The Propagation Dynamics Of A Nonlinear Competitive Lotka-Volterra Model

In this thesis,we mainly study the propagation dynamics of a Lotka-Volterra reactiondiffusion competition system with nonlinear competition terms.The main contents are as follows:The first chapter introduces the research background,significance,mathematical symbols and preliminary knowledge used throughout this thesis.In the second chapter,we study the existence and linear stability of the constant equilibria.By analyzing the existence of zero points of the functions and using stability theory,we obtain the existence and linear stability conditions of the constant equilibria of the system in five cases.In the third chapter,we study the Lyapunov stability of the bistable traveling wave solution by using the comparison principle and the upper-lower solution method.In the forth chapter,we prove the global exponential stability of the bistable traveling wave solution by a squeezing technique and the comparison principle.Since the system under consideration does not meet the conditions required in the known references,we can overcome the difficulties caused by the nonlinear reaction terms in the two coupled equations by developing new analytical techniques,constructing appropriate upper and lower solutions and establishing appropriate squeezing schemes.In the fifth chapter,we study the asymptotic spreading speed in the case of monostability.The existence conditions of single or multiple spreading speeds are obtained by using the theory of monotone dynamic system.For the case of a single spreading speed,it is proved that the speed is equal to the minimum wave speed of the traveling wave solution,and the explicit conditions for the linear selection of the single spreading speed is obtained by constructing the upper solutions.Furthermore,for the case of multiple spreading speeds,the range of the asymptotic spreading speed of each species and the conditions for the existence of multiple spreading speeds are obtained by using the comparison principle.Finally,the theoretical results are demonstrated by numerical simulation.In the sixth chapter,we summarize the main work and the points of innovation,and put forward the further research.