Behaviors Of The Solutions Of Two Species Mutualistic Models With Diffusion

With the development of modern science and technology,mathematics has been extensively applied both in natural sciences such as engineering,medicine, ecology,and in social sciences such as economics,finance.Especially,mathematics is playing an increasing important role in ecology.Ecology produces interesting problems,mathematics provides models and ways to understand them. and ecology returns to verify the mathematical models.A great deal of research has been done to sophisticated models in ecology,most models can be described mathematically by nonlinear parabolic and elliptic partial differential equations.This presentation is devoted to reaction diffusion systems describing ecological models.To make it more readable and systematic,we concerns primarily with two-species mutualistic models and the qualitative properties of these models are extensively studied.It consists of five parts.In the first part,we briefly introduce the background and history about the related work.The second part is concerned with a system of semilinear parabolic equations with a free boundary,which arises in a mutualistic ecological model.The local existence and uniqueness of a classical solution are obtained by straightening the free boundary and using Schauder fixed point theorem.The global existence of the solution is given by establishing a priori estimates.The asymptotic behavior of the free boundary problem is studied.Our results show that the free boundary problem admits a global slow solution if the inter-specific competitions are strong, while if the inter-specific competitions are weak there exist the blowup solution and global fast solution.The third part deals with strongly-coupled elliptic systems with self-diffusion and cross-diffusion.To overcome the difficulty caused by cross-diffusion,we will change the strongly-coupled problem into a weakly-coupled problem by using a translation.The existence of coexistence follows from the upper and lower solutions and corresponding iterations.Our results show that this strongly coupled mutualistic model possesses at least one coexistence state if the birth rate is big and self-diffusions and intra-specific competitions are strong.Finally,a numerical simulation is presented to illustrate the main results.Because of the periodicity of the birth and death rates:rates of interactions and environmental carrying capacities on seasonal scale.nonlinear periodic diffusion equations arise naturally in ecological models.In the forth part,the cooperating two-species Lotka-Volterra model with periodic coefficients is discussed. The existence and asymptotic behavior of T-periodic solutions for the periodic reaction diffusion system under homogeneous Dirichlet boundary conditions are investigated.We show that the problem admits a maximal T-periodic solution and a minimal T-periodic solution.An numerical simulation is also given to illustrate the periodicity of the model.The blowup theory of nonlinear diffusion equations is also one of important contents.The fifth part studies a parabolic system with diffusion and selfdiffusion in a two species mutualistic model.The global existence of the solution is given first.For the sake of convenience,we present the blowup result results for the parabolic system without self-diffusion.Then we give the blowup result of the corresponding parabolic equation with self-diffusion and finally the sufficient conditions are given for the parabolic system with self-diffusion to blow up in finite time by complex calculations.An numerical simulation is also given to illustrate the blowup results.Finally,by summing up the given conclusions:we try to make further consideration for future research.