Traveling Waves And Speed Of Asymptotic Spreading For A Time Periodic Lotka-volterra Cooperative System

In population dynamics,it is very common to consider the effect of periodicity caused by seasonal factors.Comparing with the case of constant coefficients,the study on pe-riodic systems is not abundant.In the current thesis,we study the speed of asymptotic spreading and traveling wave solutions of a Lotka-Volterra type cooperative system with time period coefficients.Firstly,we study the existence and nonexistence of the periodic traveling wave solu-tions for this system,where the traveling wave solutions describe an evolutionary process that two cooperative species invade a new habitat.By sub-and super-solutions and monotone iteration technique,we transform the existence of the traveling wave solution to the existence of a pair of sub-and super-solutions.By constructing proper upper and lower solutions,we obtain the existence of the traveling wave solutions connecting 0 with the positive periodic solution when the wave speed is large.In addition,the nonexistence of traveling wave solutions is confirmed by the theory of asymptotic spreading.Secondly,we try to formulate the long time behavior of Cauchy problem by traveling wave solutions,which is the stability of traveling wave solutions.With proper initial value,we describe the long time behavior of the initial value problem by the squeezing technique based on the comparison principle.These results show that the time period traveling wave solution can still determine the long time behavior of the corresponding initial value problem.Finally,we investigate the asymptotic spreading of the system by the well known theory of asymptotic spreading of scalar equations and comparison principle.When two cooperative species are invaders,our conclusion indicates that one species will have a spreading speed that is larger than the uncoupled case.When the new species is intro-duced into the habitat and the other species is the aborigine,it is proved the new species will have a spreading speed that is faster than the uncoupled case.Both cases imply the nontrivial effect of coupled nonlinearity in asymptotic spreading.