de Aguiar et al. (2004). Invasion and extinction in the mean field approximation

de Aguiar et al. (2004). Invasion and extinction in the mean field approximation for a spatial host-pathogen model. Journal of Statistical Physics, 114, 1417-1451.

Abstract: 

We derive the mean field equations of a simple spatial host-pathogen, or predator-prey, model that has been shown to display interesting evolutionary properties. We compare these equations, and the equations including pair-correlations, with the low-density approximations derived by other authors. We study the process of invasion by a mutant pathogen, both in the mean field and in the pair approximation, and discuss our results with respect to the spatial model. Both the mean field and pair correlation approximations do not capture the key spatial behaviors—the moderation of exploitation due to local extinctions, preventing the pathogen from causing its own extinction. However, the results provide important hints about the mechanism by which the local extinctions occur.

Comparison with mean-field approx: 
A toy model with just one type of agent with a growth and death rate is well-approximated by the mean-field approximation.
The host-pathogen model is less well described by the mean field than the toy model previously discussed. In particular, the number of independent correlations between pairs of sites is larger in the host-pathogen model than in the toy model, indicating that the mean field equations for the former should not be as accurate as for the latter.

Mean field equation for two types of pathogens, with different transmission rates.
Approximate equations a mutant can always invade if it has higher transmissibility. The exact mean field equations, which account for discrete time and one event per cell per time, show that if the transmission rates are close, both types can co-exist in the population.

Pair approximations.
Pair approximations can do a pretty good job of reproducing the dynamics in a mixed population of a single pathogen type, though they do not get the exact values of the simulation.

The dynamics of invasion in the pair approximation. 
The big question:
In the spatial version of the model it is known that the system may evolve spontaneously towards an evolutionarily stable pathogen type that cannot be invaded even by more transmissible ones. The basic reason for the appearance of this evolutionarily stable type of intermediate transmissibility lies in the self-organized spatial structure of the population. Hosts and pathogens are distributed patchily. Mutant pathogens can arise that spread more quickly than susceptible hosts are replenished, but these pathogens cause the extinction of the host and themselves only locally. Can the dynamics in the pair approximation account for this spatial phenomenon?

Results:
(Fig. 7) The qualitative resemblance between the pair approximation and the lattice calculations is very good. The number of infected individuals, however, is almost three times smaller in the lattice calculation. Also, the population of y1 eventually die in the lattice model, although it takes about 8000 generations (but stably co-exist in the pair approximation).