Vainstein et al., 2007, Does mobility decrease cooperation?

Ref: Vainstein, M. H., Silva, A. T. C., & Arenzon, J. J. (2007). Does mobility decrease cooperation? Journal of Theoretical Biology, 244, 722-728.

The model:
R = 1, P = S = 0, and T = b > 1.
An equal number of cooperators and defectors are placed randomly in a 2-D square lattice of length L, with a number of boundary conditions, such as the total density p.
Each individual plays the PD with its nearest 4 neighbors (if any), accumulates the corresponding payoff, and then may either move or try to reproduce. This corresponds ot von Neumann space. Updating appears to be synchronous.
Reproduction step: each player compares its total payoff with the ones of its neighbors and changes strategy, following the one among them with the greatest payoff.
Diffusion (movement) step: Each agent makes an attempt to jump to a site chosen randomly within its four nearest neighbors. Provided the site is empty, the site is accepted with probability m.

The claim:
Mobility may allow cooperation to evolve. They used a variation on the Spatial PD with cellular automata, with agents playing all closest neighbors in von Neumann space, and synchronous updating. Either before or after "reproduction," agents moved with a probability m into an adjacent space. The order of operations (called COD or CDO) strongly influences the population dynamics and the resulting population ratio of cooperators.
They claim the probability of future encounters increases when mobility is small but positive.

One nice thing about this model is that there is no correlation between cooperation and movement, and thus allows that previous mobility could have a role in the evolution of cooperation. They offer real-life examples from microorganisms with extracellular metabolism.

They point out the complicated relationship between diffusion and the variables m and p, and suggest that this relationship is not simple, and requires further research.

Comments
They pay no attention to the fact that order of diffusion and reproduction plays a huge role in the population dynamics. They also only investigate this scenario in a limited range of conditions. Since this model is essentially identical to a limited case of what was explored by Nowak and May (1993), they would do better to provide a more thorough investigation. The major finding, though, is that in the spatial PD with cellular automata, having a population density less than 1 with possible movement can increase the final stability of cooperators.
Another criticism: they only start with 50% cooperators in all conditions. Nowak and May generally started with 10%, assuming that cooperators need to be able to invade. This is one thing that is easy to test.
Also, like Nowak and May, they assume synchronous updating. Using asynchronous updating can change the character of their results.