Stability of oscillations in the spatially extended May-Leonard model

TL;DR

This work analyzes the stability of exact spatially uniform time-periodic oscillations in the diffusive May-Leonard model for three competing species. Using a Floquet framework and perturbation theory, it proves the absence of long-wave modulational instability for the uniform oscillations and derives a universal expression for the eigenvalue splitting in the small-wavenumber limit, showing diffusion-difference controls stability through $R=r\sum (D_i-D_j)^2$. It also demonstrates a finite-wavenumber period-doubling instability in the case $D_u=1$, $D_v=D_w=0$, and shows how additional diffusion can stabilize or destabilize the pattern depending on parameters, with implications for pattern formation near heteroclinic cycles. The results are extended to a broader symmetric class of systems and highlight how conservation laws and invariant manifolds shape spatiotemporal dynamics in competing-species models.

Abstract

The May-Leonard model for three competing species, symmetric with respect to cyclic permutation of the variables and extended by diffusive terms, is considered. Exact time-periodic solutions of the system have been found, and their stability with respect to spatially periodic disturbances is studied. The stability of solu tions with respect to longwave spatial modulations is revealed. A period doubling instability breaking the spatial uniformity is found.

Figures (10)

• Figure 1: Schematic of the system dynamics at $\alpha+\beta=2$. The triangle is the heteroclinic cycle and the closed trajectories on the plane $u+v+w=1$ correspond to periodic solutions with two different values of $A$.
• Figure 2: The solution $(u(t),v(t),w(t))$ for $A=2\cdot 10^{-3}$.
• Figure 3: The dependence of the period $T$
• Figure 4: Plot of the stability parameter $\alpha$ as a function of $x=1/27-A$.
• Figure 5: Dependence