Rock-Paper-Scissors & Unintended Consequences

I’ve done a bit of research on the concept of cyclic dominance chains in ecological systems (i.e., rock-paper-scissors in the natural world). A lot of our intuition goes out the window when a normal hierarchy of dominance loops back on itself.

Now, this seems unrelated to the idea of unintended consequences, but I want to draw out the connection I’m trying to develop. The idea comes from concepts I learned studying optimal behavior and seriously thinking about what it means to “help a species” survive. Just as a simple thought experiment, suppose you really like Rock.If we remove all of Paper (Rock’s predator) then Scissors (Rock’s prey) will have nothing to predate on. Therefore, removal of Paper leads to extinction of Scissors which leads to extinction of your precious Rocks! Okay… If we overstock the area with Scissors then there will be a cycle where Paper goes near extinction, this lack of predator and abundance of prey should help out Rock immensely.

Let’s find out what may happen with a simple model (and I mean really simplified):

(1)   \begin{align*} \frac{dR}{dt}&=RS-RP,\\ \frac{dP}{dt}&=PR-PS,\\ \frac{dS}{dt}&=SP-SR. \end{align*}

Let R(0)+P(0)+S(0)=1, i.e., the entire area is filled with the combination of the three.  Since \frac{dR}{dt}+\frac{dP}{dt}+\frac{dS}{dt}=0 we have that for any t that R(t)+P(t)+S(t)=R(0)+P(0)+S(0)=1.  This gives us fixed points for the system at (R,P,S)=(1,0,0), (0,1,0), (0,0,1), and (1/3,1/3,1/3) and that the space where the dynamics exist is a triangle

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Each corner of this phase space is a saddle and the coexistence equilibrium is a center. The interpretation of this is that if any population is brought to zero (when we’re on an edge of the phase space) then the system will now exhibit a typical hierarchy and only a single species will persist. Inside the triangle the populations will oscillate, switching off which is dominant in the system.

I’ve been exploring various means that someone may use to “help out” a species. Most of the straightforward ideas actually hinder the species one wants to help.  For instance, if one facilitates the interactions Rock has with Scissors in order to aid the Rock population then an environment where paper is the most fit species has been created (their predator has been reduced and their prey has increased). There are still a lot of questions however with respect to transient and non equilibrium behaviors that need to be explored; stay tuned.

  •  Annor* ASB, Buch* KR, Rodriguez-Pinzon* DA, Lin MR, Okuneye K, Morin BR.  Female Centered Mate Selections as an Explanatory Mechanism for Dimorphic Solutions in a Rock-Paper-Scissors Game. MTBI  Technical Reports, 2015. (link)
  • Merckens J*, van de Geijn B*, Hociota I*, Tadaya D*, Morin B. The Dynamics of a Spatial Cyclic Competition System. MTBI Technical Reports, 2009. (link)
  • Morin B. Rock, Paper, Scissors, Urns and the Male Side Blotched Lizard. Master’s Thesis, Adviser: Mina Ossiander. Oregon State University, May, 2009.