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The replicator dynamics for multilevel selection in evolutionary games

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Daniel Cooney at University of Illinois Urbana-Champaign
Daniel Cooney
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Abstract and Figures

We consider a stochastic model for evolution of group-structured populations in which interactions between group members correspond to the Prisoner’s Dilemma or the Hawk–Dove game. Selection operates at two organization levels: individuals compete with peer group members based on individual payoff, while groups also compete with other groups based on average payoff of group members. In the Prisoner’s Dilemma, this creates a tension between the two levels of selection, as defectors are favored at the individual level, whereas groups with at least some cooperators outperform groups of defectors at the between-group level. In the limit of infinite group size and infinite number of groups, we derive a non-local PDE that describes the probability distribution of group compositions in the population. For special families of payoff matrices, we characterize the long-time behavior of solutions of our equation, finding a threshold intensity of between-group selection required to sustain density steady states and the survival of cooperation. When all-cooperator groups are most fit, the average and most abundant group compositions at steady state range from featuring all-defector groups when individual-level selection dominates to featuring all-cooperator groups when group-level selection dominates. When the most fit groups have a mix of cooperators and defectors, then the average and most abundant group compositions always feature a smaller fraction of cooperators than required for the optimal mix, even in the limit where group-level selection is infinitely stronger than individual-level selection. In such cases, the conflict between the two levels of selection cannot be decoupled, and cooperation cannot be sustained at all in the case where between-group competition favors an even mix of cooperators and defectors.
Steady state densities for various relative selection strengths λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} when γ=2.5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma = 2.5$$\end{document} (G(x) maximized by x=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x=1$$\end{document}), computed from the result of Proposition 3.2. (Left) Steady state corresponding to initial distribution with Hölder exponent θ0=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _0 = 1$$\end{document} near x=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x=1$$\end{document}. (Right) Steady state corresponding to initial distribution with Hölder exponent θ0=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _0 = 2$$\end{document} near x=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x=1$$\end{document}
Steady state densities for various relative selection strengths λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} when γ=2.5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma = 2.5$$\end{document} (G(x) maximized by x=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x=1$$\end{document}), computed from the result of Proposition 3.2. (Left) Steady state corresponding to initial distribution with Hölder exponent θ0=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _0 = 1$$\end{document} near x=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x=1$$\end{document}. (Right) Steady state corresponding to initial distribution with Hölder exponent θ0=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _0 = 2$$\end{document} near x=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x=1$$\end{document}
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Steady state densities for various relative selection strengths λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} when γ=1.5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma = 1.5$$\end{document} (G(x) maximized by x=0.75\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x=0.75$$\end{document}), computed from the result of Proposition 3.2. (Left) Steady state corresponding to initial distribution with Hölder exponent θ0=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _0 = 1$$\end{document} near x=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x=1$$\end{document}. (Right) Steady state corresponding to initial distribution with Hölder exponent θ0=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _0 = 2$$\end{document} near x=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x=1$$\end{document}. Dotted line indicates that average group payoff G(x) is maximized at 75% cooperators
Steady state densities for various relative selection strengths λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} when γ=1.5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma = 1.5$$\end{document} (G(x) maximized by x=0.75\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x=0.75$$\end{document}), computed from the result of Proposition 3.2. (Left) Steady state corresponding to initial distribution with Hölder exponent θ0=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _0 = 1$$\end{document} near x=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x=1$$\end{document}. (Right) Steady state corresponding to initial distribution with Hölder exponent θ0=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _0 = 2$$\end{document} near x=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x=1$$\end{document}. Dotted line indicates that average group payoff G(x) is maximized at 75% cooperators
… 
Comparison of the type of group composition x with maximum group reproduction rate G(x) to the peak abundance for the steady-state x^λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{x}}_{\lambda }$$\end{document} as λ→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \rightarrow \infty $$\end{document}, plotted in terms of the parameter γ=S+T-2P\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma = S + T - 2P$$\end{document}. For γ≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \ge 2$$\end{document} both peak group fitness and most abundant group type are all-cooperator groups, while for γ<2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma < 2$$\end{document}, the most fit group type γ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tfrac{\gamma }{2}$$\end{document} exceeds the most abundant group type at steady-state γ-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma - 1$$\end{document}
