Stochastic prey-predator model with small random immigration

Jawdat Alebraheem; Mogtaba Mohammed; Ismail M. Tayel; Muhamad Hifzhudin Noor Aziz; Jawdat Alebraheem; Mogtaba Mohammed; Ismail M. Tayel; Muhamad Hifzhudin Noor Aziz

doi:10.3934/math.2024725

AIMS Mathematics

2024, Volume 9, Issue 6: 14982-14996. doi: 10.3934/math.2024725

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Stochastic prey-predator model with small random immigration

1.
Department of Mathematics, College of Science, Majmaah University, Majmaah 11952, Saudi Arabia
2.
University of Gezira, College of Mathematical Sciences and computer Wad Medani, Sudan
3.
Institute of Mathematical Sciences, Faculty of Science Universiti Malaya 50603, Kuala Lumpur, Malaysia

Received: 24 January 2024 Revised: 18 March 2024 Accepted: 03 April 2024 Published: 25 April 2024
MSC : 92-11, 60H10, 34D20

In this paper, we introduce a novel stochastic prey-predator model under random small immigration. Mainly, we prove boundedness for the solution of the model using probabilistic and analytic types of inequalities. Furthermore, possible conditions on the immigration for achieving stochastic square stability are obtained. The immigration of both prey and predator is assumed to be either constant and stochastically perturbed or proportional to the population and stochastically perturbed. In all cases, we arrived at the fact that stability can only be achieved if the immigration is small enough. We also show that as random immigration increases, the dynamic becomes destabilized and could lead to chaos. Lastly, we perform a computational analysis in order to verify the obtained theoretical results.
- stochastic stability,
- prey-predator model,
- boundedness,
- small random immigration
Citation: Jawdat Alebraheem, Mogtaba Mohammed, Ismail M. Tayel, Muhamad Hifzhudin Noor Aziz. Stochastic prey-predator model with small random immigration[J]. AIMS Mathematics, 2024, 9(6): 14982-14996. doi: 10.3934/math.2024725

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Abstract

In this paper, we introduce a novel stochastic prey-predator model under random small immigration. Mainly, we prove boundedness for the solution of the model using probabilistic and analytic types of inequalities. Furthermore, possible conditions on the immigration for achieving stochastic square stability are obtained. The immigration of both prey and predator is assumed to be either constant and stochastically perturbed or proportional to the population and stochastically perturbed. In all cases, we arrived at the fact that stability can only be achieved if the immigration is small enough. We also show that as random immigration increases, the dynamic becomes destabilized and could lead to chaos. Lastly, we perform a computational analysis in order to verify the obtained theoretical results.

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