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Stochastic dynamics in finite populations: Difference between revisions
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Stochastic dynamics in finite populations (view source)
Revision as of 13:37, 9 March 2012
, 9 March 2012→Rock-Paper-Scissors game
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==Rock-Paper-Scissors game== | ==Rock-Paper-Scissors game== | ||
[[Image:Stochastic dynamics - noise term Cxx, no mutations.png|300px|thumb| | [[Image:Stochastic dynamics - noise term Cxx, no mutations.png|300px|thumb|Value of the element <math>\mathcal C_{xx}(x,y,z)</math> of the noise matrix <math>\mathcal C(\mathbf x)</math> for <math>d = 3</math> strategies and <math>\mu = 0</math>. <math>\mathcal C_{xx}(x,y,z)</math> determines how the noise in the <math>x</math>-direction affects the <math>x</math>-coordinate. In the case of <math>\mu = 0</math>, this noise vanishes for <math>x\to0</math>. For <math>y\to0</math> and <math>z\to0</math> we recover the usual multiplicative noise from one-dimensional evolutionary processes.]] | ||
Comparisons between the deterministic dynamics in infinite populations, the stochastic dynamics in finite populations and individual based simulations focus on the [[Rock-Paper-Scissors game]] with a generic payoff | |||
<math>\begin{matrix}~&\begin{matrix}\ \ R\quad & S\quad & P\quad\end{matrix} \\ | <math>\begin{matrix}~&\begin{matrix}\ \ R\quad & S\quad & P\quad\end{matrix} \\ | ||
\begin{matrix}R\\S\\P\end{matrix}& | \begin{matrix}R\\S\\P\end{matrix}& | ||
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</math> | </math> | ||
According to the [[replicator equation]] the game exhibits saddle node fixed points at <math>x = 1, y = 1</math>, and <math>z = 1-x-y = 1</math> as well as an interior fixed point at <math>\textstyle\hat{\mathbf x} = \left(\frac12,\frac13,\frac16\right)</math> independent of the parameter <math>s</math>. For <math>s > 1</math>, <math>\hat x</math> is a stable focus and an unstable focus for <math>s<1</math>. In the non-generic case <math>s=1</math> the dynamics exhibits closed orbits. | |||
<math>\hat{\mathbf x} = \left(\frac12,\frac13,\frac16\right)</math> | |||
{{-}} | {{-}} | ||