Comparison of the type of group composition x with maximum group reproduction rate G(x) to the peak abundance for the steady-state x^λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{x}}_{\lambda }$$\end{document} as λ→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \rightarrow \infty $$\end{document}, plotted in terms of the parameter γ=S+T-2P\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma = S + T - 2P$$\end{document}. For γ≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \ge 2$$\end{document} both peak group fitness and most abundant group type are all-cooperator groups, while for γ<2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma < 2$$\end{document}, the most fit group type γ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tfrac{\gamma }{2}$$\end{document} exceeds the most abundant group type at steady-state γ-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma - 1$$\end{document}
… 
Color indicates the fraction of cooperators in the group type with peak abundance for the steady-state x^λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{x}}_{\lambda }$$\end{document} (as calculated in Eq. 3.15) for various values of λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} and γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document}. Other parameters are fixed as β=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta = 1$$\end{document} and α=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha = 1$$\end{document}, so group average payoff is maximized at x∗=minγ2,1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x^* = \min \left( \tfrac{\gamma }{2},1\right) $$\end{document} and peak abundance satisfies x^λ→minγ-1,1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{x}}_{\lambda } \rightarrow \min \left( \gamma - 1,1\right) $$\end{document} as λ→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \rightarrow \infty $$\end{document}. Vertical slices of heatmap can be interpreted in the same way as the green curve for x^λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{x}}_{\lambda }$$\end{document} in Fig. 5 for a fixed value of γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} (color figure online)
Color indicates the fraction of cooperators in the group type with peak abundance for the steady-state x^λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{x}}_{\lambda }$$\end{document} (as calculated in Eq. 3.15) for various values of λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} and γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document}. Other parameters are fixed as β=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta = 1$$\end{document} and α=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha = 1$$\end{document}, so group average payoff is maximized at x∗=minγ2,1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x^* = \min \left( \tfrac{\gamma }{2},1\right) $$\end{document} and peak abundance satisfies x^λ→minγ-1,1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{x}}_{\lambda } \rightarrow \min \left( \gamma - 1,1\right) $$\end{document} as λ→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \rightarrow \infty $$\end{document}. Vertical slices of heatmap can be interpreted in the same way as the green curve for x^λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{x}}_{\lambda }$$\end{document} in Fig. 5 for a fixed value of γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} (color figure online)
… 
The mean fraction of cooperators M1f\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_1^f$$\end{document} and the fraction of cooperators in the most abundant group composition for the steady-state x^λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{x}}_{\lambda }$$\end{document} (as calculated in Eq. 3.16) for various values of λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document}. Other parameters are fixed as γ=32\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma = \frac{3}{2}$$\end{document}, β=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta = 1$$\end{document}, α=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha = 1$$\end{document}, so group average payoff is maximized at x∗=34\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x^* = \tfrac{3}{4}$$\end{document} and peak abundance satisfies x^λ→12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{x}}_{\lambda } \rightarrow \frac{1}{2}$$\end{document} as λ→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \rightarrow \infty $$\end{document}. We note that the mean M1fθ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_1^{f_{\theta }}$$\end{document} is initially larger than x^λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{x}}_{\lambda }$$\end{document} but also tends to 12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tfrac{1}{2}$$\end{document} as λ→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \rightarrow \infty $$\end{document}
+1
The mean fraction of cooperators M1f\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_1^f$$\end{document} and the fraction of cooperators in the most abundant group composition for the steady-state x^λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{x}}_{\lambda }$$\end{document} (as calculated in Eq. 3.16) for various values of λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document}. Other parameters are fixed as γ=32\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma = \frac{3}{2}$$\end{document}, β=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta = 1$$\end{document}, α=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha = 1$$\end{document}, so group average payoff is maximized at x∗=34\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x^* = \tfrac{3}{4}$$\end{document} and peak abundance satisfies x^λ→12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{x}}_{\lambda } \rightarrow \frac{1}{2}$$\end{document} as λ→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \rightarrow \infty $$\end{document}. We note that the mean M1fθ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_1^{f_{\theta }}$$\end{document} is initially larger than x^λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{x}}_{\lambda }$$\end{document} but also tends to 12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tfrac{1}{2}$$\end{document} as λ→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \rightarrow \infty $$\end{document}
