Non-Markovian Rock-Paper-Scissors games

TL;DR

This work addresses how non-Markovian waiting times, modeled by power-law and gamma distributions, alter the extinction-fixation dynamics in the zero-sum rock-paper-scissors game. It introduces generalized MF rate equations with memory kernels $\\Theta_{PL}$ and $\\Theta_G$, derived from non-exponential waiting-time distributions, and complements them with Gillespie-based simulations (including Laplace-Gillespie) to compute fixation probabilities under memory. The main finding is that memory effects characterized by high coefficients of variation $\\text{CV}$ can drastically modify fixation outcomes, breaking the classic law of the weakest and, in some regimes, causing symmetry breaking among competing species; fixation heatmaps quantify these shifts. Overall, the results demonstrate that waiting-time structure and memory profoundly influence non-Markovian evolutionary dynamics, with potential experimental relevance in microbial cyclic competition and other ecological systems.

Abstract

There is mounting evidence that species interactions often involve long-term memory, with highly-varying waiting times between successive events and long-range temporal correlations. Accounting for memory undermines the common Markovian assumption, and dramatically impacts key ingredients of population dynamics including birth, foraging, predation, and competition processes. Here, we study a critical aspect of population dynamics, namely non-Markovian multi-species competition. This is done in the realm of the zero-sum rock-paper-scissors (zRPS) model that is broadly used in the life sciences to metaphorically describe cyclic competition between three interacting species. We develop a general non-Markovian formalism for multi-species dynamics, allowing us to determine the regions of the parameter space where each species dominates. In particular, when the dynamics are Markovian, the waiting times are exponentially distributed and the fate of the zRPS model in large well-mixed populations is encoded in a remarkably simple condition, often referred to as the ``law of the weakest'' (LOW), stating that the species with the lowest growth rate is the most likely to prevail. We show that the survival behavior and LOW of the zRPS model are critically affected by non-exponential waiting times, and especially, by their coefficient of variation. Our findings provide key insight into the influence of long waiting times on non-Markovian evolutionary processes.

Figures (7)

• Figure 1: (a-c) Dynamics in the the ternary simplex (phase space) for the zRPS model with exponential WTD in (a). In (b) and (c) the last two reactions of \ref{['eq:reac']} have an exponential WTD, while the first reaction has a power-law WTD (\ref{['eq:Psi-PL']}) with $(k_A,\alpha_{\rm A}) = (0.8,2.5)$ in (b), and a gamma WTD (\ref{['eq:Psi-G']}) with $(k_A,\alpha_{\rm A}) = (0.8, 0.8)$ in (c). In (a-c): $k_{\rm A}=0.8, k_{\rm B}=k_{\rm C}=1$ and $N=100$. Gray dotted lines: stochastic trajectories (single realization, clockwise dynamics) represent $(n_A\!,\!n_B\!,\!n_C)/N$, with initial conditions $(1/3,1/3,1/3)$. Red thick lines: deterministic outermost orbits, see Appendix C. Each corner corresponds to the fixation of the labeled species. (d) RGB diagram used to color code the fixation heatmaps, where the letters 'A' (red), 'B' (green) and 'C' (blue) denote the winning species associated with each color, see text.
• Figure 2: An illustration of a gamma distribution (blue line) for $\alpha_{\rm A}=0.1$ and $\Lambda_{\rm A}=0.25$, such that the mean (black triangle) equals $0.4$. The red line depicts an exponential distribution with the same mean. In contrast, the medians (green squares) differ significantly: $0.277$ (exponential WTD) and $0.0024$ (gamma WTD).
• Figure 3: Mean field steady-state concentrations of $A$, $B$ and $C$ versus $\alpha_{\rm A}^{-1}$ and $\alpha_{\rm A}$ (a,c) and $k_{\rm A}$ (b,d). In (a-b) and (c-d) the first reaction of (\ref{['eq:reac']}) has a power-law and gamma WTD, respectively, while the second and third reactions have exponential WTDs. Markers are mean field values (see legend) as obtained by averaging over stochastic simulations (see details in Appendix C). In all panels dashed lines show the analytical results: Eq. (\ref{['mean-field-PL']}) in (a,b) and Eq. (\ref{['coex-gamma-full']}) in (c,d). The solid lines in (a,b) show the exact expression from numerically solving Eqs. (\ref{['REPL']}) for $\dot{a} = \dot{b}=\dot{c}=0$. Parameters are: $k_{\rm A}=k_{\rm B}=k_{\rm C}=1$ (a), $\alpha_{\rm A} = 3$ and $k_{\rm B}=k_{\rm C}=1$ (b), $k_{\rm A}=k_{\rm B}=k_{\rm C}=1$ (c), and $\alpha_{\rm A} = 0.4$ and $k_{\rm B}=k_{\rm C}=1$ (d).
• Figure 4: RGB fixation heatmaps for power-law WTD (\ref{['eq:Psi-PL']}) (a-c) and gamma WTD (\ref{['eq:Psi-G']}) (d-f) versus $\alpha_{\rm A}$ and $k_{\rm A}$, for $N = 999$ (a-c) and $N=300$ (d-f) and $k_{\rm B}=k_{\rm C}=1$. Here, red, green and blue denote the fixation of $A$, $B$ and $C$ while cyan denotes the fixation of $B$ and $C$ with (approximately) equal probability, see diagram of Fig. \ref{['fig1']}(d). In (a) and (d) the WTDs for the second and third reactions of (\ref{['eq:reac']}) are exponential. In (b-c) the second and third reactions of (\ref{['eq:reac']}) have power-law WTDs with $\alpha_{\rm B} \!=\!\alpha_{\rm C}\!=\! 10$ (b) and $\alpha_{\rm B} \!=\!\alpha_{\rm C}\!=\! 1.5$ (c). In (e-f) the second and third reactions of (\ref{['eq:reac']}) have gamma WTDs with $\alpha_{\rm B}=\alpha_{\rm C}= 0.9$ (e) and $\alpha_{\rm B}=\alpha_{\rm C} = 0.5$ (f). In (a-b) and (d-e) we compare our results to the theoretical curve for $k^*(\alpha)$ (dashed lines): Eq. (\ref{['kstar-PL']}) for (a-b) and Eq. (\ref{['kstar-full']}) for (d-e); the dotted lines denote $k_{\rm A}=1$.
• Figure 5: RGB fixation heatmaps (based on the diagram of Fig. \ref{['fig1']}) for power-law (a) and gamma (b) WTDs of the first reaction of \ref{['eq:reac']}: mean interevent time $\langle \tau_{\rm A}\rangle$ vs. coefficient of variation ${\rm CV_{\rm A}}$. Here $N \!=\! 999$, $k_{\rm B} \!=\! k_{\rm C} \!=\! 1$, and $\alpha_{\rm A}\!\leq \!1$ in (b).