We study the evolution of cooperation in communities described in terms of graphs, such that individuals occupy the vertices and engage in single rounds of the Prisoner's Dilemma with those individuals with whom they are connected through the edges of those graphs. We find an overwhelming dominance of cooperation whenever graphs are dynamically generated through the mechanisms of growth and preferential attachment. These mechanisms lead to the appearance of direct links between hubs, which constitute sufficient conditions to sustain cooperation. We show that cooperation dominates from large population sizes down to communities with nearly 100 individuals, even when extrinsic factors set a limit on the number of interactions that each individual may engage in.

We introduce a class of small-world networks--homogeneous small-worlds--which, in contrast with the well-known Watts-Strogatz small-worlds, exhibit a homogeneous connectivity distribution, in the sense that all nodes have the same number of connections. This feature allows the investigation of pure small-world effects, detached from any associated heterogeneity. Furthermore, we use at profit the remarkable similarity between the properties of homogeneous small worlds and the heterogeneous small-worlds of Watts-Strogatz to assess the separate roles of heterogeneity and small-world effects. We investigate the dependence on these two mechanisms of the threshold for epidemic outbreaks and also of the coevolution of cooperators and defectors under natural selection. With respect to the well-studied regular homogeneous limits, we find a subtle interplay between these mechanisms. While they both contribute to reduce the threshold for an epidemic outburst, they exhibit opposite behavior in the evolution of cooperation, such that the overall results mask the true nature of their individual contribution to this process.

We study the system of equations describing a stationary thermoconvective flow of a non-Newtonian fluid. We assume that the stress tensor S has the formwhere u is the vector velocity, P is the pressure, θ is the temperature and μ, p and τ are the given coefficients depending on the temperature. D and I are respectively the rate of strain tensor and the unit tensor. We prove the existence of a weak solution under general assumptions and the uniqueness under smallness conditions.

On a class of nonlocal nonlinear elliptic problems RAIRO-Modélisation mathématique et analyse numérique, tome 26, n o 3 (1992), p. 447-467. © AFCET, 1992, tous droits réservés. L'accès aux archives de la revue « RAIRO-Modélisation mathématique et analyse numérique » implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ [X] MATHEMATICALMOOEUJHGAHDHUMERICALAHALYSIS MOMUSATIOH MATHÉMATIQUE ET ANALYSE NUMÉRIQUE

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