Sympatric speciation by symmetry-breaking: The three-clade case

TL;DR

This paper extends the concept of symmetry-breaking speciation to three clades, analyzing how populations diverge under certain dynamical laws and identifying conditions for stable multi-clade states.

Contribution

It generalizes symmetry-breaking speciation to three-clade systems using polynomial-based differential equations and provides strategies to determine steady states and their stability.

Findings

Higher coupling reduces the number of stable states.

Larger populations increase the likelihood of three-clade divergence.

The model's limitations are discussed when zero roots occur.

Abstract

In this paper we expand the concept of biological speciation by symmetry breaking of Golubitsky and Stewart to the case of three clades in which N populations following the same dynamical laws can separate. The underlying differential equation is based on a fifth order polynomial of a trait variable with first or second order coupling. We present some general strategies to find all possible steady states and their stabilities. Numerical data are given for a specific system. We show the locations of three-clade distributions in dependence on the coupling and an environmental parameter. The results show a decrease of the number of stable states with higher coupling and a higher probability of ending in a three-clade state for larger N. Limits and potentials of the approach if zero roots for the trait variable occur are discussed.