Extinction in stochastic predator-prey population model with Allee effect on prey

Miljana JovanoviĆ; Marija KrstiĆ

doi:10.3934/dcdsb.2017129

Article Contents

2017, Volume 22, Issue 7: 2651-2667. Doi: 10.3934/dcdsb.2017129

This issue Previous Article Long-time behavior of stochastic reaction-diffusion equation with dynamical boundary condition Next Article Existence and stability of periodic oscillations of a rigid dumbbell satellite around its center of mass

Extinction in stochastic predator-prey population model with Allee effect on prey

Miljana JovanoviĆ^, and
Marija KrstiĆ

University of Niš, Faculty of Sciences and Mathematics, Višegradska 33,18000 Niš, Serbia

* Corresponding author: Miljana Jovanović
* Corresponding author: Miljana Jovanović

Received: July 2016

Revised: December 2016

Published: October 2017

The authors were supported by the Grant No 174007 of MNTRS.

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Abstract

This paper presents the analysis of the conditions which lead the stochastic predator-prey model with Allee effect on prey population to extinction. In order to find these conditions we first prove the existence and uniqueness of global positive solution of considered model using the comparison theorem for stochastic differential equations. Then, we establish the conditions under which extinction of predator and prey populations occur. We also find the conditions for parameters of the model under which the solution of the system is globally attractive in mean. Finally, the numerical illustration with real life example is carried out to confirm our theoretical results.

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Mathematics Subject Classification: Primary:60H10, 92D25;Secondary:92D40.

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Figure 1. Deterministic and stochastic trajectories of moose and wolf population described by (2) with parameters (12), $\alpha=0.0002$, $A_1=0.5$, $\sigma_1^2=\sigma_2^2=0.005$

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Figure 2. Left: Deterministic and stochastic trajectories of moose population described by (2) with parameters (12), $\alpha=0.0002$, $A_1=5$, $\sigma_1^2=\sigma_2^2=0.005$; Right: Stochastic trajectories of moose and wolf populations in which we can observe behavior of these populations in 140 years

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Figure 3. Deterministic and stochastic trajectories of moose population described by (2) with parameters (12), $\alpha=0.01$, $A_1=0.5$ and different intensities of noise

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Figure 4. Deterministic and stochastic trajectories of moose and wolf population described by (2) with parameters (12), $\alpha=0.01$, $A_1=5$, $\sigma_1^2=0.14$, $\sigma_2^2=0.005$

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Figure 5. Stochastic trajectories of moose and wolf population described by (2) with three different initial values

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