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Journal of Mathematical Biology (2019) 79:101–154
https://doi.org/10.1007/s00285-019-01352-5
Mathematical Biology
The replicator dynamics for multilevel selection in
evolutionary games
Daniel B. Cooney1
Received: 6 November 2018 / Revised: 25 March 2019 / Published online: 8 April 2019
© Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract
We consider a stochastic model for evolution of group-structured populations in which
interactions between group members correspond to the Prisoner’s Dilemma or the
Hawk–Dove game. Selection operates at two organization levels: individuals compete
with peer group members based on individual payoff, while groups also compete with
other groups based on average payoff of group members. In the Prisoner’s Dilemma,
this creates a tension between the two levels of selection, as defectors are favored at
the individual level, whereas groups with at least some cooperators outperform groups
of defectors at the between-group level. In the limit of infinite group size and infinite
number of groups, we derive a non-local PDE that describes the probability distribution
of group compositions in the population. For special families of payoff matrices, we
characterize the long-time behavior of solutions of our equation, finding a threshold
intensity of between-group selection required to sustain density steady states and the
survival of cooperation. When all-cooperator groups are most fit, the average and most
abundant group compositions at steady state range from featuring all-defector groups
when individual-level selection dominates to featuring all-cooperator groups when
group-level selection dominates. When the most fit groups have a mix of cooperators
and defectors, then the average and most abundant group compositions always feature
a smaller fraction of cooperators than required for the optimal mix, even in the limit
where group-level selection is infinitely stronger than individual-level selection. In
such cases, the conflict between the two levels of selection cannot be decoupled, and
cooperation cannot be sustained at all in the case where between-group competition
favors an even mix of cooperators and defectors.
Keywords Multilevel selection ·Evolutionary game theory ·Replicator dynamics
This research was supported by NSF Grants DMS-1514606 and GEO-1211972 and by ARO Grant
W911NF-18-1-0325.
BDaniel B. Cooney
dcooney@math.princeton.edu
1Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ, USA
123
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... By considering two-player, two-strategy games with a range of payoff matrices, it is possible to formulate a variety of social dilemmas in which cooperation is socially beneficial, but in which individual-level replicator dynamics can favor dominance of defection, coexistence between cooperators and defectors, or bistability of alldefector and all-cooperator states [18,19]. Depending on the payoff structure of underlying games, it is also possible to explore scenarios in which the average payoff of group members is maximized by a group composed only of cooperators, as well as games for which an intermediate level of cooperation can maximize the collective payoff for a group [20]. ...
... Luo and Mattingly considered the case of two types of individuals in which one type had a fixed advantage under individual-level replication and the other type conferred a collective advantage to their group, showing that beneficial group-level outcomes could be achieved in the long-time behavior a PDE model of multilevel selection when competition among groups was sufficiently strong [52]. Subsequent extensions of these two-level birth-models have explored fixation probabilities in finite populations [38], the existence of quasistationary distributions in a diffusive PDE scaling limit of the two-level stochastic process [42,54], and the formulation of individual-level and group-level replication rates based on two-player, two-strategy social dilemma games played within each group [20,55]. The resulting hyperbolic PDE models for multilevel selection have been further generalized to study multilevel dynamics with individual-level and group-level replication rates described by arbitrary functions of the fraction of cooperators within each group [56], and results for these generalized models have been applied to explore synergistic effects of group-level competition and within-group mechanisms for promoting the evolution of cooperation [57,58] and to study models of protocell evolution and the origin of chromosomes [59]. ...
... For these two-level replicator equation models, it was possible to use the method of characteristics to determine the long-time behavior for these models of multilevel selection. A particularly interesting feature of the two-level replicator model was a phenomenon described as a "shadow of lower-level selection", in which the long-time group-level replication rate could not exceed the replication rate of the all-cooperator group, even for scenarios in which the replication rate of groups was maximized by intermediate levels of cooperation [20,55,56]. One question of interest for this paper is whether this behavior is limited to the case of previously studied two-level replicator equations, or whether this long shadow cast by lower-level selection can hold for a broader class of PDE models of multilevel selection that incorporate frequency-dependent competition at the group level. ...
