In this model each site can be occupied by one of *k* species.
An individual of type *i* produces a new offspring of its
own type at rate 1, and sends it to a neighbor *y* chosen at
random.

If the target site *y* is occupied by type *j* then
*y* will switch to type *j* with probability p[i,j] and
remain unchanged otherwise.

This model is a continuous time version of a system introduced by Silvertown et al (1992), who investigated a special case with

**A Single Dominant Species.** We say that type *i*
dominates type *j* if p[i,j] > p[j,i]. Durrett and Levin (1997),
using techniques of Grannan and Swindle (1991) and Mountford and
Sudbury (1992), have shown

**Theorem.** If type 1 is dominant over each other type then it
takes over the system. That is, if we
start with infinitely many sites in state 1 then for each *x*
P( state_t[x] = 1 ) converges to 1 as *t* tends to infinity.

**Math Exercise.** Show that the mean field ODE:

has the same behavior. (Hint: u[1] in increasing in *t*
and cannot converge to a limit < 1.)

**s3 Exercise.** Fill in parameters with p[1,2] > p[2,1]
and p[1,3] > p[3,1] and watch the 1's win.

More interesting behavior occurs if we consider a

**Cyclic Three Species System.** Suppose p[1,3] = b[1], p[2,1] = b[2],
p[3,2] = b[3], and the other p[i,j]=0. In this case

defines a fixed point for the ODE. Furthermore, the function

H(x[1], x[2], x[3]) = v[1] log x[1] + v[2] log x[2] + v[3] log x[3]

is constant along solutions of the ODE, so the fixed point is surrounded by periodic orbits. For example, when b[1] = 0.3, b[2] = 0.7, and b[3] = 1.0, the fixed point is at (0.5, 0.15, 0.35) and the ODE looks like:

Based on the behavior of the ODE, it is natural to

**Conjecture.** If all the b[i] > 0 then coexistence occurs.

**s3 Exercise.** Simulate the model using the parameter values
b[1] = 0.3, b[2] = 0.7, b[3] = 1.0 to confirm that in this case
there is an equilibrium state with an interesting spatial structure.

**Related Work.** If one considers the cyclic model with
p[1,3] = p[2,1] = p[3,2] = 1 but attempted takeovers are
successful with probability q and result in an empty site with
probability 1-q, one gets a stochastic spatial analogue of ODE's
considered by Gilpin (1975) and May and Leonard (1975).
Orbits in the modified ODE spiral outward to the boundary
but there is still coexistence in the spatial model. Other variations
of the spatial model have been considered by Tainaka (1995).

Gilpin, M.E. (1975) Limit cycles in competition communities.
*Am. Nat.* **108**, 207-228

May, R.M. and Leonard, W.L. (1975) Nonlinear aspects of competition
between three species. *SIAM J. Appl. Math.* **29**, 243-253

Grannan, E. and G. Swindle (1991). Rigorous results on mathematical
models of catalyst surfaces. *J. Stat. Phys.* **61**, 1085-1103

Mountford, T.S. and Sudbury, A. (1992). An extension of a result of
Swindle and Grannan on the poisoning of catalyst surfaces. *J. Stat.
Phys.* **67**, 1219-1222

Silvertown, J., Holtier, S., Johnson, J. and Dale, P. (1992).
Cellular automaton models of interspecific competition for space - the effect
of pattern on process. *J. Ecol.* **80**, 527-534

Tainaka, K. (1995) Indirect effect in cyclic voter model models. {\it Physics Letters A.} {\bf 207}, 53-57

Durrett, R. and Levin, S. (1997) Spatial aspects of interspecific
competition. * J. Theoret. Biol.*, to appear