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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 13:52, 9 March 2012
, 9 March 2012→Rock-Paper-Scissors game
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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 | 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} \\ | <center><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}& | ||
\begin{pmatrix}0 & \frac{s}{2} & -1 \\ | \begin{pmatrix}0 & \frac{s}{2} & -1 \\ | ||
-1 & 0 & 2+s \\ | -1 & 0 & 2+s \\ | ||
\frac{1+s}{3} & -1 & 0\end{pmatrix}\end{matrix} | \frac{1+s}{3} & -1 & 0\end{pmatrix}\end{matrix}. | ||
</math> | </math></center> | ||
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. | 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. | ||
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<div class="lab_description SDE"> | <div class="lab_description SDE"> | ||
[[Image:RSP - ODE.svg|left|200px]] | [[Image:RSP - ODE.svg|left|200px]] | ||
==== [[EvoLudoLab: Rock-Paper-Scissors - ODE|Replicator dynamics]]==== | ====[[EvoLudoLab: Rock-Paper-Scissors - ODE|Replicator dynamics]]==== | ||
In the limit <math>N\to\infty</math> with <math>s=1.4</math> and without mutations, <math>\mu=0</math>, <math>\hat x</math> is an attractor of the replicator dynamics. The figure shows a sample trajectory that spirals towards the interior fixed point <math>\hat x</math>. | |||
{{-}} | {{-}} | ||
</div> | </div> | ||
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[[Image:RSP - ODE Mutations.svg|left|200px]] | [[Image:RSP - ODE Mutations.svg|left|200px]] | ||
==== [[EvoLudoLab: Rock-Paper-Scissors - ODE|Replicator-Mutator dynamics]]==== | ==== [[EvoLudoLab: Rock-Paper-Scissors - ODE|Replicator-Mutator dynamics]]==== | ||
Including mutations in the replicator dynamics gives rise to the replicator-mutator dynamics. Mutations destabilizes the interior fixed point <math>\hat x</math> an can give rise to stable limit cycles. The image shows a sample trajectory for <math>\mu=0.002</math>. | |||
{{-}} | {{-}} | ||
</div> | </div> | ||
