Spreading and vanishing in a diffusive prey–predator model with variable intrinsic growth rate and free boundary

We study the spreading and vanishing phenomena in a diffusive prey–predator system with variable intrinsic growth rate and free boundary. In this model, the free boundary represents the spreading front and is caused only by the prey, and the variable intrinsic growth rate is allowed to tend to zero and decay “very fast” as t→∞t→∞ or x→∞x→∞. Our main attention is on the effect of variable intrinsic growth rate on the solution and attempt to find some new techniques to deal with the variable intrinsic growth rate. We first study the long time behavior of (u,v)(u,v) for the vanishing case (h∞<∞h∞<∞). Then we find the criteria for spreading and vanishing. At last, the long time behavior of (u,v)(u,v) for the spreading case (h∞=∞h∞=∞) is discussed. Theorem 2.2, Theorem 2.3 and Theorem 4.1 together establish a spreading–vanishing dichotomy.