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Stochastic dynamics in finite populations: Difference between revisions
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Stochastic dynamics in finite populations (view source)
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==Rock-Paper-Scissors game== | ==Rock-Paper-Scissors game== | ||
[[Image:Stochastic dynamics - noise term Cxx, no mutations.png|300px|thumb| | [[Image:Stochastic dynamics - noise term Cxx, no mutations.png|300px|thumb|Demographic noise in evolutionary dynamics with \(d = 3\) strategic types in the absence of mutations, \(\mu = 0\). The value of the element \(\mathcal C_{xx}(x,y,z)\) of the noise matrix \(\mathcal C(\mathbf x)\) is shown. \(\mathcal C_{xx}(x,y,z)\) determines how the noise in the \(x\)-direction affects the \(x\)-coordinate. In the case of \(\mu = 0\), this noise vanishes for \(x\to0\). For \(y\to0\) and \(z\to0\) we recover the usual multiplicative noise from one-dimensional evolutionary processes.]] | ||
The [[Rock-Paper-Scissors game]] exhibits cyclic dominance among its three strategic types: Rock beats Scissors beats Paper beats Rocks etc. In evolving populations this gives rise to oscillations in the abundance of each strategic type. The amplitude of these oscillation may (i) decrease over time resulting in stable equilibrium frequencies, (ii) keep increasing and approaching a heteroclinic cycle along the boundary of the simplex \(S_3\) or result in the extinction of one type and eventual reach an absorbing homogenous state or, (iii) give rise to stable limit cycles. | The [[Rock-Paper-Scissors game]] exhibits cyclic dominance among its three strategic types: Rock beats Scissors beats Paper beats Rocks etc. In evolving populations this gives rise to oscillations in the abundance of each strategic type. The amplitude of these oscillation may (i) decrease over time resulting in stable equilibrium frequencies, (ii) keep increasing and approaching a heteroclinic cycle along the boundary of the simplex \(S_3\) or result in the extinction of one type and eventual reach an absorbing homogenous state or, (iii) give rise to stable limit cycles. | ||
