Research On A Kind Of Reaction-diffusion Problem With Nonlinear Density-dependent Terms

Biological mathematics is an interdisciplinary subject between mathematics and biology.Since 1798,Malthus proposed a mathematical model for studying population growth,mathematical models have become an important tool to solve problems in ecology.Mathematical theories along with research methods are widely implemented in such areas to model and explain some ecological phenomenon.Population ecology is a subject that investigates the birth,death,competition,predation and reciprocity of a population.Mutualism exists widely in the ecological environment.In nature,the interaction between two mutualistic populations is very complex,due to the effect of linear and nonlinear population density.At the same time,researchers have found that spatial diffusion is also an important factor of affecting the population's behavior,owing to the mobility of the population for food and reproduction.Considering the effect of diffusion and nonlinear density constraint,the paper is devoted to the qualitative analysis of a two-species mutualistic model by some theories and methods of partial differential equation.In addition,due to the influence of periodic factors such as seasons and biological habits,the population may also have periodic phenomena.The paper consists of the following sections.In chapter 1 we briefly introduce the background and the development about the related work,and present the main research contents of the thesis.In chapter 2 as the preliminary knowledge,several basic definitions are firstly introduced,Then the existence and uniqueness of the global solution are proved based on the prior estimation and maximum principle.Finally,the consistent boundedness of the solution is derived using the comparison principle.Chapter 3 deals with the long time behavior of the solution.The local stability of each equilibrium point is discussed by using the characteristic decomposition and linearization.Then the globally asymptotic stability of its positive equilibrium is presented by upper and lower solutions.Finally,it is proved that the population density has infinite grow quality when time increases infinitely.Chapter 4 discusses the mutualistic model with periodic coefficient.The existence of periodic solution and stability of initial value problem are proved by using the upper and lower solutions,as well as the corresponding iteration.In Chapter 5 numerical simulations by Matlab software have been presented to both the mutualistic diffusive problems with constant coefficients and the corresponding initial value problems with periodic coefficients.Numerical simulation is used to further illustrate the correctness of above theoretical results.Chapter 6 summarizes the main work studied in this thesis,and gives its biological explanation.Based on this,we also present further considerations.