Exact Reduction of the Generalized Lotka-Volterra Equations via Integral and Algebraic Substitutions

TL;DR

This paper presents a method for exactly reducing complex generalized Lotka-Volterra models to focus on a subset of species, preserving the original dynamics and enabling simpler, approximate models.

Contribution

It introduces an exact reduction technique for generalized Lotka-Volterra equations using integral and algebraic substitutions, allowing focus on key species while retaining full dynamics.

Findings

Exact reduction is possible under certain conditions.

Reduced models preserve original dynamics precisely.

The method facilitates simpler approximate models.

Abstract

Systems of interacting species, such as biological environments or chemical reactions, are often described mathematically by sets of coupled ordinary differential equations. While a large number $\beta$ of species may be involved in the coupled dynamics, often only $\alpha < \beta$ species are of interest or of consequence. In this paper, I explore how to build reduced models that include only those given $\alpha$ species, but still recreate the dynamics of the original $\beta$-species model. Under some conditions detailed here, this reduction can be completed exactly, such that the information in the reduced model is exactly the same as the original one, but over fewer equations. Moreover, this reduction process suggests a promising type of approximate model -- no longer exact, but computationally quite simple