On a Lotka–Volterra weak competition system with Robin and free boundary conditions

Abstract

We consider a Lotka–Volterra type weak competition model with Robin boundary conditions on the left and free boundary conditions on the right for both species. We aim to study the long time behavior of the solution and the criteria for spreading and vanishing. Under the vanishing happens (s∞≔limt→∞s(t)<∞), one species disappears and the long-term behavior of the other species is determined by the size of s∞: there exists a positive constant R1 such that the species disappears when s∞≤R1; when s∞>R1, the solution of the equation converges to the solution of the corresponding elliptic problem. For the spreading happens (s∞=∞), we get the upper and lower bounds for the solution. We also demonstrate the criteria for spreading and vanishing.