Evolutionary dynamics for persistent cooperation in structured populations

The emergence and maintenance of cooperative behavior is a fascinating topic in evolutionary biology and social science. The public goods game (PGG) is a paradigm for exploring cooperative behavior. In PGG, the total resulting payoff is divided equally among all participants. This feature still leads to the dominance of defection without substantially magnifying the public good by a multiplying factor. Much effort has been made to explain the evolution of cooperative strategies, including a recent model in which only a portion of the total benefit is shared by all the players through introducing a new strategy named persistent cooperation. A persistent cooperator is a contributor who is willing to pay a second cost to retrieve the remaining portion of the payoff contributed by themselves. In a previous study, this model was analyzed in the framework of well-mixed populations. This paper focuses on discussing the persistent cooperation in lattice-structured populations. The evolutionary dynamics of the structured populations consisting of three types of competing players (pure cooperators, defectors, and persistent cooperators) are revealed by theoretical analysis and numerical simulations. In particular, the approximate expressions of fixation probabilities for strategies are derived on one-dimensional lattices. The phase diagrams of stationary states, and the evolution of frequencies and spatial patterns for strategies are illustrated on both one-dimensional and square lattices by simulations. Our results are consistent with the general observation that, at least in most situations, a structured population facilitates the evolution of cooperation. Specifically, here we find that the existence of persistent cooperators greatly suppresses the spreading of defectors under more relaxed conditions in structured populations compared to that obtained in well-mixed populations.

Figure 1

(a) Full $s-d$ phase diagram obtained for $r=2,w=2$. Different phases are denoted by the symbols of strategies that survive finally in the equilibrium state. Solid lines indicate continuous transitions between different states. (b) A representative cross section of (a) at $d=0.4$ illustrating the frequencies of $C,D$, and $PC$ depending on $s$. (c) is a spatial pattern of the model indicating how the three strategies evolve within 400 full MCS, where $s=0.55,d=0.4$. (d) The changes of the three strategies with time as $s=0.66,d=0.4$.

Figure 2

Players with different strategies are aligned on a one-dimensional lattice, transitions only occur at the boundary sites between two runs of players with different strategies, and then the focal individual will be replaced by one of its nearest neighbors.

Figure 3

(a) and (c) The fixation probabilities for strategy $PC$ (green line) and $D$ (red line) in a one-dimensional lattice-structured population with $C,D$, and $PC$. (b) and (d) The fixation probabilities for strategy $PC$ (green line) and $D$ (red line) in a one-dimensional lattice-structured population with only $D$ and $PC$. Parameters are set as $r=2.5,d=0.4$ for all (a)–(d). Simulations are performed on $500\times 1$ lattice for the fixation probabilities of $PC$ (green solid dots) and $D$ (red solid diamonds), respectively.

Figure 4

A single $PC$ player invades successfully, reproduces inchmeal, and becomes fixed in a $500\times 1$ lattice-structured population. (a)–(c) are characteristic snapshots of the evolution dynamics over time for $PC$. The parameters are $r=2.5,w=2,s=0.45$, and $d=0.4$. Colors (blue, green, and red) show the distribution of cooperators, persistent cooperators, and defectors.

Figure 5

(a) Full $s-d$ phase diagram obtained for $r=2$. Different phases are denoted by the symbols of strategies that survive finally in the equilibrium state. Solid lines indicate continuous transitions between different states. (b) A representative cross section of (a) at $d=0.4$, illustrating the frequencies of $C,D$, and $PC$ in dependence on $s$.

Figure 6

Characteristic pictures of evolutionary dynamics of the three strategies on a $100\times 100$ square lattice with specially prepared initial distribution. Blue, green, and red show the location of cooperators $\left(C\right)$, persistent cooperators $\left(PC\right)$, and defectors $\left(D\right)$, respectively. The snapshots were taken at (a) $t=0$, (b) $t=50$, (c) $t=200$, and (d) $t=400$ full MCS, respectively. The parameter values were $r=2.0,s=0.5$, and $d=0.4$.

Figure 7

(a) Full $s-d$ phase diagram obtained for $r=3.5$. Different phases are denoted by symbols of strategies that survive finally in the equilibrium state. Solid lines indicate continuous transitions between different states. (b) A representative cross reference of (a) at $d=0.4$, illustrating the frequencies of $C,D$, and $PC$ depending on $s$.