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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.

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    $\begingroup$ The problem is that non-trivial solutions do exist 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 16:34
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    $\begingroup$ A stupid question. $u>0$ is an understandable condition. But why do you insist on $v>0$? $\endgroup$ – fedja Dec 2 '17 at 18:08
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    $\begingroup$ 1) You obviously cannot do that: the non-linear term changes! It is an 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 18:12
  • $\begingroup$ My mistake re adding constants, result of answering this while travelling. $v$ being positive is really just convenience: these solutions are travelling waves, the sign of $v$ determines the direction of travel. $\endgroup$ – Jess Boling Dec 3 '17 at 18:32
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    $\begingroup$ OK, will try to find a positive solution with arbitrary $v$ then :-) Give me some time and check this thread now and then. Have a good trip! $\endgroup$ – fedja Dec 3 '17 at 18:39
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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. :-(

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  • $\begingroup$ Perhaps there are some cohomogeneity one solutions on interesting geometries. Thank you. $\endgroup$ – Jess Boling Dec 10 '17 at 21:44
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    $\begingroup$ @JessBoling Not in the finite volume setting. You can apply the same argument directly to the original system and conclude that $\int_M \log(prs)$ is decreasing. And in the infinite volume setting the solutions are spreading to infinity due to the heat evolution for $p+r+s$. So, nothing periodic is anywhere in sight (unless you accept sign changing solutions) $\endgroup$ – fedja Dec 11 '17 at 0:12
  • $\begingroup$ That leads me to think a large class of positive initial conditions will diffuse toward constant (in space) solutions of the reaction ODE. $\endgroup$ – Jess Boling Dec 11 '17 at 6:22
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    $\begingroup$ @JessBoling Yep, and the constants will cycle over the curve conserving the sum and the product. All that we need to show for that is that we cannot send one of the components to $0$ as time tends to infinity. Do you know if that can happen? $\endgroup$ – fedja Dec 11 '17 at 7:08
  • $\begingroup$ In compact settings the maximum principle applied to the evolution of $\log⁡(rps)$ would rule that out, so long time convergence to the ODE is the way of the game. $\endgroup$ – Jess Boling Dec 12 '17 at 21:34

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