Phase-space analysis of large ODE systems using a low-dimensional conservation law

TL;DR

This paper introduces a low-dimensional conservation law approach to analyze large ODE systems modeling multiple populations, offering an efficient alternative to Monte Carlo simulations for biological data analysis.

Contribution

The authors develop a phase-space conservation law framework that simplifies the analysis of complex population dynamics modeled by large ODE systems.

Findings

Applicable to various population models including growth, migration, and competition

Provides explicit or iterative solutions for phase-space conservation laws

Offers a faster alternative to Monte Carlo simulations in biological data analysis

Abstract

Simultaneous deterministic and weakly stochastic dynamics of multiple populations described by a large system of ODE's is considered in the phase space of population sizes and ODE's parameters. We show that many practically interesting problems can be formulated as a low-dimensional phase-space conservation law and solved either explicitly or with simple iterative methods. In particular, we consider: non-interacting populations with unbounded and logistic growth, populations with randomized and biased migration, populations competing for a resource, coexisting species, and populations with phase-space interactions. The method provides an alternative to Monte Carlo simulations and may be useful in the fast analysis of biological data and/or removal of deterministic trends.