Coexistence and Related Concepts

For many models we will be interested in whether two species can coexist or whether competitive exclusion will occur. To be able to state theorems we need precise definitions.

We say that coexistence occurs if there is a stationary distribution which concentrates on configurations that have infinitely many sites in each possible state.

Markov chain theory tells us that in situations in which type i can die out, a stationary distribution can only have 0 or infinitely many sites in state i. In most cases in which coexistence occurs there will be a translation invariant stationary distribution where P( state_t[x] = i) is a constant u[i] > 0 that we will call the density of type i.

We say that clustering occurs if for each x and y the probability of seeing one type of particle at x and a different type of particle at y converges to 0 as t tends to infinity. Here our wording is chosen so that if we have a model with 0 = vacant, 1,2 = two types of particles then clustering means P( state_t[x] = 1, state_t[y] = 2 ) converges to 0 as t tends to infinity.

For our final definition we restrict our attention to models in which there are two types of particles 1's and 2's and vacant sites 0's. We say that 2's die out if P( state_t[x] = 2 ) tends to 0 for any initial configuration in which there are infinitely many 1's.