Spatial invasion of cooperation
Introduction
Cooperation is a fundamental principle of biological systems that organizes lower level entities into higher level units—genes form chromosomes, cells form organisms, and individuals form societies (Maynard Smith and Szathmáry, 1995). However, the emergence of cooperation poses an enduring challenge to evolutionary biologists: if cooperation is costly to the individual and benefits only the interaction partners, then Darwinian selection should favour non-cooperating defectors and eliminate cooperation. In the absence of supporting mechanisms, this outcome is inevitable, despite the fact that mutual cooperation is preferred over mutual defection. The most prominent mathematical metaphor to study such interactions is given by the prisoner's dilemma: in pairwise interactions, cooperation (C) provides a benefit b to the partner at some cost c to the cooperator , while defection (D) neither bears any costs nor provides any benefits. The net gains for the player's joint behaviour can be written in the form of a payoff matrix:Strictly speaking the prisoner's dilemma is defined in terms of the ranking of the four payoffs. This particular parameterization in terms of b and c is biologically intuitive and mathematically convenient. The crucial point is that defection pays more irrespective of the partner's decision and is thus the dominant strategy. Cooperators will therefore dwindle and eventually everybody ends up with a payoff of zero instead of the more favourable reward for mutual cooperation . This characterizes the conflict of interest between individuals and the group, which defines social dilemmas (Dawes, 1980, Hauert et al., 2006). Over the last decades, different mechanisms have been proposed to promote and maintain cooperation (Hamilton, 1964, Hauert et al., 2002, Hauert et al., 2007, Nowak, 2006b, Nowak and Sigmund, 1998, Trivers, 1971, Wilson and Sober, 1994) including spatially structured populations with limited local interactions (Nowak and May, 1992). If individuals are arranged on a lattice and interact only with their nearest neighbours, then cooperators may thrive by forming compact clusters, which increases interactions with other cooperators while reducing exploitation by defectors.
Spatial structure affects the evolutionary process in general games, i.e. in pairwise interactions with two strategic options (Hauert, 2002, Ohtsuki and Nowak, 2006a), and notably enables cooperators to survive in populations playing the prisoner's dilemma. Considering the equilibrium frequencies of cooperators and defectors in lattice populations demonstrates that the clustering advantages are substantial for small cost-to-benefit ratios , but are unable to offset the exploitation by defectors above a threshold value, , such that cooperators disappear (Szabó and Tőke, 1998). For increasing , the system undergoes a critical phase transition, characterized by diverging fluctuations in the cooperator and defector frequencies (Szabó and Hauert, 2002a). These results have led to the common belief that spatial structure is necessarily beneficial for cooperation. While this holds for prisoner's dilemma interactions, it is not universally applicable. In fact, in the snowdrift game—a closely related social dilemma with relaxed conditions such that cooperators and defectors can co-exist under conditions where cooperators are doomed in the prisoner's dilemma—spatial structure often turns out to be detrimental to cooperation (Doebeli and Hauert, 2005, Hauert, 2006a, Hauert and Doebeli, 2004).
In finite populations, evolution is stochastic such that the combination of selection and random drift eventually leads to the fixation of one or the other strategic type (Nowak, 2006a, Nowak et al., 2004). In such situations, cooperation is favoured if the fixation probability of a single cooperator, , in a defector population exceeds the fixation probability of a neutral mutant ( where N is the population size). For weak selection, i.e. if payoff differences between cooperators and defectors are small, is analytically accessible for various types of microscopic updating mechanisms (Ohtsuki et al., 2006, Taylor et al., 2007). In particular, for the payoff matrix (1), the condition implies that the fixation probability of a single defector, , in a cooperator population is (Taylor et al., 2007, Wild and Traulsen, 2007). Hence, if mutations are rare, the population spends more time in the cooperator state than in the defector state. In the prisoner's dilemma, if the death of a randomly chosen individual triggers a competition among its neighbours to repopulate the vacant site with a success rate proportional to their payoffs, then a particularly simple condition is obtained: evolution favours cooperation whenever holds, where k denotes the number of interaction partners.
This work complements studies on prisoner's dilemma games in structured populations by investigating the process of cooperators invading a world of defectors. We demonstrate that after an initial relaxation time, the number of cooperators always grows quadratically irrespective of the cost-to-benefit ratio , and we find that two distinct modes of growth exist: (i) for small , cooperators expand essentially as a single ever growing cluster whereas (ii) for larger , cooperators form an increasing number of small clusters with little variation in size. Our simulations confirm that the probability of invasion is essentially independent of the initial number of cooperators provided that they form at least a cluster (Hauert, 2001, Killingback et al., 1999, Page et al., 2000). In addition, our simulations show that behind the invasion front, cooperators and defectors quickly reach a local equilibrium, which supports analytical results based on pair approximation (Ellner et al., 1998, Le Gaillard et al., 2003, Ohtsuki et al., 2006, van Baalen and Rand, 1998).
Section snippets
Model
In order to investigate the invasion dynamics of cooperators in detail, consider a square lattice where every site is occupied by a single individual. Initially, all individuals are defectors, except for a cluster of cooperators in the centre of the lattice ( and 30). Each individual engages in pairwise interactions within its Moore neighbourhood, i.e. with the eight nearest neighbours reachable by a chess king's move. The payoffs accrued in these interactions determine the
Cooperator survival
Cooperation is inherently prone to exploitation by defectors and thus the survival probability of cooperators, , hinges on the cooperator's ability to offset the costs of cooperation with benefits accrued from interactions with other cooperators. A single cooperator in a sea of defectors performs poorly, and its only hope is to propagate its strategy through random drift. Since players never adopt worse performing strategies in our setup, single cooperators never survive and readily
Conclusions
Under favourable conditions, i.e. for low costs and high benefits, cooperators are able to invade a spatially extended world of defectors. The ability to form clusters enables cooperators to persist, because spatial aggregation enables more frequent interactions with other cooperators while reducing exploitation by defectors. The invasion of cooperators occurs in two phases: during the first phase, the number of cooperators increases slowly because the expansion of cooperators is partly offset
Acknowledgments
Ph.L. acknowledges financial support from the Swiss National Science Foundation, Grant PBLAA-106056. C.H. & M.A.N. are supported by the John Templeton Foundation and the NSF/NIH joint program in mathematical biology (NIH Grant R01GM078986). The Program for Evolutionary Dynamics (PED) at Harvard University is sponsored by Jeffrey Epstein.
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