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

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, I explore how to build reduced 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

Figures (5)

• Figure 1: The fraction of 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.
• Figure 3: Approximate algebraic model for $\beta=3$, $\alpha=1$.
• Figure 4: Approximate integral model for $\beta=2$, $\alpha=1$.
• Figure 5: Approximate integral model for $\beta=3$, $\alpha=1$.

Theorems & Definitions (18)

• Example 3.1
• Example 3.2
• Remark : Resulting functional form
• Remark : $(\beta,\beta-1)-reducibility$
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Remark : Entanglement
• Theorem 3.3