Exploring the refuge-induced bubbling phenomenon and harvesting in a three species food chain model that incorporates memory effect and odour effect

TL;DR

This paper develops a fractional-order predator-prey model incorporating odour-mediated refuge and harvesting, analyzing stability, bifurcations, and long-term dynamics to understand how memory and odour influence ecosystem stability.

Contribution

It introduces a novel fractional-order three-species food chain model with odour effects, refuge, and harvesting, providing new insights into stability and bifurcation phenomena.

Findings

Prey odour significantly stabilizes the system.

Strong memory enhances system stability.

Bifurcations occur under specific parameter conditions.

Abstract

In this study, an odour-mediated system is developed and studied. In an odor-mediated systems, the sense of smell or odour of species plays a critical role in the interactions between predators and prey. It is widely recognised in scientific literature that these systems are very common and essential across natural ecosystems. These systems are crucial for various behaviors, including foraging, mating, and avoiding predators. In this paper, it is assumed that the presence of prey odour aids the predator in its hunting efforts. It is assumed that both prey and intermediate predators seek refuge against their respective predators upon detecting the odour of their predators. In other words, the odour of predators assists prey species in evaluating the danger and seeking refuge for hiding. This model incorporates the prey species' harvesting as well. We also explore the impact of fading memory on the system by incorporating fractional derivatives into the model. The conditions for both the existence and local stability of the non-negative equilibria are derived. The current model system undergoes both Hopf and transcritical bifurcation when the parameter values are appropriately chosen. The dynamic behaviour of the system is showcased and thoroughly analysed using a range of diagrams, highlighting the impact of prey refuge and predator odour parameters. This paper extensively examines the long-term impacts of harvesting within the system. The extent to which prey odour influences the system is investigated, and it emerges that prey odour can play a significant function within the system. It has been observed that when individuals within the system have a strong memory, it positively affects the stability of the system. Numerical simulations are conducted in order to demonstrate and validate the usefulness of the model being considered, therefore supporting the analytical conclusions.