An Entire Solution to the Lotka–Volterra Competition-Diffusion Equations

Source abstract

We deal with a system of Lotka–Volterra competition-diffusion equations on $\mathbb{R}$, which is a competing two species model with diffusion. It is known that the equations allow traveling waves with monotone profile. In this article we prove the existence of an entire solution which behaves as two monotone waves propagating from both sides of the x-axis, where an entire solution is meant by a classical solution defined for all space and time variables. The global dynamics for this entire solution exhibits the extinction of the inferior species by the superior one invading from both sides. The proof is carried out by applying the comparison principle for the competition-diffusion equations, that is, using an appropriate pair of a subsolution and a supersolution.