Disordered environments in spatial games

Abstract

The Prisoner's dilemma is the main game theoretical framework in which the onset and maintainance of cooperation in biological populations is studied. In the spatial version of the model, we study the robustness of cooperation in heterogeneous ecosystems in spatial evolutionary games by considering site diluted lattices. The main result is that due to disorder, the fraction of cooperators in the population is enhanced. Moreover, the system presents a dynamical transition at ρ * , separating a region with spatial chaos from one with localized, stable groups of cooperators.

Does mobility decrease cooperation?

“…Remarkably, the fall of cooperation after the maximum seems not to be associated with any particular behavior of active sites, whose fraction keeps growing with the total density. Thus, although no particular sign is observed in ρ a around the maximum of ρ c , the decrease of cooperation after the maximum is related to a smaller number of empty sites that act as pinning points that slow the dynamics or even prevent that some regions of cooperators be predated, as was observed in [52]. Fig.…”

“…3, for COD dynamics, shows both the density of cooperators and strategy-changing individuals (active sites, ρ a ), as a function of the total lattice occupation ρ, after the system attained a stationary state where both quantities fluctuate around their average values. Also shown, for comparison, are the results from [52] at ρ = 1, where no movement is allowed. The transition from the region with ρ c = 0 to the cooperative one seems to be continuous and the finite fraction of active sites indicates that when ρ c = 0, both strategies, C and D, coexist.…”

“…In the former, as the name says, each step consists of combats followed by the generation of offspring done in parallel, and then diffusion, while in the later, the diffusion and offspring steps are reversed. During the diffusive step, each agent makes an attempt to jump to a site chosen randomly within its four nearest neighbours, what is accepted, provided the site is empty, with a probability m. Here we only consider local steps with a reduced dispersal range (one lattice site), m thus measuring the mobility of the agents (m = 0 reduces to the case studied in [52]). …”

“…These spatial games, where the interactions are localized and non random, have been studied and extended in many ways (see, for example, Refs. [1,2,4,9,10,15,19,22,23,24,25,29,30,31,34,37,38,40,41,44,45,46,47,49,52]). Once the population is spatially structured, a natural question concerns the effects of mobility that, along with other important biological factors, is often neglected [28]: is it possible to evolve and sustain cooperation in a population of mobile agents, where retaliation can be avoided by moving away from the former partner?…”

“…We addressed in earlier work [52] the question of the robustness of cooperation in spatial games in the presence of heterogeneous environments. By introducing quenched disorder in the lattice (random dilution) each individual would sense a locally varying social environment as the number of neighbors becomes site dependent: optimal cooperation can be achieved for weak disorder as the defects (or inaccessible regions) act as pinning fields for the strategy transition waves that cross the system, keeping the clusters of cooperators more protected from invasions.…”