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Dynamics of a prey-predator system with modified Leslie-Gower and Holling type Ⅱ schemes incorporating a prey refuge

  • * Corresponding author: P. Raynaud de Fitte

    * Corresponding author: P. Raynaud de Fitte 

The first author is supported by TASSILI research program 16MDU972 between the University of Annaba (Algeria) and the University of Rouen (France)

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  • We study a modified version of a prey-predator system with modified Leslie-Gower and Holling type Ⅱ functional responses studied by M.A. Aziz-Alaoui and M. Daher-Okiye. The modification consists in incorporating a refuge for preys, and substantially complicates the dynamics of the system. We study the local and global dynamics and the existence of cycles. We also investigate conditions for extinction or existence of a stationary distribution, in the case of a stochastic perturbation of the system.

    Mathematics Subject Classification: Primary: 92D25, 34D23, Secondary: 60H10.


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  • Figure 1.  A phase portrait of (1.2) with three equilibrium points and a cycle in the interior of $ \mathcal{A} $. The dashed lines are isoclines $ y = \frac{x(1-x)(k_1+x-m)}{a(x-m)} $ and $ y = k_2+x-m $. The grey region is the invariant attracting domain $ \mathcal{A} $

    Figure 2.  A phase portrait of (1.2) with an unstable equilibrium and a stable limit cycle

    Figure 3.  Hopf bifurcation of the system (1.2)

    Figure 4.  Solutions to the stochastic system (1.3) and the corresponding deterministic system, represented respectively by the blue line and the red line

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