Unraveling memory effects and chaos control in a fractional-order tri-trophic food web: insights from iso-spike patterns in bi-parametric plane

Abstract

Trophic interactions play a crucial role in the functioning of ecological communities. It influences the species coexistence, biodiversity, and ecosystem stability. Traditional integer-order differential models have been widely used to study these ecosystem dynamics but often overlook the critical role of memory effects and long-range interactions in ecological processes. This study introduces a fractional-order tri-trophic intraguild predation (IGP) model that integrates Caputo derivative to incorporate memory effects in the predator–prey dynamics. The model applies Leslie-Gower and Holling type-II functional responses to describe realistic predation behaviors. A key contribution of this work is the introduction of iso-spike method, a new computational approach to visualize and analyze periodic behaviors in fractional-order ecological systems. It maps the frequency and structure of periodic spikes across a bi-parametric space and identifies shrimp-shaped periodic islands—a distinctive feature in bifurcation diagrams. These islands demarcate regions where chaotic dynamics transition into stable periodic behavior due to the influence of memory. The analysis through one-parameter bifurcation diagrams and two-parameter iso-spike diagram reveals that lower fractional orders (stronger memory) induce point stability, suppressing chaotic oscillations, while higher fractional orders (weaker memory) allow chaotic dynamics to persist. These findings suggest that memory effects, represented by fractional-order derivatives, can play a pivotal role in stabilizing ecological systems. Our simulations are performed on the PARAM SHIVAY supercomputing facility (80 cores, 768 GB RAM), which provide a computationally efficient framework for large-scale exploration of dynamic regimes across extensive parameter spaces. This approach not only advances the field of fractional-order modeling but also has significant implications for real-world ecological management. By integrating memory effects through fractional calculus and employing the iso-spike method, this study opens new avenues for modeling ecological systems with complex feedback loops, long-range dependencies, and memory-driven dynamics. The findings show that fractional-order models can better understand and manage ecosystem stability and resilience against environmental changes. © The Author(s), under exclusive licence to Springer Nature B.V. 2025.