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 β of species may be involved in the coupled dynamics, often only α<β species are of interest or of consequence. In this paper, we explored how to construct models that include only those given α species, but still recreate the dynamics of the original β-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.

Highlights

  • Consider an environment in which a large number of species interact

  • Since, in all of these examples, we assumed some data about the species of interest, we can approximate this integral directly from the data, which is generated according to the original model

  • Some variables may be eliminated in exchange for this extra information about the reduced set

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Summary

Introduction

Consider an environment in which a large number of species interact. This could be, for example, a chemical reaction (with reacting chemical species) or an ecological system (with interacting organisms). In many types of model reduction of systems of differential equations, the goal is to reduce the computational cost (III) while controlling the incurred error (II). There are two defining characteristics of these methods They preserve the correspondence between the set of α species of interest as they appear in the original model and the resulting set after the reduction occurs. This property is termed species correspondence, or correspondence They create a reduced model that contains the exact same information as the original one, but with fewer equations. The resultant model is not necessarily better in a computational sense, this method reveals a path towards model reduction that preserves correspondence, closely approximates the dynamics, and is computationally quite simple.

Background
Exact Dimension Reduction
Approximate Model Reduction
Algebraic Substitutions
Integral Substitutions
Discussion

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