860
edits
Stochastic dynamics in finite populations: Difference between revisions
From EvoLudo
Stochastic dynamics in finite populations (view source)
Revision as of 18:29, 27 March 2012
, 27 March 2012no edit summary
No edit summary |
|||
| Line 4: | Line 4: | ||
Stochastic differential equations (SDE) provide a general framework to describe the evolutionary dynamics of an arbitrary number of strategic types \(d\) in finite populations, which results in demographic noise, as well as 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, \(\mu\), are not too small compared to the inverse population size \(1/N\). 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 \(\mu N\ll1\) this limits the use of SDE’s, but in this case well established alternative approximations are available based on time scale separation. | Stochastic differential equations (SDE) provide a general framework to describe the evolutionary dynamics of an arbitrary number of strategic types \(d\) in finite populations, which results in demographic noise, as well as 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, \(\mu\), are not too small compared to the inverse population size \(1/N\). 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 \(\mu N\ll1\) this limits the use of SDE’s, but in this case well established alternative approximations are available based on time scale separation. | ||
The tutorial on [[ | The tutorial on [[2×2_Games/Stochastic_dynamics|stochastic dynamics in \(2\times2\) games]] covers the simpler case with two strategic types, \(d=2\). Here we focus on the general case with \(d>2\) strategic types and illustrate our approach based on the [[Rock-Scissors-Paper game]] with mutations (\(d=3\)). The stochastic dynamics is in excellent agreement with simulation based results for sufficiently large populations. In the absence of mutations the excellent agreement extends to small population sizes. | ||
''This tutorial complements a series of [[#References|research articles]] by [http://www.evolbio.mpg.de/~traulsen/ Arne Traulsen], [http://www.inb.uni-luebeck.de/~claussen/ Jens Christian Claussen] & [http://www.math.ubc.ca/~hauert/ Christoph Hauert]'' | ''This tutorial complements a series of [[#References|research articles]] by [http://www.evolbio.mpg.de/~traulsen/ Arne Traulsen], [http://www.inb.uni-luebeck.de/~claussen/ Jens Christian Claussen] & [http://www.math.ubc.ca/~hauert/ Christoph Hauert]'' | ||
