Modified Lotka Volterra Model with Perspectives of the Piecewise Derivative

TL;DR

The paper addresses modeling predator–prey dynamics with memory and crossover behaviors by extending the Lotka–Volterra system with piecewise derivatives across multiple kernels. It introduces a formal framework of piecewise fractional operators including Caputo $_a^{C}D_t^{\delta}$, Atangana-Baleanu $_a^{ABC}D_t^{\delta}$, and Caputo-Fabrizio $_a^{CF}D_t^{\delta}$, plus several piecewise integrals, to generate three case studies with power-law, Mittag-Leffler, and fading-memory kernels. The authors derive equilibria and stability criteria, prove a Lipschitz-based theorem guaranteeing uniqueness for the fractional LV system, and apply Newton-polynomial-based numerical schemes to obtain piecewise-solution trajectories. Numerical experiments illustrate chaotic dynamics and crossovers, highlighting memory effects and stochastic perturbations in predator–prey interactions, with potential relevance for ecological modeling under nonlocal memory and randomness.

Abstract

This study uses the Lotka Volterra Predator-Prey model to offer a notion of piecewise patterns for the various piecewise derivatives. Using the piecewise derivatives, we produced numerical solutions that are referred to as the Adams-Bashforth method. The computer results show piecewise patterns in the Lotka Volterra Predator-Prey model's real-world behaviours. The Lotka-Volterra model looks into the relationships between competition and abundance between two competing species. Changes in the abundance of one species are modelled as a function of the abundance of its competitors, but the competitive mechanism is given and evaluated. This notion led some scholars to label certain mathematical expressions as "phenomenological" and to propose a different theoretical framework that gives resources special consideration. The Lotka-Volterra model, often called the predator-prey model or the Lotka-Volterra model, is a nonlinear mathematical expression that is frequently used to analyse the dynamical behaviours of biological systems in which two species interact, one as a predator and the other as prey. The variation in the populations over time is illustrated by the mathematical statement.

Figures (12)

• Figure 1: $\delta=0.91$.
• Figure 2: $\delta=0.91$.
• Figure 3: $\delta=0.91$.
• Figure 5: $\delta=0.8$.
• Figure 6: $\delta=0.8$.