Cyclic Dominance and Biodiversity in Well-Mixed Populations

Coevolutionary dynamics is investigated in chemical catalysis, biological evolution, social and economic systems. The dynamics of these systems can be analyzed within the unifying framework of evolutionary game theory. In this Letter, we show that even in well-mixed finite populations, where the dynamics is inherently stochastic, biodiversity is possible with three cyclic-dominant strategies. We show how the interplay of evolutionary dynamics, discreteness of the population, and the nature of the interactions influences the coexistence of strategies. We calculate a critical population size above which coexistence is likely.

Figure 1

Drift reversal for the four processes; the linearized Moran and Fermi process coincide within line thickness. The main panel shows the scaling with population size $N$ for fixed $w=0.45$ and $s=0.8$. Full lines and the dotted line show the numerical solutions of Eq. (11). The dashed lines are the analytical expansions Eqs. (15, 17). For small $N$, the continuum approximation to the integral leads to deviations. Inset: Average drift for varying payoff matrix with fixed $w=0.45$ and $N=1000$. The processes intersect in $\Delta H=1/(20N^2)$ at $s=1$ ($d=0$). For $N\to \infty$, this reduces to the expectation based on the replicator equation, $\Delta H=0$ in $d=0$. The intersection with the horizontal line indicates the drift reversal in finite populations.

Figure 2

Critical population size: Intensity of selection $w$ versus critical population size $N_c$, where $\Delta H$ changes its sign for fixed $s=0.8$. For $N>N_c$ the drift is towards the internal fixed point. The dashed lines refer to the Lizard’s payoff matrix [1] for MO and LU. Under strong selection $w=1$ the LU reverses at $N_c=20$. For the restricted random matrix ensemble (green circles, see text) under LU $N_c$ coincides with the case of Eq. (1). Each symbol corresponds to a single random payoff matrix. (inset: scatter plot of payoff entries in ▴ 1st, ▾ 2nd, and ▪ 3rd column).