860
edits
Stochastic dynamics in finite populations: Difference between revisions
From EvoLudo
Stochastic dynamics in finite populations (view source)
Revision as of 15:21, 12 October 2023
, 12 October 2023→Rock-Paper-Scissors game
| Line 16: | Line 16: | ||
Comparisons between the deterministic dynamics in infinite populations, the stochastic dynamics in finite populations and individual based simulations are illustrated for a generic payoff matrix | Comparisons between the deterministic dynamics in infinite populations, the stochastic dynamics in finite populations and individual based simulations are illustrated for a generic payoff matrix | ||
\ | \begin{align} | ||
\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 & {\textstyle\frac{s}{2}} & -1 \\ | \begin{pmatrix}0 & {\textstyle\frac{s}{2}} & -1 \\ | ||
-1 & 0 & 2+s \\ | -1 & 0 & 2+s \\ | ||
{\textstyle\frac{1+s}{3}} & -1 & 0\end{pmatrix}\end{matrix}. | {\textstyle\frac{1+s}{3}} & -1 & 0 | ||
\ | \end{pmatrix} | ||
\end{matrix}. | |||
\end{align} | |||
According to the [[replicator equation]] the game exhibits saddle node fixed points at \(x = 1, y = 1\), and \(z = 1-x-y = 1\) as well as an interior fixed point at \(\textstyle\hat{\mathbf x} = \left(\frac12,\frac13,\frac16\right)\) independent of the parameter \(s\). For \(s > 1\), \(\hat x\) is a stable focus and an unstable focus for \(s < 1\). In the non-generic case \(s=1\) the dynamics exhibits closed orbits. | According to the [[replicator equation]] the game exhibits saddle node fixed points at \(x = 1, y = 1\), and \(z = 1-x-y = 1\) as well as an interior fixed point at \(\textstyle\hat{\mathbf x} = \left(\frac12,\frac13,\frac16\right)\) independent of the parameter \(s\). For \(s > 1\), \(\hat x\) is a stable focus and an unstable focus for \(s < 1\). In the non-generic case \(s=1\) the dynamics exhibits closed orbits. | ||
