Dynamical analysis of multiplicative environmental noise in a stochastic discrete-time fractional-order delayed predator-prey model

Fairouz Tchier; Shahid Khan; Jongsuk Ro; Maslina Darus; Fairouz Tchier; Shahid Khan; Jongsuk Ro; Maslina Darus

doi:10.3934/math.2026837

AIMS Mathematics

2026, Volume 11, Issue 7: 20577-20605. doi: 10.3934/math.2026837

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Dynamical analysis of multiplicative environmental noise in a stochastic discrete-time fractional-order delayed predator-prey model

Fairouz Tchier ^{1
,},
Shahid Khan ^{2
,},
Jongsuk Ro ^{3
,
,},
Maslina Darus ^{4
,}

1.
Mathematics Department, College of Science, King Saud University, P.O. Box 22452, Riyadh 11495, Saudi Arabia; Email: ftchier@ksu.edu.sa
2.
Department of Mathematics, Abbottabad University of Science and Technology, Pakistan; Email: shahidmaths761@gmail.com
3.
Department of Intelligent Energy and Industry, Chung-Ang University, Seoul 06910, Republic of Korea; Email: jsro@cau.ac.kr
4.
Department of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia; Email: maslina@ukm.edu.my

Received: 23 April 2026 Revised: 14 June 2026 Accepted: 23 June 2026 Published: 13 July 2026
MSC : 34D20, 34D23

The focus of this study was to investigate a discrete-time stochastic predator-prey model. This model included environmental noise, temporal delay, and a nonlinear Ivlev-type functional response. First of all, we proved boundedness, positivity, first, and second moment boundedness, characteristic equation, and Jury conditions. Further, using the characteristic equation and Jury conditions, we determined equilibrium, exponential mean-square stability, and local stability which prevented the system from becoming unstable as a result of random shocks. Additionally, we studied fractional memory effects on the system and demonstrated how the delay parameter exhibited Neimark–Sacker (discrete Hopf) bifurcation. The graphical analysis and the numerical simulations validated the theoretical results.
- discrete-time predator-prey model,
- Ivlev functional response,
- weak Allee effect,
- fractional-order memory,
- Neimark-Sacker bifurcation,
- mean-square stability,
- multiplicative noise
Citation: Fairouz Tchier, Shahid Khan, Jongsuk Ro, Maslina Darus. Dynamical analysis of multiplicative environmental noise in a stochastic discrete-time fractional-order delayed predator-prey model[J]. AIMS Mathematics, 2026, 11(7): 20577-20605. doi: 10.3934/math.2026837

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Abstract

The focus of this study was to investigate a discrete-time stochastic predator-prey model. This model included environmental noise, temporal delay, and a nonlinear Ivlev-type functional response. First of all, we proved boundedness, positivity, first, and second moment boundedness, characteristic equation, and Jury conditions. Further, using the characteristic equation and Jury conditions, we determined equilibrium, exponential mean-square stability, and local stability which prevented the system from becoming unstable as a result of random shocks. Additionally, we studied fractional memory effects on the system and demonstrated how the delay parameter exhibited Neimark–Sacker (discrete Hopf) bifurcation. The graphical analysis and the numerical simulations validated the theoretical results.

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