A free boundary problem of the ratio-dependent prey-predator model

Abstract

This paper deals with a free boundary problem for the ratio-dependent prey–predator model over a one-dimensional habitat, in which the free boundary represents the spreading front and is caused only by predator. In this problem, it is assumed that the species can only invade further into the new environment from the right end of the initial region, and the spreading front expands at a speed that is proportional to the predator’s population gradient at the front. The main objective is to realize the dynamics/variations of the prey, predator, and the free boundary. We prove a spreading–vanishing dichotomy for this model, namely the predator species either successfully spreads to infinity as at the front and survives in the new environment, or it fails to establish and dies out in the long run while the prey species stabilizes at a positive equilibrium state. The long-time behavior of solution and criteria for spreading and vanishing are obtained.

Keywords:

• ratio-dependent prey–predator model
• free boundary problems
• spreading–vanishing dichotomy
• long-time behavior

AMS Subject Classifications:

• 35K51
• 35R35
• 92B05
• 35B40

Notes

This work was supported by NSFC [grant number 11371113].