The Studies Of Efficient Numerical Methods For Population Models

Ecology,basically is the study of the relation between species and their environment. In recent years there have been more and more studies of mathematical models in population ecology which shows their importance in understanding the dynamic processes and in making practical predictions.Single species models are relevant to laboratory studies.However,in the real world,they can reflect a variety of effects on population dynamics.Multi-species models have been used in a lot of areas,such as predator-prey and competition interactions,renewable resource management,evolution of pesticide resistant strains,ecological control of pests,multi-species societies. plant-herbivore systems and so on.The age structured population models have been investigated by many authors. The earliest age-structured models without spatial diffusion are McKendrick-von Foerster's linear model[32,56]and Gurtin-MacCamy's nonlinear one[34].We also refer to[18.29,30,44,50]and the references cited therein for more investigations on the problems without spatial diffusion.On the other hand.modeling the effects of spatial diffusion on age-structured population models is of particular importance.Spatial diffusion was first introduced into age-structured population models by Gurtin[33]and Rotemberg[66].and a large number of important extensions have been studied by many authors[2,3,23,35,46,49,51,52,55,58].In Chapter 1,we analyze an age structured population model without spatial diffusion,which is for marine invertebrates.The population dynamics of marine invertebrates is very important and interesting,which describes the population growth of invertebrates having sessile adults and pelagic larvae,such as barnacles.The process of the population growth contains two different life stages,in which the sessile adults are adhered to a limited area and produce larvae while the larvae can freely move from one area to another.In Chapter 2,we consider the age-structured population growth model with nonlinear diffusion and reaction,which represents the dynamics of a species remaining confined to space domain for all time,such as fish(see,for example,[23,36]).In many population models,birth rates are considered to act instantaneously. However,in reality,there may be a time delay taking account of the time to reach maturity,the finite gestation period and so on.Many new models incorporating delayed effects have been studied.Smith in[70]and Smith and Thieme in[71]derived a scalar delayed differential equation for the population with immature and mature age classes. Smith's approach was used in[74]to obtain a system of 'delay differential equations for mature population.Furthermore,in[75],they derived a non-local reaction-diffusion equation with time delay in a continuous unbounded one-dimensional spatial domain. Moreover,in[54],D.Liang and J.H.Wu considered a species living in a spatially transporting one-dimensional field and derived a reaction advection diffusion equation with time delay and nonlocal effect.In[55],new reaction-diffusion equation models with delayed nonlocal reaction were developed in two-dimensional bounded domains combining different boundary conditions.In Chapter 3,we consider the population model with local delay.The model we study can be easily modified to be the wellknown diffusive Nicholson's blowflies model,which has been widely studied in ninny papers,such as[57],[72],[73]and[78]for the finite domain case.As ecological information on life history and habitat characteristics has become more sophisticated,models have become more realistic,and the numerical analysis of simulation models,has become more important.The author has obtained some results on the development of efficient numerical methods for population models mentioned above.The dissertation is divided into three chapters.In Chapter 1.we study the discontinuous Galerkin methods(DG) for the age structured population model of marine invertebrates.Roughgarden et al.in[67]first proposed an age structured population model for sessile invertebrates living in a local area.The model can be seen as an open system model since the reproduction cycle is not closed.Some extensions have been studied by several authors in[8,39,48, 62,79].On the other hand,Roughgarden and Iwasa in[40,68]proposed the closed age-structured model and extended it to the multi-species and multi-habitats one. For the problem with a finite life-span,near the maximum age,the mortality function may be unbounded and the solution may be stiff or irregular.Thus.it is important in numerical computation to use meshes of different sizes and to refine the meshes at different time levels.For this purpose,the discontinuous Galerkin methods can be effectively applied to approximate the solution of the population growth model. The discontinuous Galerkin method was first introduced and applied to a neutron transport equation by Reed and Hill in[65].Since then,the discontinuous Galerkin methods were studied for different kinds of problems in many papers,for example, [1,7,9,22,26,41],etc..The discontinuous Galerkin methods own not only some advantages of finite element methods,but also those of finite difference methods.The methods are well