Steady-State and Dynamical Behavior of a PDE Model of Multilevel Selection with Pairwise Group-Level Competition
Preprint
Full-text available
  • Nov 2024
Evolutionary competition often occurs simultaneously at multiple levels of organization, in which traits or behaviors that are costly for an individual can provide collective benefits to groups to which the individual belongs. Building off of recent work that has used ideas from game theory to study evolutionary competition within and among groups, we study a PDE model for multilevel selection that considers group-level evolutionary dynamics through a pairwise conflict depending on the strategic composition of the competing groups. This model allows for incorporation of group-level frequency dependence, facilitating the exploration for how the form of probabilities for victory in a group-level conflict can impact the long-time support for cooperation via multilevel selection. We characterize well-posedness properties for measure-valued solutions of our PDE model and apply these properties to show that the population will converge to a delta-function at the all-defector equilibrium when between-group selection is sufficiently weak. We further provide necessary conditions for the existence of bounded steady state densities for the multilevel dynamics of Prisoners' Dilemma and Hawk-Dove scenarios, using a mix of analytical and numerical techniques to characterize the relative strength of between-group selection required to ensure the long-time survival of cooperation via multilevel selection. We also see that the average payoff at steady state appears to be limited by the average payoff of the all-cooperator group, even for games in which groups achieve maximal average payoff at intermediate levels of cooperation, generalizing behavior that has previously been observed in PDE models of multilevel selection with frequency-indepdent group-level competition.
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... In the limit in which the number and size of groups tend to infinity, it is possible to derive a PDE description of this two-level birth-death process, with an advection term describing the individual-level advantage of defection and a nonlocal term describing the grouplevel benefits of cooperation. This approach for PDE modeling of multilevel selection has been extended to incorporate individual-level and group-level replication rates arising from payoffs achieved from games played within groups [47,48], and has been further extended to describe arbitrary replication rates [49] and has been applied to incorporate the effects of within-group population structure [50] and to model biological scenarios including the evolution of protocells and the origin of chromosomes [51]. These PDE models often provide analytically tractable ways to understand the tradeoffs between individual-level and group-level incentives, highlighting the conditions under which between-group competition can allow for the establishment of long-time cooperation in a population. ...
... In this paper, we take inspiration from the stochastic model by Boyd and coauthors to formulate a PDE describing the evolution of altruistic punishment via multilevel selection with pairwise between-group competition. While existing work on PDE models for multilevel selection based on the approach of Luo and coauthors have considered group-level replication events occurring at a rate depending only on the strategic competition of the replicating group [44,45,47,49], the model we consider will describe group-level replication based on pairwise conflicts between groups and allows us to study frequencydependent between-group competition. Our PDE models feature within-group replicator dynamics in which individuals are favored when their payoff is above average within their group, while competition at the group level takes places through pairwise conflict between groups, with the probability that a group wins a pairwise conflict determined by a function of the strategic competition of the two competing groups. ...
... (3.9) so the dynamics take the form of a generalized two-level replicator equation in this case [47,49]. ...
Exploring the Evolution of Altruistic Punishment with a PDE Model of Cultural Multilevel Selection
Preprint
Full-text available
  • May 2024
Two mechanisms that have been used to study the evolution of cooperative behavior are altruistic punishment, in which cooperative individuals pay additional costs to punish defection, and multilevel selection, in which competition between groups can help to counteract individual-level incentives to cheat. Boyd, Gintis, Bowles, and Richerson have used simulation models of cultural evolution to suggest that altruistic punishment and pairwise group-level competition can work in concert to promote cooperation, even when neither mechanism can do so on its own. In this paper, we formulate a PDE model for multilevel selection motivated by the approach of Boyd and coauthors, modeling individual-level birth-death competition with a replicator equation based on individual payoffs and describing group-level competition with pairwise conflicts based on differences in the average payoffs of the competing groups. Building off of existing PDE models for multilevel selection with frequency-independent group-level competition, we use analytical and numerical techniques to understand how the forms of individual and average payoffs can impact the long-time ability to sustain altruistic punishment in group-structured populations. We find several interesting differences between the behavior of our new PDE model with pairwise group-level competition and existing multilevel PDE models, including the observation that our new model can feature a non-monotonic dependence of the long-time collective payoff on the strength of altruistic punishment. Going forward, our PDE framework can serve as a way to connect and compare disparate approaches for understanding multilevel selection across the literature in evolutionary biology and anthropology.
