Boundedness And Asymptotic Behavior Of Solutions In Two-species Chemotaxis Model

In the natural world,there are many biological phenomena that can be described by some non-linear evolution systems.By studying the nature of systems to predict the evolution process between populations,which has become a hot issue in biomathematics.In the ecosystem,each species is an interdependent relationship,and the evolution process of each population is the result of the participation of multiple species.Therefore,this thesis intends to study some properties of solutions to a chemotaxis systems with two-species and one-stimuli and sensitivity function,including the global existence,boundedness,and asymptotic behavior.The research of this model has certain significance both in terms of mathematical theory and real life.The thesis consists of four chapters.In Chapter one,introduction.It mainly summarizes the actual background and development status of the problems studied in this thesis.Starting from the background of the Keller-Segel chemotaxis model,we summarize the latest research progress and related issues of the model,and briefly state the research work of this thesis.In Chapter two,a parabolic-parabolic-parabolic chemotaxis model of two biological species and a single chemical signal substance with a nonlinear chemotactic sensitivity function is discussed.Under the homogeneous Neumann boundary condition,by constructing the energy function,and then using the parabolic equation thermal semigroup theory and the classic Moser iteration method,it is proved that when the chemotactic sensitivity function and the parameters in the equation meet certain conditions,the system has a unique global bounded solution.In Chapter three,considering the asymptotic behavior of the solutions to the model mentioned above.It is proved the global existence solution of this model is exponentially decaying by constructing Lyapunov function when it meets sensitivity function satisfying some conditions and the initial mass is appropriately small.For large initial mass,globally asymptotic is also obtained.In Chapter four,the main fingdings of this thesis are reviewed,and some questions will be proposed to the future research.