Integrability of a Family of Lotka--Volterra Three Species Biological System

TL;DR

This paper investigates the integrability of a three-species Lotka-Volterra system, identifying conditions for integrability and demonstrating non-existence of certain integrals in chaotic parameter regimes.

Contribution

It provides a comprehensive analysis of the system's integrability, showing it is integrable when two parameters are zero and non-integrable otherwise, with detailed algebraic surface analysis.

Findings

System is integrable when two parameters are zero.

Non-existence of polynomial, rational, and Darboux integrals for positive parameters.

Chaotic behavior correlates with non-integrability.

Abstract

The aim of this study is to analyze the integrability problem of Lotka--Volterra three species biological system. The system which considered in this work is a biological plausibility or a chemical model. The system has a complex dynamical behavior because it is chaotic system. We, first show that the system is a complete integrable when two of the involved parameters in the system are zero. Second, thorough invariant algebraic surfaces and exponential factors, the nonintegrability problems have been investigated. Particularly, we show the non-existence of polynomial, rational, formal series, and Darboux first integrals when parameters are strictly positive.