Joint evolution of altruistic cooperation and dispersal in a metapopulation of small local populations

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Abstract

We investigate the joint evolution of public goods cooperation and dispersal in a metapopulation model with small local populations. Altruistic cooperation can evolve due to assortment and kin selection, and dispersal can evolve because of demographic stochasticity, catastrophes and kin selection. Metapopulation structures resulting in assortment have been shown to make selection for cooperation possible. But how does dispersal affect cooperation and vice versa, when both are allowed to evolve as continuous traits? We found four qualitatively different evolutionary outcomes. (1) Monomorphic evolution to full defection with positive dispersal. (2) Monomorphic evolution to an evolutionarily stable state with positive cooperation and dispersal. In this case, parameter changes selecting for increased cooperation typically also select for increased dispersal. (3) Evolutionary branching can result in the evolutionarily stable coexistence of defectors and cooperators. Although defectors could be expected to disperse more than cooperators, here we show that the opposite case is also possible: Defectors tend to disperse less than cooperators when the total amount of cooperation in the dimorphic population is low enough. (4) Selection for too low cooperation can cause the extinction of the evolving population. For moderate catastrophe rates dispersal needs to be initially very frequent for evolutionary suicide to occur. Although selection for less dispersal in principle could prevent such evolutionary suicide, in most cases this rescuing effect is not sufficient, because selection in the cooperation trait is typically much stronger. If the catastrophe rate is large enough, a part of the boundary of viability can be evolutionarily attracting with respect to both strategy components, in which case evolutionary suicide is expected from all initial conditions.

Highlights

► We study a structured metapopulation model with realistic population dynamics. ► The joint evolution of cooperation and dispersal is investigated. ► Evolutionary branching can result in the coexistence of cooperators and defectors. ► Novel difference in the dispersal behaviour of cooperators and defectors is observed. ► Rescuing effect of dispersal does not necessarily prevent evolutionary suicide.

Introduction

Cooperation — how can it emerge and be maintained when threatened by free-riders, individuals who do not cooperate themselves, but enjoy all the benefits from the cooperation of others? Various mechanisms promoting cooperation have been proposed. Potential cooperation events may be repeated many times between the same individuals. If an individual decides not to cooperate, the partner is likely to remember such defecting act, and not to give help in the future either. Therefore, direct reciprocity may promote cooperation (Trivers, 1971). In case pairwise interactions are repeated, but not between the same individuals, by defecting one may get a bad reputation, and that way risk the possibility of receiving help in the future. Therefore, indirect reciprocity (Alexander, 1979, Sugden, 1986, Alexander, 1987, Nowak and Sigmund, 1998a, Nowak and Sigmund, 1998b, Nowak and Sigmund, 2005) may promote cooperation, depending on the social norm (Ohtsuki and Iwasa, 2006).

Assortment has also been proposed to be a mechanism promoting cooperation (Fletcher and Doebeli, 2009). Whatever the mechanism causing assortment is, if a cooperating individual is more likely to interact with cooperating individuals than a defecting individual is, the direct benefit received from them may promote cooperation. Assortment can occur for example in populations that are not well-mixed, and in which cooperation is typically possible only among nearby individuals. In such a population individuals are likely to have some relatives around. Giving help to relatives is often beneficial from an evolutionary point of view, and thus kin selection (Hamilton, 1964a, Hamilton, 1964b) may promote cooperation. Metapopulations, collections of local populations connected by dispersal (Levins, 1969, Levins, 1970), provide a natural setup for assortment. This has given motivation to study the evolution of cooperation in metapopulation models or in other spatially heterogeneous populations (Le Galliard et al., 2003, Parvinen, 2011).

Dispersal is a key feature in metapopulations, and its evolution has been widely studied in various types of models (for references see this and the following paragraphs). Already Hamilton and May (1977) observed that kin selection can promote dispersal even in stable habitats, see also (Motro, 1982a, Motro, 1982b, Motro, 1983, Frank, 1986): If the relatedness between a focal individual and other individuals in the present patch is high, by dispersing the focal individual reduces kin competition. In large local populations this mechanism is not effective, because the relatedness between individuals in a large local population is essentially zero. When large local populations show equilibrium dynamics, dispersal is typically selected against (Hastings, 1983, Parvinen, 1999), because there is no benefit to disperse between patches with similar conditions. Temporal heterogeneity has been shown to be a mechanism promoting dispersal, because it results in sparsely populated patches, into which immigration typically is beneficial. Such temporal heterogeneity can result, e.g., from non-equilibrium local population dynamics (Doebeli, 1995, Doebeli and Ruxton, 1997, Holt and McPeek, 1996, Parvinen, 1999) or catastrophes (Gyllenberg and Metz, 2001, Gyllenberg et al., 2002, Parvinen, 2002, Parvinen, 2006, Ronce et al., 2000). In case of small local populations, local population dynamics is necessarily stochastic resulting in temporal heterogeneity, which together with kin selection promotes dispersal (Metz and Gyllenberg, 2001, Parvinen et al., 2003, Parvinen and Metz, 2008, Parvinen et al., 2012).

