Invasion Entire Solutions In A Time Periodic Lotka-Volterra Competition-Diffusion System

Traveling wave solutions and entire solutions for the Lotka-Volterra competition-diffusion system in the autonomous case have been studied intensively, while there are still very few studies devoted to time periodic traveling waves for systems with periodic reaction terms. Unlike the autonomous case, the presence of time periodic nonlinearities poses significant difficulties and requires new approaches. We deal with the time periodic Lotka-Volterra competition-diffusion system and study the so-called entire solutions in the monostable case in this paper.Firstly, by the Harnack’s inequality and the comparison principle coupled with the sub-and supersolution, we study the exponential decay estimate of the time periodic traveling waves as it tends towards its limiting state.Secondly, we establish the exact exponential decay rate of the time periodic traveling waves by employing a dynamical system approach. Using the moving coordinate frame z=x-ct as an evolution variable, we utilize the Laplace transform method and spectral theory to obtain the exponential decay rate.Finally, we study the entire solution. We firstly obtain some estimates by using the exponential asymptotic behaviors, secondly we construct a pair of appropriate (irregular) supersolution and subsolution, we then establish the existence and various qualitative properties of an entire solution by using the comparison argument, which behaves as two pairs of time periodic wave fronts with different speeds coming from the both sides of x-axis. This indicates that the entire solution exhibits the extinction of the inferior species v by the superior one u invading from both sides of x-axis, which provides a new invasion way of u to v other than traveling waves in a time periodic environment.