A Lotka-Volterra type model analyzed through different techniques

TL;DR

This paper analyzes a modified Lotka-Volterra predator-prey model using various techniques, including stability analysis, numerical schemes, and fractional derivatives, providing insights into its mathematical properties and solution behaviors.

Contribution

It introduces a fractional derivative extension of the Lotka-Volterra model and investigates its well-posedness, stability, and solution existence, comparing different analytical and numerical methods.

Findings

Model is well-posed with non-negativity and conservation laws

Numerical schemes Euler and Mickens are effective for the model

Fractional model solutions are well-posed with proven existence and uniqueness

Abstract

We consider a modified Lotka-Volterra model applied to the predator-prey system that can also be applied to other areas, for instance the bank system. We show that the model is well-posed (non-negativity of solutions and conservation law) and study the local stability using different methods. Firstly we consider the continuous model, after which the numerical schemes of Euler and Mickens are investigated. Finally, the model is described using Caputo fractional derivatives. For the fractional model, besides well-posedness and local stability, we prove the existence and uniqueness of solution. Throughout the work we compare the results graphically and present our conclusions. To represent graphically the solutions of the fractional model we use the modified trapezoidal method that involves the modified Euler method.