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... Questions like these about group-structured populations can be addressed via mathematical models that take state-dependent birth and death rates at both levels explicitly into account. For example, the partial differential equation models of group-structured populations developed in Luo (2014), Cooney (2019), Cooney (2020), Cooney and Mori (2022) have been used to study a number of interesting questions about groupstructured populations, including the evolution of cooperation, as well as transitions of the dominant selection regime from the individual to the group level or vice versa. However, these models do not address what happens when there is a continuum of individual-level and group-level phenotypes. ...
... As we show in other work (Simon et al. 2024), this effect increases the between group variance, making group-level evolution more effective, which in turn requires stronger individual-level selection, i.e., larger s, to balance. We also note that the multi-level partial differential equation models of Luo (2014), Cooney (2019Cooney ( , 2020, Cooney and Mori (2022) could provide an alternative approach to determining the relative strength of the individual-level and the group-level selection regimes, but such models would have to be modified to incorporate games with continuous strategies and non-linear payoffs. Figure 5 shows what happens when the between group game remains the branching snowdrift game (9, 11), but the within-group game is a prisoner's dilemma (10). ...
Evolutionary branching in multi-level selection models
Article
Full-text available
  • Oct 2024
  • J MATH BIOL
We study a model of group-structured populations featuring individual-level birth and death events, and group-level fission and extinction events. Individuals play games within their groups, while groups play games against other groups. Payoffs from individual-level games affect birth rates of individuals, and payoffs from group-level games affect group extinction rates. We focus on the evolutionary dynamics of continuous traits with particular emphasis on the phenomenon of evolutionary diversification. Specifically, we consider two-level processes in which individuals and groups play continuous snowdrift or prisoner’s dilemma games. Individual game strategies evolve due to selection pressure from both the individual and group level interactions. The resulting evolutionary dynamics turns out to be very complex, including branching and type-diversification at one level or the other. We observe that a weaker selection pressure at the individual level results in more adaptable groups and sometimes group-level branching. Stronger individual-level selection leads to more effective adaptation within each group while preventing the groups from adapting according to the group-level games.
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... Evidence of these conflicts has been 30 found in fields as wide-ranging as experimental evolution [4][5][6], animal behavior[7], development [8], and 31 molecular genetics [9]. Additionally, theory has predicted that these cross-scale processes can give rise 32 to complex, non-intuitive evolutionary dynamics [10, 11]. Nonetheless, our empirical understanding of 33 these dynamics lags behind the theory due to the general difficulty in simultaneously tracking the fitness 34 of replicators across different levels of organization. ...
Intracellular competition shapes plasmid population dynamics
Preprint
Full-text available
  • Feb 2025
Conflicts between levels of biological organization are central to evolution, from populations of multicellular organisms to selfish genetic elements in microbes. Plasmids are extrachromosomal, self-replicating genetic elements that underlie much of the evolutionary flexibility of bacteria. Evolving plasmids face selective pressures on their hosts, but also compete within the cell for replication, making them an ideal system for studying the joint dynamics of multilevel selection. While theory indicates that within-cell selection should matter for plasmid evolution, experimental measurement of within-cell plasmid fitness and its consequences has remained elusive. Here we measure the within-cell fitness of competing plasmids and characterize drift and selective dynamics. We achieve this by the controlled splitting of synthetic plasmid dimers to create balanced competition experiments. We find that incompatible plasmids co-occur for longer than expected due to methylation-based plasmid eclipsing. During this period of co-occurrence, less transcriptionally active plasmids display a within-cell selective advantage over their competing plasmids, leading to preferential fixation of silent plasmids. When the transcribed gene is beneficial to the cell, for example an antibiotic resistance gene, there is a cell-plasmid fitness tradeoff mediated by the dominance of the beneficial trait. Surprisingly, more dominant plasmid-encoded traits are less likely to fix but more likely to initially invade than less dominant traits. Taken together, our results show that plasmid evolution is driven by dynamics at two levels, with a transient, but critical, contribution of within-cell fitness.
