Here the possible states are 0 = vacant, 1, 2 = two prey species, 3 = predator.

Types i = 1, 2 behave like a contact process, dying at a constant rate d[i] and being born at vacant sites at rate b[i] times the fraction of neighbors in state 1.

3's die at a contant rate d[3], are born at sites occupied by 1's at at rate b[3] times the fraction of neighbors in state 3. and are born at sites occupied by 1's at at rate b[4] times the fraction of neighbors in state 3.

Finally, there is stirring at rate *nu*: for
each pair of neighboring sites *x* and *y*
we exchange the values at *x* and at *y* at rate *nu*.

In the absence of predators, this system reduces to the competing contact process, where the stronger species (identified by the larger of the two ratios b[i]/d[i]) will competitively exclude the other. However, if the predators feeding rate on the stronger species is larger, its presence may stabilize the competition between the two species.

One way of seeing this is to consider the mean field ODE:

Here one can solve three equations in three unknowns to find
conditions for an interior fixed point but a more frutiful approach
is to derive conditions from an **invadability analysis**. Half of this
may be described as follows.

By results for predator prey systems, 2's and 3's can coexist if (b[2]-d[2])/b[2] > d[3]/b[4] and when this holds their equilibrium densities will be v[2] = d[3]/b[4] and

Examining the behavior of the ODE near (0,v[2],v[3]) we see that 1's can invade the (2,3) equilibrium if

b[1] - d[1] - b[1]v[2] - ( b[1] + b[3] )v[3] > 0

In a similar way one can derive conditions for the (1,3) equilibrium
to exist and for the 2's to be able to invade it. When both sets
of conditions hold we say there is **mutual invadability**.
It is easy to prove that in this case that the ODE has an
interior fixed point. By extending the methods of Durrett (1993),
Nikhil Shah (1997) has shown

**Theorem.** If mutual invadability holds then coexistence occurs
fo fast stirring.

To get a feel for the resulting phase diagram, set b[3] = 4, b[4] = 3/2, all the d[i] = 1, and vary b[1] and b[2]. The formulas above imply that 1 and 3 coexist if b[1] > 4/3, 2 and 3 coexist if b[2] > 3, and finally all three species can coexist inside the region bounded by the equations

The last few lines can be summarized in the following picture:

Note that there is a region where all three species can coexist but 2's and 3's cannot. Upon reflection this is not surprising: it simply says the 2's are not a sufficiently good food source to maintain the predator by themselves.

**s3 Exercise.** For a concrete example of predator mediated
coexistence in our spatial model, run the model with its default
values: b[1] = 4, d[1] = 1, b[2] = 3, d[2] = 1, b[3] = 4,
b[4] = 1.5, d[3] = 1, fraction of stirring steps = 0.5. This is one
of the points in our picture above. Searching for your
own values for which coexistence occurs should give you an
appreciation of the information about the phase diagram
in the theorem.

**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.

Durrett, R. (1993) Predator-prey systems. Pages 37--58 in *Asymptotic
problems in probability theory: stochastic models and diffusions on
fractals.* Edited by K.D. Elworthy and N. Ikeda, Pitman Research Notes
83, Longman Scientific, Essex, England

Shah, N. (1997) Predator-mediated coexistence. Ph.D. Thesis
Cornell U. *In preparation.*