Spreading and vanishing in a diffusive prey-predator model with variable intrinsic growth rate and free boundary. (English) Zbl 1381.35253
Summary: 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 \(t \to \infty\) or \(x \to \infty\). 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 \((u, v)\) for the vanishing case (\(h_\infty < \infty\)). Then we find the criteria for spreading and vanishing. At last, the long time behavior of \((u, v)\) for the spreading case (\(h_\infty = \infty\)) is discussed. Theorems 2.2, 2.3 and 4.1 together establish a spreading-vanishing dichotomy.
MSC:
| 35R35 | Free boundary problems for PDEs |
| 35K91 | Semilinear parabolic equations with Laplacian, bi-Laplacian or poly-Laplacian |
| 35Q92 | PDEs in connection with biology, chemistry and other natural sciences |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35K51 | Initial-boundary value problems for second-order parabolic systems |
| 35K57 | Reaction-diffusion equations |
Keywords:
diffusive prey-predator model; variable intrinsic growth rate; free boundary; spreading and vanishing; long time behaviorReferences:
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