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... The Replicator Dynamic Equation describes changes in the proportion of strategy adoption over time within a population, based on the relative performance of the average payoff of a specific strategy compared to the overall expected utility [35]. Following the calculation process outlined by Tanimoto [36], the Replicator Dynamic Equations for the contractor and the owner are represented by Equations (7) and (8), respectively. ...
An Evolutionary Game Theory-Based Framework for Analyzing Behavioral Strategies in Contractor–Owner Conflicts over Additional Construction Costs
Article
Full-text available
  • Feb 2025
The construction industry faces increasing conflicts over additional construction costs due to economic uncertainties, such as global pandemics and wars. These disputes often lead to project delays, legal actions, and even construction halts, causing significant financial and operational losses for stakeholders. To address these challenges, this study develops a simulation model based on evolutionary game theory (EGT) to identify the key influencing factors and applies the Analytic Hierarchy Process (AHP) to analyze and manage the conflicts between contractors and owners in private construction projects. The model quantifies decision-making dynamics by calculating the relative importance of various factors under different scenarios. A proof-of-concept simulation of the model reveals that cooperative evolution dynamics significantly decrease when the cost-sharing ratio reaches 0.5 for contractors and 0.9 for owners. Furthermore, the sensitivity analysis indicates that exceeding cost-sharing thresholds undermines cooperation, increasing the risk of disputes. Through this simulation, this study concludes that fostering mutual trust and informed decision-making on cost-sharing ratio significantly reduces project disputes and enhances the stakeholders’ profitability. The developed model and its framework serve as valuable tools for providing project stakeholders with actionable insights aimed at fostering strategic behaviors that minimize dispute-driven financial risks in construction projects.
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... In this form, replicator dynamics has become a general model for the evolution of populations under frequency-dependent selection, "flexible enough to cover a great deal of evolution models, suggesting a unifying view of replicator selection from the primordial soup up to animal societies" (Schuster and Sigmund 1983). The mathematical analysis of general replicator equations has been extensive (Hofbauer andSigmund 1988, 1998;Hofbauer and Garay 2003;Weibull 1995;Hauert et al. 2002;Cressman 2003;Nowak and Sigmund 2004;Ohtsuki and Nowak 2006;Cressman and Tao 2014;Cooney 2019), testifying to its fundamental role in the development of evolutionary game theory. Stochastic versions of the replicator equation have a long tradition. ...
Persistence and neutrality in interacting replicator dynamics
Article
Full-text available
  • Jan 2025
  • J MATH BIOL
We study the large-time behavior of an ensemble of entities obeying replicator-like stochastic dynamics with mean-field interactions as a model for a primordial ecology. We prove the propagation-of-chaos property and establish conditions for the strong persistence of the N-replicator system and the existence of invariant distributions for a class of associated McKean–Vlasov dynamics. In particular, our results show that, unlike typical models of neutral ecology, fitness equivalence does not need to be assumed but emerges as a condition for the persistence of the system. Further, neutrality is associated with a unique Dirichlet invariant probability measure. We illustrate our findings with some simple case studies, provide numerical results, and discuss our conclusions in the light of Neutral Theory in ecology.
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... After establishing this foundation, we focus on the Replicator (REP) update rule, also within the framework of pairwise interactions. The Replicator dynamics, widely applied in evolutionary games [26,[58][59][60][61][62], is very important since it operates under the Darwinian assumption that the growth rate per capita depends on how well a strategy performs relative to the average population performance. ...
Anomalous behavior of Replicator dynamics for the Prisoner's Dilemma on diluted lattices
Preprint
Full-text available
  • Sep 2024
In diluted lattices, cooperation is often enhanced at specific densities, particularly near the percolation threshold for stochastic updating rules. However, the Replicator rule, despite being probabilistic, does not follow this trend. We find that this anomalous behavior is caused by structures formed by holes and defectors, which prevent some agents from experiencing fluctuations, thereby restricting the free flow of information across the network. As a result, the system becomes trapped in a frozen state, though this can be disrupted by introducing perturbations. Finally, we provide a more quantitative analysis of the relationship between the percolation threshold and cooperation, tracking its development within clusters of varying sizes and demonstrating how the percolation threshold shapes the fundamental structures of the lattice.
