General Properties of a System of $S$ Species Competing Pairwise

TL;DR

This paper analyzes a general class of pairwise interacting multi-species systems, revealing fundamental properties, fixed points, and the role of the interaction matrix, with implications for understanding complex ecological and evolutionary dynamics.

Contribution

It generalizes previous models to arbitrary species number, deriving simple rate equations and identifying key properties related to the interaction matrix's null space.

Findings

Difference between even and odd number of species cases

Criteria for existence of fixed points and invariant variables

Relation of Lotka-Volterra equations to pairwise interactions

Abstract

We consider a system of $N$ individuals consisting of $S$ species that interact pairwise: $x_m+x_\ell \rightarrow 2x_m\,\,$ with arbitrary probabilities $p_m^\ell $. With no spatial structure, the master equation yields a simple set of rate equations in a mean field approximation, the focus of this note. Generalizing recent findings of cyclically competing three- and four-species models, we cast these equations in an appealingly simple form. As a result, many general properties of such systems are readily discovered, e.g., the major difference between even and odd $S$ cases. Further, we find the criteria for the existence of (subspaces of) fixed points and collective variables which evolve trivially (exponentially or invariant). These apparently distinct aspects can be traced to the null space associated with the interaction matrix, $p_m^\ell $. Related to the left- and right- zero-eigenvectors, these appear to be "dual" facets of the dynamics. We also remark on how the standard Lotka-Volterra equations (which include birth/death terms) can be regarded as a special limit of a pairwise interacting system.