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

Safia Slimani; Paul Raynaud de Fitte; Islam Boussaada

doi:10.3934/dcdsb.2019042

Article Contents

2019, Volume 24, Issue 9: 5003-5039. Doi: 10.3934/dcdsb.2019042

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

1.
Normandie Univ, Laboratoire Raphaël Salem, UMR CNRS 6085, Rouen, France
2.
PSA & Inria DISCO & Laboratoire des Signaux et Systèmes, Université Paris Saclay, CNRS-CentraleSupélec-Université Paris Sud, 3 rue Joliot-Curie, 91192 Gif-sur-Yvette cedex, France

^* Corresponding author: P. Raynaud de Fitte
^* Corresponding author: P. Raynaud de Fitte

Received: June 2018

Revised: September 2018

Early access: February 2019

Published: July 2019

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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Abstract

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.

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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} $

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

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Figure 3. Hopf bifurcation of the system (1.2)

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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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