Stirring is a mechanism that exchanges the values of neighboring sites.
When stirring happens at fast rate, it makes adjacent sites independent.
A result of DeMasi, Ferrari, and Lebowitz (1986) when applied to the
quadratic contact process, implies that if we use scale the square
lattice by multiplying by *epsilon* and stir at rate
*epsilon*^{-2}, i.e., *epsilon* to the minus 2 power, then
in the limit as *epsilon* tends to 0 the density of occupied
sites satisfies

In words, the Laplacian Lu results from diffusion of particles, while rapid stirring makes adjacent sites independent, so densities evolve according to the mean field ODE.

The mean field ODE for the contact process has a two nontrivial fixed
points p1 < p2
when *delta* < 1/4. This might suggest that as stirring
becomes more rapid the critical value *delta*_c approaches 1/4.
However, results of Noble (1992) and Durrett and Neuhauser (1994) show

**Theorem.** As *epsilon* to 0, the critical value converges
to 2/9. Furthermore, the density of the stationary distribution which
is the limit starting from all 1's converges to p2, the larger fixed
point of the mean field ODE.

To explain the value 2/9's we recall that in one dimension the limiting
reaction diffusion equation has **travelling wave solutions** u(x,t)
= h (x-vt) that keep their shape but move at velocity v. Further,
if adopt the convention that h tends to p2 at minus infinity and 0
at plus infinity then v > 0, i.e., the region where the density is near
p2 expands, if and only if *delta* < 2/9.

This strategy has been used to facilitate the study of other models. See the pages on predator prey systems, predator mediated coexistence and catalyst surfaces.

**s3 Exercise.** Consider the quadratic contact process,
setting the fraction of stirring steps = 0.5 and *delta* = 0.15.
The process an equilibrium distribution in which about 70% of the sites are
occupied. This is not quite as large as the 81.6% predicted by
solving u(1-u) = *delta*, however it is a drastic improvement
over the rapid extinction that occurs in the process without stirring.

**s3 Footnote.** To increase the effect of single stirring steps
we exchange the value at x with a site y chosen at random from the square
of radius 3 centered at x.

DeMasi, A., Ferrari, P. and Lebowitz, J. (1986) Reaction diffusion
equations for interacting particle systems. *J. Stat. Phys.*
**44**, 589-644

Noble, C. (1992) Equilibrium behavior of the sexual reproduction process
with rapid diffusion. *Ann. Probab.* **20**, 724-745

Durrett, R. and Neuhauser, C. (1994) Particle systems and reaction
diffusion equations. *Ann. Probab.* **22**, 289-333