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Exploring the Evolution of Altruistic Punishment with a PDE Model of Cultural Multilevel Selection
Article
  • Mar 2025
  • B MATH BIOL
Two mechanisms that have been used to study the evolution of cooperative behavior are altruistic punishment, in which cooperative individuals pay additional costs to punish defection, and multilevel selection, in which competition between groups can help to counteract individual-level incentives to cheat. Boyd, Gintis, Bowles, and Richerson have used simulation models of cultural evolution to suggest that altruistic punishment and pairwise group-level competition can work in concert to promote cooperation, even when neither mechanism can do so on its own. In this paper, we formulate a PDE model for multilevel selection motivated by the approach of Boyd and coauthors, modeling individual-level birth-death competition with a replicator equation based on individual payoffs and describing group-level competition with pairwise conflicts based on differences in the average payoffs of the competing groups. Building off of existing PDE models for multilevel selection with frequency-independent group-level competition, we use analytical and numerical techniques to understand how the forms of individual and average payoffs can impact the long-time ability to sustain altruistic punishment in group-structured populations. We find several interesting differences between the behavior of our new PDE model with pairwise group-level competition and existing multilevel PDE models, including the observation that our new model can feature a non-monotonic dependence of the long-time collective payoff on the strength of altruistic punishment. Going forward, our PDE framework can serve as a way to connect and compare disparate approaches for understanding multilevel selection across the literature in evolutionary biology and anthropology.
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Fission as a source of variation for group selection
Article
  • Jun 2024
  • EVOLUTION
Without heritable variation natural selection cannot effect evolutionary change. In the case of group selection, there must be variation in the population of groups. Where does this variation come from? One source of variation is from the stochastic birth–death processes that occur within groups. This is where variation between groups comes from in most mathematical models of group selection. Here, we argue that another important source of variation between groups is fission, the (generally random) group-level reproduction where parent groups split into two or more offspring groups. We construct a simple model of the fissioning process with a parameter that controls how much variation is produced among the offspring groups. We then illustrate the effect of that parameter with some examples. In most models of group selection in the literature, no variation is produced during group reproduction events; that is, groups “clone” themselves when they reproduce. Fission is often a more biologically realistic method of group reproduction, and it can significantly increase the efficacy of group selection.
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Evolutionary games and matching rules
Article
Full-text available
  • Sep 2018
  • INT J GAME THEORY
This study considers evolutionary games with non-uniformly random matching when interaction occurs in groups of n≥2n\ge 2 individuals using pure strategies from a finite strategy set. In such models, groups with different compositions of individuals generally co-exist and the reproductive success (fitness) of a specific strategy varies with the frequencies of different group types. These frequencies crucially depend on the matching process. For arbitrary matching processes (called matching rules), we study Nash equilibrium and ESS in the associated population game and show that several results that are known to hold for population games under uniform random matching carry through to our setting. In our most novel contribution, we derive results on the efficiency of the Nash equilibria of population games and show that for any (fixed) payoff structure, there always exists some matching rule leading to average fitness maximization. Finally, we provide a series of applications to commonly studied normal-form games.
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Stochasticity, Selection, and the Evolution of Cooperation in a Two-Level Moran Model of the Snowdrift Game
Article
Full-text available
The Snowdrift Game, also known as the Hawk-Dove Game, is a social dilemma in which an individual can participate (cooperate) or not (defect) in producing a public good. It is relevant to a number of collective action problems in biology. In a population of individuals playing this game, traditional evolutionary models, in which the dynamics are continuous and deterministic, predict a stable, interior equilibrium frequency of cooperators. Here, we examine how finite population size and multilevel selection affect the evolution of cooperation in this game using a two-level Moran process, which involves discrete, stochastic dynamics. Our analysis has two main results. First, we find that multilevel selection in this model can yield significantly higher levels of cooperation than one finds in traditional models. Second, we identify a threshold effect for the payoff matrix in the Snowdrift Game, such that below (above) a determinate cost-to-benefit ratio, cooperation will almost surely fix (go extinct) in the population. This second result calls into question the explanatory reach of traditional continuous models and suggests a possible alternative explanation for high levels of cooperative behavior in nature.
