Construction and application of exact solutions of the diffusive Lotka-Volterra system: a review and new results

TL;DR

This review compiles all known exact solutions of the diffusive Lotka-Volterra system, highlighting their importance in modeling various scientific processes and providing a foundation for testing analytical and numerical methods.

Contribution

It is the first comprehensive review and presentation of exact solutions for the diffusive Lotka-Volterra system, expanding understanding and application across multiple scientific fields.

Findings

Compilation of all known exact solutions to the diffusive Lotka-Volterra system

Demonstration of the applicability of solutions in modeling biological and physical processes

Provision of solutions as benchmarks for testing numerical and analytical methods

Abstract

This review summarizes all known results (up to this date) about methods of integration of the classical Lotka-Volterra systems with diffusion and presents a wide range of exact solutions, which are the most important from applicability point of view. It is the first attempt in this direction. Because the diffusive Lotka-Volterra systems are used for mathematical modeling enormous variety of processes in ecology, biology, medicine, physics and chemistry, the review should be interesting not only for specialists from Applied Mathematics but also those from other branches of Science. The obtained exact solutions can also be used as test problems for estimating the accuracy of approximate analytical and numerical methods for solving relevant boundary value problems.