Spatial Patterns of Prisoner’s Dilemma Game in Metapopulations

Abstract

Because to defect is the evolutionary stable strategy in the prisoner's dilemma game (PDG), understanding the mechanism generating and maintaining cooperation in PDG, i.e. the paradox of cooperation, has intrinsic significance for understanding social altruism behaviors. Spatial structure serves as the key to this dilemma. Here, we build the model of spatial PDG under a metapopulation framework: the sub-populations of cooperators and defectors obey the rules in spatial PDG as well as the colonization-extinction process of metapopulations. Using the mean-field approximation and the pair approximation, we obtain the differential equations for the dynamics of occupancy and spatial correlation. Cellular automaton is also built to simulate the spatiotemporal dynamics of the spatial PDG in metapopulations. Join-count statistics are used to measure the spatial correlation as well as the spatial association of the metapopulation. Simulation results show that the distribution is self-organized and that it converges to a static boundary due to the boycotting of cooperators to defectors. Metapopulations can survive even when the colonization rate is lower than the extinction rate due to the compensation of cooperation rewards for extinction debt. With a change of parameters in the model, a metapopulation can consist of pure cooperators, pure defectors, or cooperator-defector coexistence. The necessary condition of cooperation evolution is the local colonization of a metapopulation. The spatial correlation between the cooperators tends to be weaker with the increase in the temptation to defect and the habitat connectivity; yet the spatial correlation between defectors becomes stronger. The relationship between spatial structure and the colonization rate is complicated, especially for cooperators. The metapopulation may undergo a temporary period of prosperity just before the extinction, even while the colonization rate is declining.

Spatial games and the maintenance of cooperation in an asymmetric Hawk-Dove game

“…These simulations also showed that there are a variety of viable strategies if the benefit to cost ratio is adjacent to 1/2, and the patterns this variety are chaotic or oscillate based on the change of related factors if the benefit to cost ratio is greater than 1/2. These findings have several implications that may help us more fully understand the dynamic of meta-populations [37]. Furthermore, the results potentially implicate that the dynamics of spatially extended systems possesses a wide variety of forms in both biological and social systems.…”

The effect of landscape heterogeneity on host–parasite dynamics

“…We, therefore, adopt two other methods for analyzing the host-parasite dynamics. First, we analyzed the spatial dynamics and equilibriums of the model using numerical solutions (Hui and MeGeoch 2007). The value of P r and Q r/r¢ at time step t = 3000 were assumed to be the equilibriums of the model.…”

“…Many methods have been used to study the spatial heterogeneity in ecological systems Okuyama 2007;Webb et al 2007aWebb et al , 2007bBolker and Pacala 1997;North and Ovaskainen 2007), among which the spatially stochastic simulation is often used (Hui and MeGeoch 2007;Webb et al 2007b;Okuyama 2007;Su et al 2008b). In particular, the role of spatially structured heterogeneity in the eco-epidemiological system has been examined successfully using this method (Hiebeler 2005;North and Ovaskainen 2007;Melbourne et al 2007).…”

“…A potential problem with such a method is that the analysis is often restricted to direct computer simulations, making mathematical analysis impossible. This dilemma can be resolved by using the method of moment-closure approximations (e.g., pair approximation) introduced by Matsuda et al (1992); see also Iwasa 2000), which has been successfully applied in a wide range of ecological, epidemiological, and evolutionary systems (Sat" o et al 1994;Keeling and Rand 1996;Ovaskavinen et al 2002;Hiebeler 2005;Hui and MeGeoch 2007;Okuyama 2007;Su et al 2008b). Pair approximation is a method for constructing a system of ordinary differential equations for global and local densities, and dealing with them as separate state variables that change over time (Iwasa 2000).…”

Why mussels stick together: spatial self-organization affects the evolution of cooperation

“…To what extent cooperation evolves in a population depends on the nature and intensity of interactions between individuals (Hamilton 1963;Axelrod and Hamilton 1981;Nowak and May 1992;Rainey and Rainey 2003;Doebeli and Hauert 2005;Santos and Pacheco 2005;Ohtsuki et al 2006;Santos et al 2006;West et al 2007;Masuda 2007;Van Dyken and Wade 2012). The number of cooperating individuals an organism interacts with likely determines the effectiveness of its cooperation strategy and may affect the degree of cooperativeness that evolves within a population (Vainstein and Arenzon 2001;Zhang et al 2005;Ohtsuki et al 2006;Hui and McGeoch 2007).…”