The dynamics of the agent representations and choices interacts with the diffusion dynamics of emitted pollution. Agents occupy the cells of a two dimensional grid that represent space. Polluting cars generate pollution at a constant rate s. This pollution diffuses in space according to Fick's law, and part of it simply evaporates locally ( in the sense that it disappears). Pollution, P( r,t) in r at time t, obeys the following partial differential equation:
where D is the diffusion constant, 
the laplacian of
P, ev the evaporation term. s, the source term,
is only present when the occupant of the cell bought a polluting car.
The s term strongly couples the dynamics of pollution to
that of the agents' choices. The range of possible pollution
levels correspond to fixed
configuration of choices. Pollution is zero when
all agents always bought non-polluting cars and maximum and equal to
when they all have always bought polluting cars.
, the maximum pollution, reflects the cost of pollution,
and is the natural scale on which to compare differences in the
prior utilities of the agents.
However, since the agents only use local information, the gradient of pollution times the range of polling is the relevant parameter to compare differences in prior utilities. The agents base their choices on the information provided by neighbors. In homogeneous regions of either brand purchasers, the pollution gradient is small and the risk aversion parameter makes the agent choose the same brand as their neighbors. A pollution gradient is present in frontier regions. Pollution is higher on the side of the boundary where polluting cars were bought which lowers the utilities of the polluting cars. On the other side pollution is lower, and the utility of non-polluting cars is thus increased. The agents at the boundary get their information on polluting (resp. non-polluting) cars from purchasers of polluting (resp. non-polluting) cars in the polluted (resp. non-polluted) region. The posterior utilities are thus split according to the difference in pollution betweeen the two regions. Given this type of information, the agent at the boundary might agree to pay the extra cost of a non-polluting car that compensate for the difference in pollution he has been able to poll. The maximum gradient is obtained when one supposes that a half plane has always been occupied by polluting cars and the other half by non-polluting cars. In this case the gradient of pollution is:
The difference in pollution experienced by neighboring agents is simply the product of the gradient by the range of polling. For reasonable parameter values, this product is much less than saturation pollution, which explains why the agents never agree to pay an extra cost for the anti-polluting device equal to saturation pollution.