitive,and we can stabilize the positive or zero equilibrium by controlling the reproduction in the second group.