Figure 1

Elements of the noise matrix $\mathcal{C}(x,y,z)$ for processes fulfilling Eq. (8) with $d=3$ strategies for $\mu =0$ (left) and $\mu >0$ (right). (a) The element ${\mathcal{C}}_{xx}$ determines how the noise in the $x$ direction affects the $x$ coordinate. In the case of $\mu =0$, this noise vanishes for $x\to 0$. For $y\to 0$ and $z\to 0$ we recover the usual multiplicative noise from one-dimensional evolutionary processes [8]. (b) For nonvanishing mutations ${\mathcal{C}}_{xx}$ increases at $x\to 0$ and $x\to 1$ and exceeds the value $\frac{1}{2\sqrt{N}}$ (contour line) in the interior of the simplex, which is the maximum value without mutations. The element ${\mathcal{C}}_{yy}$ follows from the transformation $x\leftrightarrow y$. (c) The element ${\mathcal{C}}_{xy}$ determines how the noise in the $x$ direction affects the $y$ coordinate (or vice versa). This term is always negative, as explained in the text. For $x\to 0$ or $y\to 0$, ${\mathcal{C}}_{xy}$ vanishes in the case of $\mu =0$. For $z\to 0$ we find ${\mathcal{C}}_{xy}=-{\mathcal{C}}_{xx}$, which ensures that the sum of the noise in the $x$ coordinate ${\mathcal{C}}_{xx}+{\mathcal{C}}_{xy}$ vanishes on this edge of the simplex: If the noise increases $x$ for $z=0$ it has to decrease $y$ by the same amount. (d) In the case of $\mu >0$, the noise no longer vanishes at the boundaries of the simplex but the expressions are too cumbersome to display explicitly.Reuse & Permissions

Figure 2

Comparison of trajectories for the (a) deterministic case (replicator equation), (b) stochastic differential equation [Eq. (5)], and (c) individual based simulations for an RSP interaction with weakly attracting interior fixed point $\widehat{\mathbf{x}}$ with $s=1.4$ [cf. Eq. (16)] and $N=1000$ [in (b) and (c)]. Stochastic trajectories start close to $\widehat{\mathbf{x}}$ and end at time $T=349.9$ (b) and $T=395.8$ (c). Simulations can also be performed online at [43].Reuse & Permissions

Figure 3

Probability of absorption and time to absorption (in the absence of mutations, $\mu =0$). (a) and (c) Probability to reach one of the three absorbing homogenous states of rock (solid line, $\blacksquare$), scissors (dashed line, $▴$), or paper (dotted line, $⧫$) as well as (b) and (d) the associated time to absorption for rock as a function of the population size $N$ with a stable (top row, $s=1.4$) and unstable (bottom row, $s=0.8$) interior fixed point $\widehat{\mathbf{x}}$. If $\widehat{\mathbf{x}}$ is an attractor [(a) and (b)] it may take exceedingly long times for large $N$ until the population reaches an absorbing state—even though this will inevitably occur. Because of this results are only shown up to $N=3000$. No such limitations occur if $\widehat{\mathbf{x}}$ is a repellor. The symbols indicate results from individual based simulations. Error bars and gray shaded areas indicate the standard deviation of the mean for simulations and stochastic calculations, respectively. Simulation results and results based on the stochastic differential equation [Eq. (5)] were both averaged over ${10}^5$ independent runs.Reuse & Permissions

Figure 4

Average frequency (solid line) and standard deviation (gray shaded area) of the three strategies rock [(a) and (b)], scissors [(c) and (d)], and paper [(e) and (f)] as a function of the mutation rate $\mu$ for small (left column, $N=100$) and large populations (right column, $N={10}^5$) with a stable interior fixed point $\widehat{\mathbf{x}}$, $s=1.4$, based on Eq. (5). For comparison, symbols and error bars depict results from individual based simulations. No simulation results are shown for $N={10}^5$ because of prohibitive computational efforts. Simulations and Eq. (5) are in excellent agreement for $\mu \gtrsim 1/N$ but for smaller $\mu$ substantial deviations occur (see text for details). Stochastic fluctuations decrease with increasing $N$ and for $N={10}^5$ the population spends most of the time in the close vicinity of $\widehat{\mathbf{x}}$. Simulation results and results based on the stochastic differential equation [Eq. (5)] were both averaged over ${10}^{10}$ time steps after a relaxation time of ${10}^6$ steps ($dt=0.01$ for the SDE).Reuse & Permissions

Figure 5

Performance comparison of individual based simulations (IBSs) vs stochastic differential equations (SDEs). (a) Ratio of the CPU times $CP{U}_{\text{SDE}}/CP{U}_{\text{IBS}}$ as a function of the population size $N$ and the number of strategic types $d$. The bold contour indicates equal performance. For small $N$ and large $d$ IBSs are faster (left of solid line), but for larger $N$ and smaller $d$ SDEs are faster (right of solid line). Each contour indicates a performance difference of one order of magnitude. Note that IBSs for $d=2,3$ are based on analytical calculations of the eigenvalues and eigenvectors of $\mathcal{B}\left(\mathbf{x}\right)$, which requires substantially less time and hence explains the discontinuity in the contours. (b) Computational time with $d=10$ as a function of $N$ for IBSs and SDEs. As a reference for the scaling $N^2$ and a constant are shown. (c) Computational time with $N=5000$ as a function of $d$ for IBSs and SDEs. As a reference for the scaling $d^{1/2}$ and $d^3$ are shown. For a proper scaling argument much larger $d$ are required but already $d=100$ far exceeds typical evolutionary models and hence is only of limited relevance in the current context. All comparisons use a constant payoff matrix and the local update process [8] [such that ${\mathcal{A}}_k\left(\mathbf{x}\right)=0$ and $\gamma ({\pi }_j,{\pi }_k)=1/2$], a mutation rate of $1/N$ and are based on at least 1000 time steps as well as at least 1 min running time. CPU time is measured in milliseconds required to calculate 1000 time steps. The time increment for the SDEs is $dt=0.01$.Reuse & Permissions