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

## Abstract

**:**

## 1. Introduction

- I
- Model validation: Does the mathematical model adequately represent the scientific system in question?
- II
- Model reduction: How much error is incurred by the use of the reduced model compared to the original, or high-fidelity, model?
- III
- Computational implementation: Is the reduced model less computationally expensive than the original model and, if so, by how much?

**integral**and

**algebraic**substitution. There are two defining characteristics of these methods. First, they preserve the correspondence between the set of $\alpha $ 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 simply correspondence. Second, they create a reduced model that contains the exact same information as the original one, but with fewer equations. Variables are eliminated without loss of information. This means that the resulting derivatives (after reduction) are equivalent to those of the original system and therefore that the solutions of the original and the reduced equations are the same dynamical system. In terms of the points above, (I) the model is an exact representation of the true system, and (II) the reduction incurs zero error. Although 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.

## 2. Background: The Generalized Lotka–Volterra Equations

## 3. Exact Dimension Reduction

**Definition**

**1**

**Definition**

**2**

**Definition**

**3**

**Example**

**1.**

**Example**

**2.**

**Remark**

**1**

**Remark**

**2**

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Remark**

**3**

**Theorem**

**1.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Conjecture**

**1**

## 4. Approximate Model Reduction

#### 4.1. Algebraic Substitutions

**Example**

**3**

**Example**

**4**

#### 4.2. Integral Substitutions

**Example**

**5**

**Example**

**6**

## 5. Discussion

## Supplementary Materials

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The fraction of the interaction that must be zero for exact reduction decreases nonlinearly as $\alpha \to \beta $.

**Figure 2.**Approximate algebraic model for $\beta =2$, $\alpha =1$. The darker blue band represents the 50% confidence interval (CI) and the lighter blue the 95% CI.

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Morrison, R.E. Exact Reduction of the Generalized Lotka–Volterra Equations via Integral and Algebraic Substitutions. *Computation* **2021**, *9*, 49.
https://doi.org/10.3390/computation9050049

**AMA Style**

Morrison RE. Exact Reduction of the Generalized Lotka–Volterra Equations via Integral and Algebraic Substitutions. *Computation*. 2021; 9(5):49.
https://doi.org/10.3390/computation9050049

**Chicago/Turabian Style**

Morrison, Rebecca E. 2021. "Exact Reduction of the Generalized Lotka–Volterra Equations via Integral and Algebraic Substitutions" *Computation* 9, no. 5: 49.
https://doi.org/10.3390/computation9050049