Stochastic dynamics in finite populations: Difference between revisions

Stochastic differential equations (SDE) provide a general framework to describe the evolutionary dynamics of an arbitrary number of types in finite populations, which results in demographic noise, and to incorporate mutations. For large, but finite populations this allows to include demographic noise without requiring explicit simulations. Instead, the population size only rescales the amplitude of the noise. Moreover, this framework admits the inclusion of mutations between different types, provided that mutation rates, <math>\mu</math>, are not too small compared to the inverse population size <math>1/N</math>. This ensures that all types are almost always represented in the population and that the occasional extinction of one type does not result in an extended absence of that type. For <math>\mu N\ll1</math> this limits the use of SDE’s, but in this case well established alternative approximations are available based on time scale separation. We illustrate our approach by a [[Rock-Scissors-Paper game]] with mutations, where we demonstrate excellent agreement with simulation based results for sufficiently large populations. In the absence of mutations the excellent agreement extends to small population sizes.

[[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.]]

<center><math>\begin{matrix}~&\begin{matrix}\ \ R\quad & S\quad & P\quad\end{matrix} \\

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.

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>.

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>.