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Suppose you have a stochastic Petri net with two species A and B and the following transitions
A+B -> 2A at rate u
A+B -> 2B at rate u
A -> B at rate v
B -> A at rate v
Now, assuming I am not horribly confused about something, and haven't miscalculated...
The total number of A'a and B's is a constant N. Let the number of A's be n so the number of B's is N-n. This system is simple enough to find the equilibrium distribution, which is a Beta-binomial_distribution with $\alpha = \beta = v/u$. If u>v, this is U-shaped, and if u>>v you get what I think of as 'bistable' system. The system will spend a long time with n close to 0, then 'flip' quite quickly to n close to N, stay there for some time then flip back, and so on. This is true for any N>2.
But the rate equation for the proportion x=n/N is just
$d x/d t = v(1-2x)$
which just says that x tends to x/2.
What bothers me is that even when N is a zillion, the continuous approximation gives a very poor idea of what is going on.
BTW, I can't see a better category than Chat for a post like this.