The Rock-paper-scissors flow is the following reaction-diffusion system
$$r_t = \Delta r + rs-rp,$$
$$p_t = \Delta p + pr-ps,$$
$$s_t = \Delta s + sp-sr.$$
We can assume $r,p,s\geq 0$, $r+p+s$ is constant, and $1=\int_M (r+p+s) dV$.
Are there travelling wave solutions in one spacial variable of the form
Here $u(y)\geq0$ is some function and $v>0$ is some constant. Such a solution would represent distributions of rock-paper-scissors which pursue each other around a circle.
It suffices to find a non-trivial, three periodic solution to the following non-linear delay differential equation
Most of the references I've found on DDEs are very numerical in flavor or cover DDEs in rather specific forms.