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

$$r(x,t)=r(x-3,t)=p(x-2,t)=s(x-1,t)=u(x-vt)?$$

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

$$u''(y)+vu'(y)+u(y)(u(y-2)-u(y-1))=0.$$

Most of the references I've found on DDEs are very numerical in flavor or cover DDEs in rather specific forms.

doexist in general (which rules out any trivial arguments) but their non-negativity is a big issue. So far I haven't managed to find anything non-negative. I prefer $2\pi$-periodic functions with delays $2\pi/3$ and $4\pi/3$ and the same equation. It seems to me like we have enough free parameters to rescale the period.. $\endgroup$ – fedja Dec 2 '17 at 16:34cannotdo that: the non-linear term changes! Itisan issue, and a big one at that. 2) That "convenience" is quite a headache too: so far I have a whole family of (sign changing) solutions with $v<0$ but $v>0$ necessarily pushes us to high frequency ranges since then the multiplier and the derivative term work in the same direction on low frequencies, and handling that requires working in higher dimension than $4$, which I prefer to abstain from on my old dying laptop. $\endgroup$ – fedja Dec 3 '17 at 18:12arbitrary$v$ then :-) Give me some time and check this thread now and then. Have a good trip! $\endgroup$ – fedja Dec 3 '17 at 18:39