Memory and mutualism in species sustainability: a time-fractional Lotka-Volterra model with harvesting

TL;DR

This paper introduces a fractional-order predator-prey model with mutualism and harvesting, demonstrating that memory effects stabilize populations and promote sustainable harvesting through dampened oscillations and quicker equilibrium attainment.

Contribution

It extends classical Lotka-Volterra models by incorporating fractional derivatives to account for memory, analyzing stability with mutualism and harvesting, and providing physical explanations for the effects.

Findings

Harvesting is sustainable under the model.

Memory effects dampen oscillations and stabilize populations.

Fractional derivatives lead to quicker equilibrium than integer derivatives.

Abstract

We first present a predator-prey model for two species and then extend the model to three species where the two predator species engage in mutualistic predation. Constant effort harvesting and the impact of by-catch issue are also incorporated. Necessary sufficient conditions for the existence and stability of positive equilibrium points are examined. It is shown that harvesting is sustainable, and the memory concept of the fractional derivative damps out oscillations in the population numbers so that the system as a whole settles on an equilibrium quicker than it would with integer time derivatives. Finally, some possible physical explanations are given for the obtained results. It is shown that the stability requires the memory concept in the model.