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In this paper we investigate a free boundary problem for the diffusive Leslie-Gower prey-predator model with double free boundaries in one space dimension. This system models the expanding of an invasive or new predator species in which the free boundaries represent expanding fronts of the predator species. We first prove the existence, uniqueness and regularity of global solution. Then provide a spreading-vanishing dichotomy, namely the predator species either successfully spreads to infinity as $t\to∞$ at both fronts and survives in the new environment, or it spreads within a bounded area and dies out in the long run. The long time behavior of $(u, v)$ and criteria for spreading and vanishing are also obtained. Because the term $v/u$ (which appears in the second equation) may be unbounded when $u$ nears zero, it will bring some difficulties for our study.

In this paper, a diffusive prey-predator model with strong Allee effect growth rate and a protection zone $\Omega _0$ for the prey is investigated. We analyze the global existence, long time behaviors of positive solutions and the local stabilities of semi-trivial solutions. Moreover, the conditions of the occurrence and avoidance of overexploitation phenomenon are obtained. Furthermore, we demonstrate that the existence and stability of non-constant steady state solutions branching from constant semi-trivial solutions by using bifurcation theory. Our results show that the protection zone is effective when Allee threshold is small and the protection zone is large.

This paper is devoted to study the dynamical properties of a Leslie-Gower prey-predator system with strong Allee effect in prey. We first gives some estimates, and then study the dynamical properties of solutions. In particular, we mainly investigate the unstable and stable manifolds of the positive equilibrium when the system has only one positive equilibrium.

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