Selection of the angular speed of rotating waves in segregated reaction-diffusion systems with asymmetric competition

We investigate the existence of segregated rotating waves, arising in the singular limit of competition-diffusion systems of the type \[ \partial_t u_i -\partial_{xx} u_i = f(u_i)-\beta u_i \sum_{j \neq i} a_{ij} u_j,\qquad x\in\mathbb{S}^1,\ t>0, 1\le i,j\le k, \] as $\beta\to+\infty$.

Here $k\ge3$, the reaction $f$ is of Fisher-KPP (logistic) type, and the competition coefficients $a_{ij}>0$ are not necessarily symmetric.

Assuming that, for every $i$, \[ \dfrac{a_{i+1,i}}{a_{i,i+1}}=\lambda>0, \] we provide a complete characterization of the rotating waves enjoying an equivariant structure, where each density is a suitable rotation of any other one: such waves exist if and only if $\lambda$ belongs to an explicit range, in which case the angular velocity $\omega=\omega(\lambda)$ is uniquely prescribed, as is the rotating profile.

In particular, stationary solutions (with $\omega=0$) exist only in the symmetric case $\lambda=1$.

This marks a strong difference with the same problem with either Dirichlet or Neumann boundary conditions, where it is known that no periodic in time solution exists, also in the asymmetric case, sheding more light on some conjectures and open problems concerning the long time behavior of competition-diffusion systems.