Promotion of cooperation in deme-structured populations with growth-merging dynamics

TL;DR

This work analyzes how spatial structure promotes cooperation in deme-structured populations undergoing cycles of growth, merging and dilution under hard selection. It derives analytical conditions for the cooperator fraction to rise during deterministic logistic growth and for cooperation to be favored on average across full cycles via a multi-level Price equation framework, then computes the fixation probability of cooperative mutants under weak selection. The results reveal that a positive between-deme growth effect must overcome within-deme costs, and that small bottleneck sizes are essential to sustain variance across demes, enabling Simpson-like global increases in cooperation. Numerical simulations show that stochastic growth further enhances cooperation, underscoring the importance of demographic noise in these dynamics. Overall, hard selection is essential for promoting cooperation in deme-structured populations and helps reconcile previous findings from growth–merging cycles and spatial graph models.

Abstract

The spatial structure of populations may promote the emergence and maintenance of cooperation. Cooperation in the prisoner's dilemma is favored under specific update rules in evolutionary graph theory models with one individual per node of a graph, but this effect vanishes in models with well-mixed demes connected by migrations under soft selection. In contrast, experiments and models involving cycles of growth, merging and dilution have shown that spatial structure can favor cooperation. Here, we reconcile these findings by studying deme-structured populations under growth-merging-dilution dynamics, corresponding to a clique (fully connected graph) under hard selection. We obtain analytical conditions for the cooperator fraction to increase during deterministic logistic growth, and to increase on average under dilution-growth-merging cycles, in the weak selection regime. Furthermore, we analytically express the fixation probability of cooperators under weak selection, yielding a criterion for cooperative mutants to have a higher fixation probability than neutral ones. Finally, numerical simulations show that stochastic growth further promotes cooperation. Overall, hard selection is essential for cooperation to be promoted in deme-structured populations.

Figures (5)

• Figure 1: Representation of one full cycle of growth-merging-dilution of the model for $D=3$ demes. Blue markers represent cooperators, while red markers represent defectors. The initial state of the demes is shown on the left. The demes first undergo a local growth phase, followed by a merging where all individuals are put together. Then, the dilution step consists in three binomial samplings that determine the new bottleneck compositions of the three demes.
• Figure 2: Growth phase: evolution of the fraction of cooperators in a deme and in the population.(a) Excess fraction $\Delta x_1(\tau) = x_1(\tau) - x_1(0)$ of cooperators in one deme, namely deme $1$, and size $N_1$ of this deme, versus growth time $\tau$. We show the numerical solution of Eq. \ref{['equation_diff']} and the analytical approximation of Eq. \ref{['approximation']}, obtained in the weak selection regime. We indicate the long-time asymptotic value $\Delta x_{1,\text{lim}}$ of $\Delta x_1$, obtained from Eq. \ref{['approximation']}, and the carrying capacity $K$ (horizontal dotted lines). (b) Excess fraction $\Delta X (\tau)= X(\tau) - X(0)$ of cooperators in the total population versus growth time $\tau$. As in (a), we show the numerical solution and the analytical approximation, here given by Eq. \ref{['Delta_X_Final']}. In addition, we show the two terms from Eq. \ref{['Delta_Xa']} and Eq. \ref{['Delta_Xb']} into which $\Delta X(\tau)$ can be decomposed. We also show the largest time $\tau_0>0$ such that $\Delta X (\tau) \ge 0$ (vertical dotted line). In both panels, results are obtained for a population of $D=5$ demes, each with initial size $B=20$ and carrying capacity $K = 1000$, using initial compositions $x_1(\tau=0)=1/10$, $x_2(0)=1/4$, $x_3(0)=1/2$, $x_4(0)=3/4$ and $x_5(0)=9/10$, fitness intensity $w=10^{-3}$, benefit $b=10$ and cost $c=1$.
• Figure 3: Impact of the benefit-to-cost ratio and of the variance in deme compositions on the evolution of cooperator fraction during growth. We show a phase diagram, representing whether $\Delta X >0$ or $\Delta X<0$, as a function of the benefit-to-cost ratio $b/c$ and the variance $\mathrm{Var}(x_i)$ of the initial deme compositions. The separation between the two zones, namely $\Delta X = 0$, is given by Eq. \ref{['Condition_Growth']} of the Supplement. Results are obtained for a population of $D=5$ demes, each with initial size $B=20$ and carrying capacity $K = 1000$, using different initial compositions with mean $\langle x_i \rangle = 0.5$, fitness intensity $w=10^{-3}$, growth time $t=2$ and cost $c=1$.
• Figure 4: Fixation of cooperation. We show the fixation probability $p$ of cooperation, starting from a single cooperator in the whole population, versus the benefit-to-cost ratio $b/c$, both for our structured population and for a well-mixed ("WM") population with the same total bottleneck size $DB$ and carrying capacity $DK$. Round markers: numerical simulation ("simu.") results with deterministic growth ("det."); solid lines: corresponding analytical predictions ("ana.") from Eq. \ref{['fixation_probability']}. The threshold of $b/c$ beyond which cooperator fixation is predicted to be promoted (with deterministic growth), given by Eq. \ref{['Condition_Finale']}, is shown as a vertical dotted blue line. Cross markers: numerical simulation results with stochastic growth ("stoch."). The fixation probability of a neutral mutant, $p=1/(DB)$, is shown for reference (horizontal dashed line). Parameter values: $D=5$, $B=10$, $K=1000$ for a structured population; $D=1$, $B=50$, $K=5000$ for a well-mixed population; for both, $t=3$, $w=5 \times 10^{-4}$ and $c=1$. Simulation results are obtained over $N=10^8$ realizations.
• Figure S1: Exponential growth phase: evolution of the fraction of cooperators in a deme and in the population. Same as Fig. \ref{['Growth_Phase']} in the main text, but with exponential growth. (a) Excess fraction $\Delta x_1 (\tau)= x_1(\tau) - x_1(0)$ of cooperators in one deme, namely deme $1$, and size $N_1(\tau)$ of this deme, versus growth time $\tau$. We show the numerical solution of Eq. \ref{['equation_diff']} for $K=10^8$ and our analytical approximation of Eq. \ref{['sol_x_exp']} and Eq. \ref{['sol_N_exp']}, obtained in the weak selection regime. (b) Excess fraction $\Delta X = X(\tau) - X(0)$ of cooperators in the total population versus growth time $\tau$. As in (a), we show the numerical solution and the analytical approximation in the weak selection regime. Here, the latter is given by Eq. \ref{['Delta_X_Final_Exponential']}. In addition, we show the two terms into which we decompose Eq. \ref{['Delta_X_Final_Exponential']}. As in Fig. \ref{['Growth_Phase']} in the main text, results are obtained for a population of $D=5$ demes, each with initial size $B=20$, using initial compositions $x_1(\tau=0)=1/10$, $x_2(0)=1/4$, $x_3(0)=1/2$, $x_4(0)=3/4$ and $x_5(0)=9/10$, fitness intensity $w=10^{-3}$, benefit $b=10$ and cost $c=1$.