Eco-evolutionary dynamics of social dilemmas
Introduction
The theory of evolution is based on Darwinian selection, mutation and drift. These forces along with neo-Darwinian additions of phenotypic variability, frequency-dependence and, in particular, cooperative interactions within and between species, form the basis for major transitions in evolution (Maynard Smith and Szathmáry, 1995, Nowak and Sigmund, 2004). Ecological effects such as varying population densities or changing environments are typically assumed to be minimal because they often arise on faster timescales such that only ecological averages matter for evolutionary processes. Consequently, evolutionary and ecological dynamics have been studied independently for long. While this assumption is justified in some situations, it does not apply whenever timescales of ecological and evolutionary dynamics are comparable (Day and Gandon, 2007). In such cases, ecological and evolutionary feedback may contribute to the unfolding of the evolutionary process. Empirically, effects of changes in population size are well documented (Dobson and Hudson, 1995, Bohannan and Lenski, 1999, Hudson et al., 1998, Fenner and Fantini, 1999, Bohannan and Lenski, 2000) and has now lead to a burgeoning field in evolutionary theory, which incorporates ecological variation (May and Anderson, 1983, Frank, 1991, Heesterbeek and Roberts, 1995, Roberts et al., 1995, Kirby and Burdon, 1997, Gandon and Nuismer, 2009, Salathé et al., 2005, Quigley et al., 2012, Gokhale et al., 2013, Song et al., 2015).
In particular, the independent study of ecological and evolutionary processes may not be able to capture the complex dynamics that often emerge in the combined system. Such potentially rich eco-evolutionary dynamics has been explored theoretically and, more recently, empirically confirmed (Post and Palkovacs, 2009, Hanski, 2011, Sanchez and Gore, 2013). Population genetics and adaptive dynamics readily embrace ecological scenarios (see e.g. Pagie and Hogeweg, 1999, Aviles, 1999, Yoshida et al., 2003, Day, 2005, Hauert et al., 2006b, Day and Gandon, 2006, Day and Gandon, 2007, Lion and Gandon, 2009, Jones et al., 2009, Gandon and Day, 2009, Wakano et al., 2009, Cremer et al., 2011) whereas the traditional focus of evolutionary game theory lies on trait frequencies or constant population sizes (Taylor and Jonker, 1978, Hofbauer and Sigmund, 1998, Nowak et al., 2004). Here we propose ways to incorporate intricacies of ecological dynamics along with environmental variation in evolutionary games.
Evolutionary game dynamics is typically assumed to take place in a population of individuals with fixed types or ‘strategies’, which determine their behaviour in interactions with other members of the population (Maynard Smith and Price, 1973, Zeeman, 1981). Payoffs determine the success of each strategy. Strategies that perform better than the average increase in abundance. This is the essence of the replicator equation (Hofbauer and Sigmund, 1998) but neglects that evolutionary changes may alter the population dynamics or vice versa. Traditionally the population consists of two strategies whose frequencies are given by and . In order to incorporate ecological dynamics we assume that and are (normalized) densities of the two strategies with (Hauert et al., 2006a). Consequently, provides a measure for reproductive opportunities, e.g. available space. Ecological dynamics is reflected in the change of the population density, . The evolutionary dynamics of the strategies is affected by intrinsic changes in population density as well as extrinsic sources such as seasonal fluctuations in the interaction parameters and hence the payoffs. For example, in epidemiology the coevolutionary dynamics of virulence and transmission rate of pathogens depends on ecological parameters of the host population. More specifically, changes in the mortality rate of hosts evokes a direct response in the transmission rate of pathogens while virulence covaries with transmission (Day and Gandon, 2006). Another approach to implement eco-evolutionary feedback is, for example, to explicitly model spatial structure and the resulting reproductive constraints (Lion and Gandon, 2009, Alizon and Taylor, 2008, Le Gaillard et al., 2003, Van Baalen and Rand, 1998), which then requires approximations in terms of weak selection or moment closures to derive an analytically tractable framework. In contrast, while our model neglects spatial correlations, it enables a more detailed look at evolutionary consequences arising from intrinsically and extrinsically driven ecological changes.
