Mathematical Modelling of a Delay Prey-Predator System using Caputo Fractional Derivatives

Abstract

In this paper introduces a nonlinear fractional-order prey–predator model to describe the interaction between one prey species and two predator species in an ecological system. These model are formulated with the use of Caputo fractional derivative with time delay to represent memory and hereditary effects that naturally arise in biological processes. The model includes prey growth dynamics and nonlinear interaction terms among the three species. Local stability of the system is examined at different equilibrium points using the Jacobian matrix method. Numerical solutions of the fractional delay system are obtained using the fractional Adams–Bash forth –Moulton predictor–corrector scheme. The influence of different fractional orders 0 < ? <1is investigated and the results are compared with those of the classical integer-order model. Validation is carried out through comparisons with the fourth-order Runge – Kutta and classical Adams–Bash forth–Moulton methods. Numerical simulations confirm the accuracy, stability, and effectiveness of the proposed approach.

Introduction

Applied mathematics plays a crucial role in modelling natural and socio-economic systems, particularly in ecology, where predator–prey models are widely used to study population dynamics, stability, and long-term behaviour. Classical models such as the Lotka–Volterra system have been extended to incorporate realistic factors including harvesting, time delays, nonlinear functional responses, and multi-species interactions. Numerous studies have analyzed these extensions, focusing on optimal harvesting strategies, equilibrium stability, and numerical solution techniques. More recently, fractional-order models have gained attention because they capture memory and hereditary effects that cannot be represented by integer-order systems.

Motivated by these developments, the present work formulates a fractional-order delayed prey–predator model consisting of one prey and two predator populations. The model incorporates harvesting effects, constant time delay, and memory effects using the Caputo fractional derivative of order 0<ω<10 < \omega < 10<ω<1. Fractional calculus is employed to provide a more realistic description of population dynamics through nonlocal temporal behaviour.

The study conducts a local stability analysis of the equilibrium points of the corresponding delayed Lotka–Volterra system using the Jacobian matrix approach. Four equilibrium points are identified: the trivial extinction state, prey-only equilibrium, prey–first predator coexistence, and prey–second predator coexistence. The analysis shows that the extinction and prey-only equilibria are unstable, the prey–first predator equilibrium is locally asymptotically stable under specific parameter conditions, and the prey–second predator equilibrium is not asymptotically stable.

To solve the fractional-order system numerically, the Fractional Adams–Bashforth–Moulton (FABM) predictor–corrector method is employed. This method is derived using the Volterra integral formulation of fractional differential equations and is well suited for handling delay and memory effects. Numerical simulations are performed and validated by comparison with the classical fourth-order Runge–Kutta (RK4) method.

The results demonstrate that the FABM method provides accurate, stable, and computationally efficient solutions for the fractional delayed prey–predator model. Overall, the study highlights the effectiveness of fractional-order modelling and advanced numerical schemes in capturing complex ecological dynamics involving delays, harvesting, and memory effects.

Conclusion

In this paper, a delayed Caputo fractional-order prey–predator model was analyzed to describe population interactions with memory and time-delay effects. The local asymptotic stability of the equilibrium points was examined using the Jacobian matrix approach. Numerical solutions were obtained by employing the Fractional Adams–Bash forth–Moulton (FABM) predictor–corrector method. The numerical simulations demonstrate that the FABM scheme produces stable and accurate approximations for the nonlinear fractional system considered. The results confirm that fractional-order models offer a robust mathematical framework for studying complex dynamical behaviours in prey–predator interactions that are not adequately captured by classical integer-order formulations.

References

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Copyright © 2026 B. Priyadharshini , D. Sivakumar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.