Mobility, fitness collection, and the breakdown of cooperation

Anatolij Gelimson, Jonas Cremer, and Erwin Frey

Phys. Rev. E 87, 042711 – Published 15 April 2013

Abstract

The spatial arrangement of individuals is thought to overcome the dilemma of cooperation: When cooperators engage in clusters, they might share the benefit of cooperation while being more protected against noncooperating individuals, who benefit from cooperation but save the cost of cooperation. This is paradigmatically shown by the spatial prisoner's dilemma model. Here, we study this model in one and two spatial dimensions, but explicitly take into account that in biological setups, fitness collection and selection are separated processes occurring mostly on vastly different time scales. This separation is particularly important to understand the impact of mobility on the evolution of cooperation. We find that even small diffusive mobility strongly restricts cooperation since it enables noncooperative individuals to invade cooperative clusters. Thus, in most biological scenarios, where the mobility of competing individuals is an irrefutable fact, the spatial prisoner's dilemma alone cannot explain stable cooperation, but additional mechanisms are necessary for spatial structure to promote the evolution of cooperation. The breakdown of cooperation is analyzed in detail. We confirm the existence of a phase transition, here controlled by mobility and costs, which distinguishes between purely cooperative and noncooperative absorbing states. While in one dimension the model is in the class of the voter model, it belongs to the directed percolation universality class in two dimensions.

Received 22 May 2012

DOI:https://doi.org/10.1103/PhysRevE.87.042711

Authors & Affiliations

Anatolij Gelimson^1,2, Jonas Cremer^1,3, and Erwin Frey¹

¹Arnold Sommerfeld Center for Theoretical Physics and Center for NanoScience, Department of Physics, Ludwig-Maximilians-Universität München, Theresienstrasse 37, D-80333 München, Germany
²The Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, Oxford OX1 3NP, United Kingdom
³Center for Theoretical Biological Physics, University of California at San Diego, La Jolla, California 92093, USA

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Vol. 87, Iss. 4 — April 2013

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Images

Figure 1
Lattice occupation (vertical axis) vs time (horizontal axis) of the one-dimensional dynamics for vanishing mobility, $M = 0$ . The lattice size is $N = 500$ . Starting with a random distribution of $50 %$ cooperators [dark gray (blue)] and $50 %$ free riders [light gray (red)] on the lattice, clusters are formed. Preferentially, the clusters of one type grow, such that only cooperators or only free riders remain in the end. (a) The cooperation costs $c$ are $0.1$ . For these low costs, cooperators take over the population. (b) $c = 0.347$ ; the cooperation costs are very close to the critical value $c_{0}$ . The cluster extension and the cluster lifetime both show a power-law divergence as $c \to c_{0}$ for increasing $N$ . (c) $c = 0.5$ ; for high costs, cooperators die out and only free riders survive.Reuse & Permissions
Figure 2
Snapshots of the two-dimensional lattice and cluster formation for vanishing mobility, $M = 0$ , at different times $t$ . Starting with a random distribution of half cooperators [dark gray (blue)] and half free riders [light gray (red)] on the lattice (a), clusters are formed (b). Preferentially, the clusters of one type grow (c), such that only cooperators or only free riders remain in the end. This simulation was performed on a $100 \times 100$ lattice; the cooperation costs were $c = 0.1$ .Reuse & Permissions
Figure 3
Stationary fraction $x$ of cooperators as a function of the costs of cooperation $c$ and the mobility $M$ for (a) the one-dimensional and (b) the two-dimensional model. There is a phase transition between stationary states where only cooperation [dark gray (blue)] or only noncooperation [light gray (red)] prevails. (a) $N = 500$ ; (b) $N = 100$ .Reuse & Permissions
Figure 4
Illustration of a simplified domain wall picture for the (a) one-dimensional and (b) two-dimensional lattice. The domain wall dynamics is constrained to the white area where transitions between all possible arrangements of cooperators [dark gray (blue)] and free riders [light gray (red)] are possible, as indicated in the graphs. The configuration of cooperators and free riders outside of the white area is considered to remain static. The one-dimensional domain wall has moved forward or backward if the dynamic sites in the white area are in the all-cooperator state $a_{3}$ or the all-free-rider state $a_{4}$ , respectively. Similarly, the section considered for the two-dimensional domain wall has moved forward or backward if the dynamic sites are in states $b_{9}$ or $b_{10}$ , respectively.Reuse & Permissions
Figure 5
One-dimensional model: Finite-size scaling for varied costs $c$ at (a) $M = 0$ , (b) $M = 0.5$ . $Δ = | c - c_{M} |$ . With the determined exponents $ν_{∥}^{\pm}$ and $ν_{⊥}^{\pm}$ , and the critical costs [ $c_{M} = 0.347 \pm 0.0008$ in case (a) and $c_{M} = 0.3620 \pm 0.0004$ in case (b)], the data for $ξ_{∥}$ and $ξ_{⊥}$ at different system sizes collapsed onto the master curves $F^{\pm}$ and $G^{\pm}$ ; see main text.Reuse & Permissions
Figure 6
Two-dimensional model: Finite-size scaling for varied costs $c$ at (a) $M = 0$ , (b) $M = 0.5$ . $Δ = | c - c_{M} |$ . With the determined exponents $ν_{∥}^{\pm}$ and $ν_{⊥}^{\pm}$ , and the critical costs [ $c_{M} = 0.163 \pm 0.0005$ in case (a) and $c_{M} = 0.1605 \pm 0.0007$ in case (b)], the data for $ξ_{∥}$ and $ξ_{⊥}$ at different system sizes collapsed onto the master curves $F^{\pm}$ and $G^{\pm}$ ; see main text.Reuse & Permissions

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