Social dilemmas occur whenever groups of cooperators perform better than groups of defectors but in mixed groups defectors outcompete cooperators (Dawes, 1980). This creates conflicts of interest between the individual and the group. In traditional (linear) public goods (PG) interactions cooperators contribute a fixed amount to a common pool, while defectors contribute nothing. In a group of size with cooperators the payoff for defectors is where denotes the multiplication factor of cooperative investments and reflects that the public good is a valuable resource. Similarly, cooperators receive , where the second equality highlights that cooperators ‘see’ one less cooperator among their co-players and illustrates that the net costs of cooperation are because a share of the benefits produced by a cooperator returns to itself. Therefore, it becomes beneficial to switch to cooperation for large multiplication factors, , but defectors nevertheless keep outperforming cooperators in mixed groups. The total investment in the PG is based on the number of cooperators in the group but the benefits returned by the common resource may depend non-linearly on the total investments. For example, the marginal benefits provided additional cooperators may decrease, which is often termed diminishing returns. Conversely, adding more cooperators could synergistically increase the benefits produced as in economies of scale. While well studied in economics (Taylor and Ward, 1982, Kollock, 1998, Schelling, 2006) such ideas were touched upon earlier in biology (Eshel and Motro, 1988) but only recently have they garnered renewed attention (Bach et al., 2006, Hauert et al., 2006b, Wakano et al., 2009, Pacheco et al., 2009, Wakano and Hauert, 2011, Archetti et al., 2011, Purcell et al., 2012, Peña et al., 2014, Peña et al., 2015).
The nonlinearity in PG can be captured by introducing a parameter , which rescales the effective value of contributions by cooperators based on the number of cooperators present (Hauert et al., 2006b). Hence, the payoff for defectors, , and cooperators, , respectively, is given by, such that the benefits provided by each additional cooperator are either discounted, , or synergistically enhanced, . The classic, linear PG is recovered for . This parametrization provides a general framework for the study of cooperation and recovers all traditional scenarios of social dilemmas as special cases (Nowak and Sigmund, 2004, Hauert et al., 2006b).
Section snippets
Eco-evolutionary dynamics
The overall population density, , can grow or shrink from 0 (extinction) to an absolute maximum of 1 (normalization). The average payoffs of cooperators and defectors, and , determine their respective birth rates but individuals can successfully reproduce if reproductive opportunities, , are available. All individuals are assumed to die at equal and constant rate, . Formally, changes in frequencies of cooperators and defectors over time are governed by the following extension of the
Environmental fluctuations
A constant feature of evolutionary as well as ecological processes is their dynamic nature. However, most evolutionary models assume a deterministic and usually constant environment in which populations evolve—either deterministically or stochastically (Taylor and Jonker, 1978, Hofbauer and Sigmund, 1998, Nowak et al., 2004, Moran, 1962). Considering variable environments is a natural way of incorporating changing ecological conditions. Stochastic or periodic fluctuations in the environment may
Discussion
Evolutionary models of social interactions traditionally assume a separation of timescales from ecological processes such that evolutionary selection always acts on ecological equilibria. However, ecological ‘equilibria’ may not simply refer to stable population sizes but also oscillatory dynamics based on stable limit cycles and a clear separation of timescales may not always apply. Nevertheless, two prominent theoretical frameworks for modelling frequency-dependent evolutionary processes
Acknowledgements
We thank Christian Hilbe, Bin Wu, and Arne Traulsen for helpful discussions. C.S.G. acknowledges financial support from the New Zealand Institute for Advanced Study (NZIAS) and the Max Planck Society. C.H. acknowledges the hospitality of the Max Planck Institute for Evolutionary Biology, Plön, Germany and financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Foundational Questions in Evolutionary Biology Fund (FQEB), grant RFP-12-10.
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2021, Journal of Theoretical BiologyCitation Excerpt :This approach is largely based on the simple matrix games, where payoff matrices describe the excess from the average growth rate in the population for the respective strategies. To add the necessary ecological details and to describe the models in terms of measurable parameters, the classical approach has been expressed in terms of the demographic vital rates (Argasinski and Broom, 2013a, 2018a,b; Zhang and Hui, 2011; Hauert et al., 2006, 2008; Huang et al., 2015; Gokhale and Hauert 2016). In this approach, instead of a single payoff function, there are separate payoff functions describing the mortality (probability of death during interaction) and fertility (offspring number resulting from the interaction).