Stochastic dynamics in finite populations: Difference between revisions

Stochastic dynamics in finite populations (view source)

328 bytes added , 11 March 2012

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 [[Image:RSP - ODE Mutations.svg|left|200px|link=EvoLudoLab: Rock-Paper-Scissors - ODE with Mutations]]
-==== [[EvoLudoLab: Rock-Paper-Scissors - ODE with Mutations|Replicator-Mutator dynamics]]====
+==== [[EvoLudoLab: Rock-Paper-Scissors - ODE with Mutations|Replicator-Mutator dynamics - Stable limit cycle]]====
-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>.
+For <math>s<1</math> the interior fixed point <math>\hat x</math> is an unstable focus. The trajectories spiral away from <math>\hat x</math> and, in the absence of mutations, approach the heteroclinic cycle along the boundary of the simplex <math>S_3</math>. With mutation rates <math>\mu>0</math>, however, the boundary of <math>S_3</math> becomes repelling, which can give rise to stable limit cycles. If the mutation rate is sufficiently high, the interior fixed point is stable again. The image shows a sample trajectory for <math>s=0.2</math>, <math>\mu=0.001</math>.
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