Multilevel selection in a resource-based model Fernando Fagundes Ferreira Escola de Artes, Ciˆ encias e Humanidades, Universidade de S˜ ao Paulo, 03828-000 S˜ ao Paulo, Brazil Paulo R. A. Campos * Departamento de F´ ısica, Universidade Federal de Pernambuco, 52171-900 Recife-PE, Brazil In the present work we investigate the emergence of cooperation in a multilevel selection model that assumes limiting resources. Following the work by Requejo and Camacho [R. J. Requejo and J. Camacho, Phys. Rev. Lett. 108, 038701 (2012)], the interaction among individuals is initially ruled by a prisoner’s dilemma (PD) game. The payoff matrix may change influenced by the resource availability and hence may also evolve to a non PD game. Furthermore, one assumes that the population is divided into groups, whose local dynamics is driven by the payoff matrix, whereas an inter-group competition results from the non-uniformity of the growth rate of groups. We study the probability that a single cooperator can invade and establish in a population initially dominated by defectors. Cooperation is strongly favoured when group sizes are small. We observe the existence of a critical group size beyond that cooperation becomes counter-selected. Although the critical size depends on the parameters of the model, it is seen that a saturation value for the critical group size is achieved. The results conform to the thought that the evolutionary history of life had repeatedly involved transitions from smaller selective units to larger selective units. PACS numbers: 02.50.Le,87.18.-h,87.23.Kg,89.65.-s Introduction Although the seminal article entitled ”The genetical evolution of social behavior I” by W. D. Hamilton [2] was published almost five decades ago, the evolution of the altruism still fascinates and challenges researchers. Strikingly, living beings have developed a behaviour ex- pressed by cooperative strategies, widespread in all levels of biological organization. There are several theories that try to explain the emergence of cooperation. The great majority are concerned with behavioural models which are addressed within the game theory framework. Few others are based on ecological models that take environ- mental features into account. Most of the approaches based on game theory aim to inquire the role of quite distinct mechanisms on the emer- gence of cooperative behavior. Among those most stud- ied mechanisms are the kin selection [2–4], reciprocal al- truism [5], indirect reciprocity [6, 7], network reciprocity [8], memory [9], migration [10, 11], imitation [12], spatial structure [13] and punishment [14]. All these concepts constitute the foundations of most evolutionary explana- tions for altruism, and are applied to the level of indi- vidual interactions, where natural selection is supposed to work on. Their extensions to larger selective units are quite controversial [15, 16]. Wilson [17] brought back the idea of multilevel (group) selection which was abandoned by evolutionary biologists decades ago. To understand this concept it is important to distinguish the competi- tion between individuals within the same group and the competition between individuals from different groups. The intra-group dynamics favours the selfish behaviour. Yet, the inter-group dynamics brings about an advan- tage for those groups that have a higher concentration of cooperators [17, 18]. Aside of mechanisms modelled through the game the- oretic approach, environmental variables may influence individual behaviour more explicitly. In a scenario where resources are scarce, selfish individuals may be fated to extinction. Since this scenario is very common in nature, it is imperative to link the strategies that describe indi- vidual behaviour with the amount of resource available in the system. Despite of its relevance few works have been concerned with the coupling between individual behavior and environmental variables [1, 19–21]. In this work we investigate how structuring and multi- level selection can boost cooperation fixation in a divided population considering a scenario of limiting resources. The dynamics inside groups is driven by the payoff ma- trix and an inter-group competition ensues from the dif- ferentiated growth rates of groups. Groups haboring a larger number of cooperators reproduce faster, though local dynamics favours the defectors. The key purpose of this work is to estimate the probability of invasion and posterior fixation of the cooperative strategy when a single cooperator is introduced into the population of defectors. To survey how important is the role played by stochasticity, the dependence of the fixation proba- bility of the cooperative behaviour on the group size is addressed. The results are strongly dependent on the net advantage of the defectors. The reminder of this paper is organized as follows. In next section we present the model. The section III is devoted to the presentation of simulation results, whereas in the last section we conclude by summarizing the main