Bifurcation analysis and dynamic behavior of a discrete predator–prey system incorporating Allee effects and environmental factors

Abdulaziz Almaslokh; A. A. Elsadany; B. Alreshidi; A. S. Zaki; Abdulaziz Almaslokh; A. A. Elsadany; B. Alreshidi; A. S. Zaki

doi:10.3934/math.2026855

AIMS Mathematics

2026, Volume 11, Issue 7: 21069-21090. doi: 10.3934/math.2026855

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Bifurcation analysis and dynamic behavior of a discrete predator–prey system incorporating Allee effects and environmental factors

1.
Department of Mathematics, Faculty of Sciences and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia
2.
Mathematics Department, Faculty of Science, Aswan University, Aswan 81528, Egypt

Received: 09 April 2026 Revised: 26 June 2026 Accepted: 03 July 2026 Published: 16 July 2026
MSC : 37N25, 37N30, 39A28, 39A30

In this paper, we study a discrete-time predator-prey model with a weak Allee effect and multiple environmental factors: Prey refuge, water availability, and moonlight intensity. A major contribution is the biologically inspired synthesis of Allee effects with multiple environmental drivers in a unified discrete-time framework. This class of discrete ecological systems is of considerable theoretical interest due to the possibility of richer dynamics than in their continuous counterparts. The paper provides a detailed study of equilibrium identification, local stability analysis, bifurcation analysis, and numerical simulations. We derive conditions for non-negative equilibria, investigate local stability, and characterize dynamical transitions from stable coexistence to oscillatory and more complex behaviors. In particular, explicit analytical conditions are obtained for the occurrence of two codimension-one bifurcations: The flip (period-doubling) bifurcation, which leads to cycles of period two and higher, and the Neimark–Sacker bifurcation, which gives rise to invariant cycles and quasiperiodic dynamics. A particular strength is the combination of analytical bifurcation arguments with numerical evidence such as bifurcation diagrams, phase portraits, and maximal Lyapunov exponents. The results demonstrate that the synergistic impact of a weak Allee effect threshold, combined with with variables such as prey refuge, water availability, moonlight intensity, and predator interference, significantly enriches the system's dynamic behavior.
- discrete predator–prey model,
- stability analysis,
- environmental factors,
- codimension-one bifurcation
Citation: Abdulaziz Almaslokh, A. A. Elsadany, B. Alreshidi, A. S. Zaki. Bifurcation analysis and dynamic behavior of a discrete predator–prey system incorporating Allee effects and environmental factors[J]. AIMS Mathematics, 2026, 11(7): 21069-21090. doi: 10.3934/math.2026855

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Abstract

In this paper, we study a discrete-time predator-prey model with a weak Allee effect and multiple environmental factors: Prey refuge, water availability, and moonlight intensity. A major contribution is the biologically inspired synthesis of Allee effects with multiple environmental drivers in a unified discrete-time framework. This class of discrete ecological systems is of considerable theoretical interest due to the possibility of richer dynamics than in their continuous counterparts. The paper provides a detailed study of equilibrium identification, local stability analysis, bifurcation analysis, and numerical simulations. We derive conditions for non-negative equilibria, investigate local stability, and characterize dynamical transitions from stable coexistence to oscillatory and more complex behaviors. In particular, explicit analytical conditions are obtained for the occurrence of two codimension-one bifurcations: The flip (period-doubling) bifurcation, which leads to cycles of period two and higher, and the Neimark–Sacker bifurcation, which gives rise to invariant cycles and quasiperiodic dynamics. A particular strength is the combination of analytical bifurcation arguments with numerical evidence such as bifurcation diagrams, phase portraits, and maximal Lyapunov exponents. The results demonstrate that the synergistic impact of a weak Allee effect threshold, combined with with variables such as prey refuge, water availability, moonlight intensity, and predator interference, significantly enriches the system's dynamic behavior.

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