Evolution of cooperation by multilevel selection

Abstract

We propose a minimalist stochastic model of multilevel (or group) selection. A population is subdivided into groups. Individuals interact with other members of the group in an evolutionary game that determines their fitness. Individuals reproduce, and offspring are added to the same group. If a group reaches a certain size, it can split into two. Faster reproducing individuals lead to larger groups that split more often. In our model, higher-level selection emerges as a byproduct of individual reproduction and population structure. We derive a fundamental condition for the evolution of cooperation by group selection: if b͞c > 1 ؉ n͞m, then group selection favors cooperation. The parameters b and c denote the benefit and cost of the altruistic act, whereas n and m denote the maximum group size and the number of groups. The model can be extended to more than two levels of selection and to include migration.finite populations ͉ prisoner's dilemma ͉ group selection ͉ fixation probability ͉ stochastic process C ompetition between groups can lead to selection of cooperative behavior. This idea can be traced back to Charles Darwin, who wrote in 1871: ''There can be no doubt that a tribe including many members who..were always ready to give aid to each other and to sacrifice themselves for the common good, would be victorious over other tribes; and this would be natural selection'' (1). The first mathematical model of group selection was proposed in 1945 by Sewall Wright (2). The enthusiastic attempt of early group selectionists to understand all of the evolution of altruism from this one perspective (3, 4) has led to vigorous criticism and a general denial of such ideas for decades (5-8). Only a small number of biologists continued to work in this area (9-19). Over many years, D. S. Wilson was the main proponent of the idea of group selection (20-22). Nowadays, there seems to be a renewed interest in the subject, as demonstrated by many empirical and theoretical studies (23-28). The current analysis of group selection is also closely related to the attempt at understanding the simultaneous effect of natural selection on multiple-levels (29-31). In our opinion, group selection is an important organizing principle that permeates evolutionary processes from the emergence of the first cells to eusociality and the economics of nations.Consider a population that is subdivided into groups. The fitness of individuals is determined by the payoff from an evolutionary game. Interactions occur between members of the same group. We model stochastic evolutionary dynamics. In any one time step, a single individual from the entire population is chosen for reproduction with a probability proportional to its fitness. The offspring is added to the same group. If the group reaches a critical size, n, it will divide into two groups with probability q. The members of the group are randomly distributed over the two daughter groups, see Fig. 1. With probability 1Ϫ q, the group does not divide, but a random individual of the group is eliminate...

Positive effects of multiple gene control on the spread of altruism by group selection

“…In accordance with [24,25,26] we get linear dependency of r on η and in accordance with [24,26] we also get linear dependency of r on μ. In discordance with [24,26] the slope of the dependencies (i.e., ∂R ∂η ) changes with different parameters n, m. The slope change is roughly in agreement with [25], although there is an inaccuracy of about 30% in its actual quantity. Furthermore, the dependencies of r on n and μ are not analyzed in [25].…”

“…This is specific for the discussed model where altruistic demes thrive and out produce other altruistic demes while selfish demes shrink and therefore produce fewer migrants. When intrademic selfishness only results in higher probability of deme extinction (but until extinction the size of the deme is constant, as is the case for example in [24], which presents a model that appears to be less realistic than ours), the values of R are significantly higher.…”

“…Now we present how r depends on number of demes n. The figure 5 shows dependencies for three different values m (average number of individuals in deme). According to several papers such as [24], [3] we would expect these functions to be of a type a/x or a + b/x. Therefore we fitted the data-points with functions of a type a + b/x with a, b constant.…”

“…Once we test the results in a more general scenario, inaccuracies naturally arise. As an example, the stochasticity of the system is neglected [25], only demes containing either only altruistic or only selfish individuals are reproduced [24] or the authors neglect of strong correlations between dependent random variables [27]. This shows that solving this problem analytically is very hard and resists numerous different approaches, no matter how inventive they are in using a wide range of mathematical tools.…”

The challenges and scope of theoretical biology

“…These features have highlighted fundamental problems, such as explaining altruism within a framework of individual-based selection (Hamilton et al, 2001;Hamilton, 1971). Recently, progress has been made through the development of theories that have either expanded fundamental selection dynamics in a hierarchical fashion, such as multilevel selection theory (Lehmann et al, 2007a), demonstrating a mathematical isomorphism between kin selection and group selection (Hamilton, 1975;Traulsen and Nowak, 2006;Lehmann et al, 2007b;Frank, 1998), or how individualbased strategies in game theory might accounts for the evolution of behavior. This last case is particularly interesting, as we have subsequently seen an expansion of this concept to account for dynamical interactions at lower levels of complexity such as viral dynamics (Nowak, 2006).…”