The Asymptotics Of Several Classes Of Traveling Wave Solutions For Reaction Diffusion Systems With Delays

Lotka-Volterra system is a extremely typical mathematical model in the field of bio-logical mathematics,which lays the foundation of the mathematical theory for the study of interspecific relationships.In this paper,we study the traveling wave solutions of two kinds of Lotka-Volterra competition systems with delays,including a temporally discrete system and a random/nonlocal dispersal system.We first study the existence of traveling wave solutions for two kinds of Lotka-Volterra competitive systems with delays.We first transform the Lotka-Volterra com-petition systems into cooperation systems.By introducing two different monotonicity conditions,we establish the corresponding comparison principle for two kinds of cooper-ation systems.The eigenvalues of the linearized equations,whose monomial coefficient of the crossings terms are zero,of two different classes of systems are solved.The appro-priate upper and lower solutions are established by using the principal eigenvalues.By applying the upper and lower solutions and a limiting argument we obtain that there exist invasion traveling wave solutions with c?c*for the systems.We then study the asymptotic behavior of traveling wave solutions of two kinds of systems with delays.The range of traveling wave solutions of the systems are discussed,and we define the bilateral Laplace transforms for any traveling wave solutions,in the range of C_[0,K]?R,R²?,of the systems.Through the Ikehara?theorem,we establish the exponential asymptotic behavior of the traveling wave solutions of the systems when c?c*and t??.And we obtain the ratio of the derivative of a traveling wave solution to the solution when the time approaches infinity.Finally,this thesis is devoted to the monotonicity of traveling wave solutions and the uniqueness of the waveforms for the systems.We prove that the ratio for any two pairs of traveling wave solutions in C_[0,K]?R,R²?are the same when t??.The relationship between the sizes of two different solutions and the monotonicity of the traveling wave solutions are discussed.Furthermore,by applying the asymptotic behavior of traveling wave solutions and the proof by contradiction,we prove the uniqueness of the waveforms.