Exclusive joint removal has a diffusing difference mode ($\mathrm{Var}(X_1-X_2)=\mathrm{Var}(X_1(0)-X_2(0))+2\lambda t$)

If individual removal is absent, simultaneous removal of one molecule from each species leaves $X_1-X_2$ unchanged. Independent constant-rate births change it by plus or minus one, each at rate $\lambda$. The difference is therefore a symmetric continuous-time random walk on the integers with variance growing as displayed. It has no normalizable stationary law. This explains why a deterministic attracting line of equilibria does not provide a stationary stochastic distribution.

// Target: probability-and-statistics.bigb