Stochastic non-autonomous Holling type-Ⅲ prey-predator model with predator's intra-specific competition

Sampurna Sengupta; Pritha Das; Debasis Mukherjee

doi:10.3934/dcdsb.2018244

Article Contents

2018, Volume 23, Issue 8: 3275-3296. Doi: 10.3934/dcdsb.2018244

This issue Previous Article Partial control of chaos: How to avoid undesirable behaviors with small controls in presence of noise Next Article Global Kneser solutions to nonlinear equations with indefinite weight

Stochastic non-autonomous Holling type-Ⅲ prey-predator model with predator's intra-specific competition

1.
Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah, West Bengal 711103, India
2.
Department of Mathematics, Vivekananda College, Thakurpukur, Kolkata-700063, India

* Corresponding author: +91 9831022438. E-mail address: prithadas02@math.iiests.ac.in (Pritha Das)
* Corresponding author: +91 9831022438. E-mail address: prithadas02@math.iiests.ac.in (Pritha Das)

Received: December 2017

Revised: February 2018

Early access: August 20, 2018

Published online: August 20, 2018

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Abstract

The objective of this article is to study the significance of dynamical properties of non-autonomous deterministic as well as stochastic prey-predator model with Holling type-Ⅲ functional response. Firstly, uniform persistence of the deterministic model has been demonstrated. Secondly, stochastic non-autonomous prey-predator system with Holling type-Ⅲ functional response is proposed. The existence of a global positive solution has been derived. Sufficient conditions for non-persistence in mean, weakly persistence in mean, extinction have been derived. Moreover the sufficient conditions for permanence of the system have been established. The analytical results are verified by numerical simulation.

Keywords:

Mathematics Subject Classification: 37B55, 92D25, 92-08, 97M10, 60H30.

Citation:

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Figure 1. Numerical simulation for the deterministic system (1) with initial condition (0.2, 0.3) by $a_1(t) = 0.1+0.01 \sin t, \ a_2(t) = 0.02+0.01\sin t$ shows the stable behavior of prey and predator

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Figure 2. Numerical simulation for the deterministic system (1) with initial condition (0.2, 0.3) by $a_1(t) = 2+0.1 \sin t,\ a_2(t) = 1+0.1\sin t$ shows the unstable behavior of prey and predator

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Figure 3. Numerical simulation for the deterministic system (1) with (0.2, 0.3) by $r(t) = 5 + 2.5 \sin t,~b(t) = 0.22+0.02\sin t,~c(t) = 0.01+0.005\sin t,~d(t)$ $ = 0.2+0.01\sin t,~a_1(t) = 0.1+0.1\sin t,~a_2(t) = 1+0.1\sin t$ shows that system is persistent

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Figure 4. Numerical simulation for the system (5) with $\frac{\sigma_1^2}{2} = \frac{\sigma_2^2}{2} = 0.21+0.02\sin t$ shows that both prey and predator population goes to extinction

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Figure 5. Numerical simulation for the system (5) with $\frac{\sigma_1^2}{2} = 0.19+0.02\sin t,~\frac{\sigma_2^2}{2} = 0.09+0.02\sin t$ shows weakly persistence in the mean of prey and extinction of predator

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Figure 6. Numerical simulation for the system (5) with $r(t) = 2.2+0.01 \sin t , \ \sigma_1 = \sigma_2 = 0.02+0.01\sin t$ shows permanence of both prey and predator

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