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

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.

Figures (13)

• Figure 1: Bifurcation diagram without coupling. The steady levels are located on the green line and the red and blue parabolas (in colour image). Solid lines indicate stable branches, dashed lines unstable branches. The horizontal light blue line indicates a specific value of $\mu=2.5.$ for which five roots exist in the following order: stable (red), unstable (green), stable (blue), unstable (blue), stable (red).
• Figure 2: Solvability of \ref{['eq:eqcond2']}.
• Figure 3: Statistics of steady states for different parameter values: total number (red), stable ones (blue), stable three-clade states (green). The left column is for first-order coupling, the right column for second-order coupling. In the first row $\mu=1.5$, in the second one $\mu=2.5$ and in the third one $\mu=3.5$.
• Figure 4: Bifurcation diagrams for varying environmental parameter $\mu$. The left column is for first-order coupling with $\beta=1$, the right column for second-order coupling with $\gamma=1$. In the top row, the distance from the origin is used as ordinate, in the bottom row the value of $\sigma_{1}$.
• Figure 5: Top row: the basins of attraction for different stable nodes on the surface of a small sphere around the origin. Green points represent initial data ending in a three-clade state; the blue and cyan points represent points ending in two-clade states, red and yellow indicate one-clade attractors. Bottom row: some trajectories starting at random points on the sphere. Here again the green points represent the family of stable three-clade states and the cyan points other stable nodes; the saddle points are indicated in yellow, the (unstable) origin in purple. The radius of the sphere was $R=0.05$ and we used first-order coupling with $\beta=-0.3$. Left column: $\mu=1.8$. Right column: $\mu=1.85$.