suited to solve problems with local behaviors.They can easily take the locally refined meshes with hanging nodes and compute adaptively.They also have very simple communication patterns between elements which make them ideal for parallel computations.Furthermore,because of no requirement of continuity across element boundaries,the basis functions on each element can be chosen to be orthogonal, which leads to well-conditioned algebraic systems for higher order finite element approximations[1,21].More recently,[47]developed the continuous-time discontinuous Galerkin method for a single linear age-structured model of population and analyzed the convergence and error estimates for the senti-discrete numerical scheme,but there is no estimates for the fully-discrete numerical scheme and the nonlinear problem.In Chapter 1.we consider the more complex age-structured population problem of the marine invertebrates which is a nonlinear coupled system.In Section 1.2,we give some notations and propose the semi-discrete and fully-discrete discontinuous Galerkin schemes for the problem.Then, in Section 1.3,we derive error estimate for semi-discrete scheme and prove the global existence of the solution for the nonlinear system of the semi-discrete discontinuous Galerkin scheme by using the broken Sobolev spaces and the Schander's fixed point theorem.Furthermore.we analyze the the error estimate for the fully-discrete scheme in Section 1.4.1.Finally,a numerical experiment is given in Section 1.4.2 t.o show the effectiveness of our scheme.The theoretical analysis of the proposed DG method for the nonlinear system of age structured population in this chapter is more difficult and has significance in both theoretical analysis and application of the nonlinear system of the population growth in biology.In Chapter 2,we focus on the age-structured population growth model with nonlinear diffusion and reaction.Many papers only obtained first order accuracy in time and age directions.In order to improve the computational accuracy and cfficiency,it is necessary to develop higher order methods in time and age variables.In this chapter, using finite difference method along the time-age characteristic line(see[42,46,58]) and the finite element method for the spatial variables,we propose two new second order numerical schemes for the nonlinear population growth problem.One is implicit and the other is explicit.We analyze the numerical schemes to the nonlinear problem by making use of the theory of variational methods,the Schauder's fixed point theorem and the technique of prior estimates.In Section 2.2,we present the second order implicit and explicit numerical schemes for the problem.Then in Section 2.3,the error estimate is analyzed for the implicit scheme,and in Section 2.4 the global existence of the solution is proved for the nonlinear implicit scheme.In Section 2.5 we analyze the error estimate for the explicit scheme.Both of them are of second order in time and age directions.Compared with first order schemes,large time step sizes can be used in our numerical schemes and thus,fewer grid points are needed to obtain the same accuracy in computation.At the end of Chapter 2,we take numerical experiments to confirm our theoretical results.The model in experiment has practical meaning,which represents the dynamics of a species such as fish(see,for example.[23,36]).Numerical results are compared with those obtained by the method in[23].Our scheme is shown clearly to be of second order in time and age directions.We consider the adaptive finite element method for population dynamics model with local delay in Chapter 3.Adaptive finite element methods have become very important theme in scientific and engineering computations for their high efficiency. And they have drawn the attention of many researchers.For pioneering work,we refer to Babu(?)ka et al.([4,5,6.10.11,12]) and for recent work we refer to[14.15.16, 24.25.26,27,60]and their citations.Among the literature,a lot of efforts have been devoted to the linear and nonlinear parabolic problems.In particular.[10].[11].[16]. [25]and[64]derived a posteriori error estimates for linear parabolic equations.[14]. [17],[59],[63]and[77]analyzed a posteriori error estimates for nonlinear parabolic equations.As is well known,the smoothness of solutions to time dependent problems frequently vary considerably over time.Therefore it is necessary and efficient to solve such kind of problems by adaptive methods.Following the idea of[16],we study the adaptive finite element method for the problem.In Section 3.1,we propose the fully discrete scheme and the auxiliary problem.In Section 3.2,we obtain the upper bound of the a posteriori error and a lower bound of the local space error respectively. In Section 3.3,the error control theorem is given to guarantee the reliability of the adaptive algorithm.Finally,in Section 3.4,we prove that the time and space adaptive algorithm in Section 3.3 will stop in a finite number of steps for any given tolerances TOL_time,TOL_space and TOL_coarse.Due to the high efficiency of the adaptive method. ore thcorctical analysis in this chapter has significant meaning in the development of efficient numerical methods for the time delay population model.