Dynamic Analysis And Traveling Wave Solutions Of Some Biological Mathematical Models

Biomathematics is an interdisciplinary subject formed by the mutual penetration of mathematics and life science,biology,agriculture,medicine,public health and other disciplines.It uses mathematical methods and techniques to solve specific practical problems in the above application fields,and conducts theoretical research on relevant mathematical methods.Therefore,the theoretical research of biomathematics model from mathematics is a subject worthy of attention and of great significance.The paper focuses on the traveling wave solution of reaction-diffusion system and the dynamic analysis of fractional differential and difference systems.The main work is as follows:In Chapter 1,we describe the background of models,the history and present conditions of the subject and mathematical methods.In Chapter 2,we investigate the system of three species ecological models involving one predator-prey subsystem coupling with a generalist predator with negative effect on the prey.Without diffusion terms,we analyze all the local dynamics of the six equilibrium points of the corresponding reaction equation,as well as the global dynamics of a boundary equilibrium point and a positive equilibrium point.With diffusion terms,we first transform the PDE system into a four-dimensional ODE system,construct a Wazewski set in space,and apply the improved high-dimensional shooting method,Lyapunov function and LaSalle invariance principle to prove the existence of traveling wave solutions.Finally,some biological implications are given and the numerical simulations are performed.In Chapter 3,we study the existence,nonexistence and minimum wave velocity of traveling wave solutions for a chemotactic model with two chemoattractants.In order to prove our main results,we transform the existence of traveling wave solutions for PDE systems to the existence of heteroclinic orbits for the corresponding ODE systems.We construct a positive invariant set in four-dimensional space by using the dynamical system theory,and obtain the existence of heteroclinic orbits in this invariant set.In particular,we analyze the monotonicity of traveling wave solutions.In Chapter 4,we first study the oscillation of the fractional-order difference equation.By using the definition and properties of the fractional-order and difference,as well as inequality techniques,we obtain some sufficient conditions for the oscillation of a fractional-order difference equation.In addition,we give an example to illustrate our main conclusions.Secondly,the global stability of the fractional Lotka-Volterra predator-prey system is studied.By constructing a positive invariant set,it is proved that the solution on the positive invariant set is unique and nonnegative.By using the generalized LaSalle invariance principle and Barbalat lemma,we analyze the global asymptotic stability of the coexistence equilibrium point of the fractional-order system with two predators and one prey of Lotka-Volterra type.In Chapter 5,we mainly summarize the work of this paper,explain our innovation,and put forward the future research prospects.