The Prisoner's Dilemma, a two-person game in which the players can either cooperate or defect, is a common paradigm for studying the evolution of cooperation. In real situations cooperation is almost never all or nothing. This observation is the motivation for the Continuous Prisoner's Dilemma, in which individuals exhibit variable degrees of cooperation. It is known that in the presence of spatial structure, when individuals "play against" (i.e. interact with) their neighbours, and "compare to" ("learn from") them, cooperative investments can evolve to considerable levels. Here, we examine the effect of increasing the neighbourhood size: we find that the mean-field limit of no cooperation is reached for a critical neighbourhood size of about five neighbours on each side in a Moore neighbourhood, which does not depend on the size of the spatial lattice. We also find the related result that in a network of players, the critical average degree (number of neighbours) of nodes for which defection is the final state does not depend on network size, but only on the network topology. This critical average degree is considerably (about 10 times) higher for clustered (social) networks, than for distributed random networks. This result strengthens the argument that clustering is the mechanism which makes the development and maintenance of the cooperation possible. In the lattice topology, it is observed that when the neighbourhood sizes for "interacting" and "learning" differ by more than 0.5, cooperation is not sustainable, even for neighbourhood sizes that are below the mean-field limit of defection. We also study the evolution of neighbourhood sizes, as well as investment level. Here, we observe that the series of the interaction and learning neighbourhoods converge, and a final cooperative state with considerable levels of average investment is achieved.

We study the ABC model ( A + B mapsto 2 B, B + C mapsto 2 C, C + A mapsto 2 A), and its counterpart: the three-component neutral drift model ( A + B mapsto 2 A or 2 B, B + C mapsto 2 B or 2 C, C + A mapsto 2 C or 2 A.) In the former case, the mean-field approximation exhibits cyclic behaviour with an amplitude determined by the initial condition. When stochastic phenomena are taken into account the amplitude of oscillations will drift and eventually one and then two of the three species will become extinct. The second model remains stationary for all initial conditions in the mean-field approximation, and drifts when stochastic phenomena are considered. We analyzed the distribution of first extinction times of both models by simulations of the master equation, and from the point of view of the Fokker-Planck equation. Survival probability vs. time plots suggest an exponential decay. For the neutral model the extinction rate is inversely proportional to the system size, while the cyclic model exhibits anomalous behaviour for small system sizes. In the large system size limit the extinction times for both models will be the same. This result is compatible with the smallest eigenvalue obtained from the numerical solution of the Fokker-Planck equation. We also studied the behaviour of the probability distribution. The exponential decay is found to be robust against certain changes, such as the three reactions having different rates.

We study the ABC model in the cyclic competition (A + B → 2B, B + C → 2C, C + A → 2A) and the neutral drift (A + B → 2B or 2A, B + C → 2C or 2B, C + A → 2A or 2C) versions, with mutations and migrations introduced into the model. When stochastic phenomena are taken into account, there are three distinct regimes in the model. (i) In the "fixation" regime, the first extinction time scales with the system size N and has an exponential distribution, with an exponent that depends on the mutation/migration probability per particle µ. (ii) In the "diversity" regime, the order parameter remains nonzero for very long times, and becomes zero only rarely, almost never for large system sizes. (iii) In the critical regime, the first passage time for crossing the boundary (one of the populations becoming zero) has a power law distribution with exponent −1. The critical mutation/migration probability scales with system size as N −1 . The transition corresponds to a crossover from diffusive behaviour to Gaussian fluctuations about a stable solution. The analytical results are checked against computer simulations of the model.

A two-dimensional Ising model with competing short range ferromagnetic and long range dipolar interactions is used to study the transition properties and phase diagram in ultrathin magnetic films. Monte Carlo simulations in systems with exchange and dipolar interactions reveal a ground state of striped phases with varying width. By raising the temperature, the domain walls are smeared out by fluctuations, leading to a random domain mesoscopic phase with no long-range order, and finally the domains are melted to the high-temperature disordered phase. Local magnetic field distributions and specific heat calculations reproduced transition points consistent with the previous phase diagram of the model. The resemblance to the phase diagram in ultrathin magnetic films, such as in Fe/Cu(100), is discussed.