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. 



Pursuit solutions to the Rock-paper-scissors flow and delay differential equations
Damn it! Such a nice equation with such an interesting behavior that I just started to understand after a few days of thinking, and the stupid positivity condition spoils all the fun. Why cannot we just trade this world for one in which burning a negative amount of fuel in a negative amount of oxidizer would produce a positive amount of heat? 


Anyway, write $a(x)=u(x), b(x)=u(x-1), c(x)=u(x-2)$. Then $a''+va'=a(b-c)$, etc. If $a,b,c>0$, we can divide and write 
$$
\frac{a''}a+v\frac{a'}a=b-c\,
$$ 
i.e.
$$
(\log a)''+\frac{(a')^2}{a^2}+v(\log a)'=b-c\,,
$$
etc.
Adding these three equations up and integrating over the period, we conclude that 
$$
\int\left(\frac{(a')^2}{a^2}+\frac{(b')^2}{b^2}+\frac{(c')^2}{c^2}\right)=0\,,
$$
so $a,b,c=\operatorname{const}$.


Case closed to the total dissatisfaction of everyone involved. :-(