Direct costs of dispersal are in general expected to select against dispersal, but also opposite effects have been observed (Comins et al., 1980, Gandon and Michalakis, 1999, Heino and Hanski, 2001). Gandon and Michalakis (1999) suggested that increases in the direct costs of dispersal may cause more and more competition between highly related philopatric individuals, which may eventually select for dispersal. Local adaptation (Kisdi, 2002, Nurmi and Parvinen, 2011) is also expected to select against dispersal, because a dispersing individual who is well adapted to the conditions in the present location, takes the risk of arriving into a patch with conditions for which it is maladapted.

It is obvious that dispersal affects the evolution of cooperation: for low dispersal rates the relatedness between individuals in a small local population can be high, and cooperation can evolve. Increasing the dispersal rate is expected to have a direct decreasing effect on relatedness, and thus to make cooperation less favourable. This is, however, not always the case (Parvinen, 2011). Although such an effect may first seem counterintuitive, note that increasing the dispersal rate has also an indirect increasing effect on relatedness through decreasing the average local population size. Furthermore, for high dispersal rates, Parvinen (2011) observed evolution to too low cooperation resulting in evolutionary suicide (Ferrière, 2000, Parvinen, 2005, Matsuda and Abrams, 1994).

Cooperation is also likely to affect the evolution of dispersal. According to first intuition, a cooperating individual could be expected to be willing to remain close to its kin and thus to disperse only moderately. On the other hand, cooperation will also affect the evolution of dispersal by increasing the local population size, which is expected to select for dispersal. These effects give motivation for the study of the interplay of dispersal and cooperation, about which only few studies exist (Hochberg et al., 2008, Pfeiffer and Bonhoeffer, 2003).

Pfeiffer and Bonhoeffer (2003) used a game-theoretical approach to study cooperation and dispersal in a spatial setting, in which cooperating cells produce ATP only with the efficient but slow method of respiration, and defectors use also the faster method of fermentation, which in total is less efficient. Population dynamics with two cell types, in case all cells are mobile, showed that cooperators dominate when cell motion is low, and defectors dominate when cell motion is high. This result is analogous to results in the model studied in this article, when only cooperation is evolving (Parvinen, 2011). Furthermore, Pfeiffer and Bonhoeffer (2003) assumed that non-dispersing (clustering) cells remain attached to the mother cell in contrast with dispersing (mobile) cells. Population dynamics of the resulting four different cell types showed that for high resource influx, non-dispersing cooperators (clustering respirator) dominate. Although their result is interesting, it leaves open the question how will dispersal and cooperation evolve, when they are continuously varying traits under the natural selection. Hochberg et al. (2008) partly investigated this question by studying a model, in which the dispersal rate of cooperators and the dispersal rate of defectors were evolving. Cooperation itself was not allowed to evolve as a continuously varying trait, but in their model 2, the relative abundance of cooperators and defectors was allowed to change. However, they did “not explicitly consider dynamics, such as group founding, group numbers, individual emigration and immigration, and competition for limiting resources within or between groups”. The novelty of this article is in that we study the joint evolution of dispersal and cooperation, when both are continuously varying traits under natural selection, in a metapopulation model with realistic (local) population dynamics.

Our choice of a model is a metapopulation model with infinitely many patches with small (finite) local populations, in which the evolution of dispersal (Cadet et al., 2003, Metz and Gyllenberg, 2001, Parvinen et al., 2003, Parvinen and Metz, 2008, Parvinen et al., 2012) and the evolution of cooperation (Parvinen, 2011) and other traits (Jesse et al., 2011) have already been studied separately. The benefit of this model type is that it is rather realistic concerning local population dynamics, in which stochastic events of birth, death, emigration and immigration are accounted for. Also, a well-defined proxy for invasion fitness is available (Metz and Gyllenberg, 2001), which allows us to perform efficient evolutionary analysis with methods of adaptive dynamics, instead of relying only on time-consuming simulations.

Section snippets

Model and methods

Our model is a structured metapopulation model with infinitely many habitat patches with small local populations (Metz and Gyllenberg, 2001). In short, our model is ecologically almost the same as the model by Parvinen (2011), but here in addition to studying the evolution of cooperation we allow also the dispersal rate to evolve. For the sake of completeness, we give the ecological details here. For the calculation of invasion fitness, see the references above.

Results

First we investigate the joint evolution of cooperation and dispersal in the absence of  cooperation in local populations of size one (Ñ=2). Fig. 2 shows zero-isoclines of the components of the fitness gradient, and thus illustrates the potential direction of monomorphic adaptive dynamics of cooperation and dispersal, as well as possible outcomes, in the absence of mutational covariance.

First consider small values of the catastrophe rate μ, Fig. 2a. Although a part of the boundary of the

Conclusion

In addition to direct (Trivers, 1971) and indirect reciprocity  (Alexander, 1979, Alexander, 1987, Nowak and Sigmund, 1998a, Nowak and Sigmund, 1998b, Nowak and Sigmund, 2005, Ohtsuki and Iwasa, 2006, Sugden, 1986), assortment (Fletcher and Doebeli, 2009) has been proposed as a general mechanism for promoting cooperation. Such assortment naturally occurs in metapopulation models, which consist of small local populations connected with dispersal. As observed previously (Parvinen, 2011),

Acknowledgment

The author wishes to thank the Academy of Finland for the financial support (project number 128323).

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