## Competing Contact Processes

Here the possible states are 0 = vacant, 1 = occupied by species 1,
2 = occupied by species 2. Each species behaves like the contact process
subject to the rule that neither site can give birth onto sites
occupied by the other:

1's die at a constant rate d[1] and are born at vacant sites at rate b[1]
times the fraction of neighbors in state 1.

2's die at a constant rate d[2] and are born at vacant sites at rate b[2]
times the fraction of neighbors in state 2.

To guess what will happen in this model we look at the mean field ODE.
Here u[0] = 1 - u[1] - u[2], so the system is two dimensional.

Suppose b[1]/d[1] and b[2]/d[2] are > 1, so the each species is individually
viable (in the ODE) and let v[i] = (b[i]-d[i])/b[i].
If b[1]/d[1] > b[2]/g[2]
then the boundary fixed point (v[1],0) is the limit starting from
any (u[1],u[2]) with each u[i] > 0. For example when b[1] = 4,
d[1] = 1, b[2] = 6, and d[2] = 2:

Based on this behavior in the ODE, it is natural to

**Conjecture.** If b[1]/d[1] > b[2]/d[2] then the 2's die out.
That is, P( state_t[x] = 2 ) tends to 0
for any initial configuration in which there are infinitely many 1's.

Neuhauser (1992) has shown (for the stronger result given here
see Durrett and Neuhauser (1997))

**Theorem.** If d[1] = d[2] and b[1] > b[2] then the 2's die out.

Biologists should not find this conclusion surprising since we have
two species competing for one resource and hence expect competitive
exclusion to occur. See, e.g., Levin (1970).

**s3 Exercise.** To convince yourself that the conjecture is
correct set b[1] = 3.0, d[1] = 1, b[2] = 4.0, and then vary d[2]
to see that 1's win when d[1] > 4/3 and 2's win when d[2] < 4/3.
Coexistence is not possible even when the
species are evenly matched. Of course in this case we cannot predict
the winner in advance. Neuhauser (1992) has shown:

**Theorem.** If d[1] = d[2] and b[1] = b[2] then clustering occurs.
That is, for any sites *x* and *y*
P( state_t[x] = 1, state_t[y] = 2 ) converges to 0 as t tends
to infinity.

Levin, S. (1970) Community equilibria and stability, an extension of
the competitive exclusion principle. *Am. Nat.* **104**,
207-228

Neuhauser, C. (1992) Ergodic theorems for the multitype contact process.
*Probab. Theory. Rel. Fields.* **91**, 467-506

Durrett, R., and Neuhauser, C. (1997) Coexistence results for some
competition models. *Ann. Applied Probab.*, **7**, No. 1,
to appear

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