Incorporating fractional operators into interaction dynamics of a chaotic biological model

Abstract

In the exploration of mathematical models in biology, the traditional use of integer-order calculus may fall short in adequately capturing elements such as randomness and memory. As a viable alternative, fractional calculus has been embraced by mathematicians, extending derivatives and integrals to non-integer orders to provide more appropriate tools for managing complexity. In this research, we investigate the effectiveness of two well-known fractional operators, namely the Caputo–Fabrizio and Atangana–Baleanu derivatives, in studying the dynamics of a nonlinear food chain model consisting of three populations. This specific biological system exhibits highly chaotic behavior driven by its sensitivity to various parameters. To approximate solutions, we employ two implicit numerical approaches based on Lagrange interpolation and product integration rules. Furthermore, we analyze the equilibrium points of the system and evaluate their stability characteristics, offering insights into the long-term behavior of the model. To verify the presence of chaotic patterns across different parameter values, we employ two chaos detection techniques—the smaller alignment index and the 0-1 test. By incorporating fractional calculus operators into the dynamic equations, we are able to obtain a more precise representation of such biological systems. Future research endeavors should focus on leveraging additional fractional operators to further enhance the accuracy of modeling nonlinear systems in the field of biology.