The Lotka-Volterra Dynamical System and its Discretization

TL;DR

This paper compares two discretization methods for the Lotka-Volterra system, demonstrating that Mickens' method preserves the system's properties while Euler's method introduces instability and inconsistency.

Contribution

It provides a comparative analysis of Euler's and Mickens' discretization methods on the classical Lotka-Volterra model, highlighting the advantages of Mickens' approach.

Findings

Euler's method causes numerical instability and dynamic inconsistency.

Mickens' method maintains the original system's properties.

Discretization choice significantly impacts system behavior.

Abstract

Dynamical systems are a valuable asset for the study of population dynamics. On this topic, much has been done since Lotka and Volterra presented the very first continuous system to understand how the interaction between two species -- the prey and the predator -- influences the growth of both populations. The definition of time is crucial and, among options, one can have continuous time and discrete time. The choice of a method to proceed with the discretization of a continuous dynamical system is, however, essential, because the qualitative behavior of the system is expected to be identical in both cases, despite being two different temporal spaces. In this work, our main goal is to apply two different discretization methods to the classical Lotka-Volterra dynamical system: the standard progressive Euler's method and the nonstandard Mickens' method. Fixed points and their stability are analyzed in both cases, proving that the first method leads to dynamic inconsistency and numerical instability, while the second is capable of keeping all the properties of the original continuous model.