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Fast cheater migration stabilizes coexistence in a public goods dilemma on networks
Article
Full-text available
  • Nov 2017
Through the lens of game theory, cooperation is frequently considered an unsustainable strategy: if an entire population is cooperating, each indi- vidual can increase its overall fitness by choosing not to cooperate, thereby still receiving all the benefit of its cooperating neighbors while no longer expending its own energy. Observable cooperation in naturally-occurring public goods games is consequently of great interest, as such systems offer insight into both the emergence and sustainability of cooperation. Here we consider a population that obeys a public goods game on a network of discrete regions (that we call nests), between any two of which individuals are free to migrate. We construct a system of piecewise-smooth ordinary differential equations that couple the within-nest population dynamics and the between-nest migratory dynamics. Through a combination of analytical and numerical methods, we show that if the workers within the population migrate sufficiently fast relative to the cheaters, the network loses stability first through a Hopf bifurcation, then a torus bifurcation, after which one or more nests collapse. Our results indicate that fast moving cheaters can act to stabilize worker-cheater coexistence within network that would otherwise collapse. We end with a comparison of our results with the dynamics observed in colonies of the ant species Pristomyrmex punctatus and in those of the Cape honeybee Apis mellifera capensis, and argue that they qualitatively agree.
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Prebiotic RNA Network Formation: A Taxonomy of Molecular Cooperation
Article
Full-text available
  • Oct 2017
Cooperation is essential for evolution of biological complexity. Recent work has shown game theoretic arguments, commonly used to model biological cooperation, can also illuminate the dynamics of chemical systems. Here we investigate the types of cooperation possible in a real RNA system based on the Azoarcus ribozyme, by constructing a taxonomy of possible cooperative groups. We construct a computational model of this system to investigate the features of the real system promoting cooperation. We find triplet interactions among genotypes are intrinsically biased towards cooperation due to the particular distribution of catalytic rate constants measured empirically in the real system. For other distributions cooperation is less favored. We discuss implications for understanding cooperation as a driver of complexification in the origin of life.
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Intense group selection selects for ideal group compositions, but selection within groups maintains them
Article
Full-text available
  • Feb 2017
A group's composition is important for its success. Colonies of the spider Anelosimus studiosus appear to have responded to this pressure by evolving the ability to maintain mixtures of docile versus aggressive individuals that help colonies avoid extinction. Here we demonstrate that colony extinction events unite the optimal group composition of all colony constituents, regardless of phenotype, with that of the colony as a whole. This is because colony extinction events explain the majority of individual mortality events in A. studiosus. Through within- and across-habitat colony manipulations, we further determined that reduction in reproductive output by individuals bearing overabundant phenotypes underlies the ability of colonies to adaptively regulate their compositions. When we experimentally created colonies with an overabundance of the docile or aggressive phenotype, individuals bearing the overabundant phenotype exhibited reduced reproductive output, which helped to move colony compositions back towards their site-specific optima. Colonies displaced from their native sites continued to recreate the patterns of reproductive output that characterized their site of origin, suggesting a genetic component to this trait. Individuals thus appear to adaptively cull their reproductive output depending on their phenotype and the composition of their colony. There is also considerable parent–offspring colony resemblance in the extent to which colonies can or do track their ideal compositions. This conveys a kind of collective heritability to this trait. Together, while group selection appears to be the principal driver of ongoing selection on group composition in A. studiosus, patterns of selection among individuals within groups appear to promote colonies' ability to track their ideal mixtures.
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A large-population limit for a Markovian model of group-structured populations
Article
  • Dec 2017
A Markovian model of group-structured (two-level) population dynamics features births, deaths, and migrations of individuals, and fission and extinction of groups. These models are useful for studying group selection and other evolutionary processes that occur when individuals live in distinct groups. We show that the sample paths of a properly scaled sequence of these models converge in an appropriate Skorohod space to a deterministic trajectory that is a unique solution to a quasilinear evolution equation. The PDE model can therefore be justified as an approximation to the Markovian one.
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