Process 1: \begin{align*} P(X(t + \Delta t) = y \,|\, X(t) = x) = \begin{cases} a \Delta t & \text{if $y=x+1$}\\ 1-(a+b)\Delta t & \text{if $y=x$}\\ b \Delta t & \text{if $y=x-1$}\\ 0 & \text{otherwise} \end{cases} \end{align*}
Process 2: \begin{align*} P(X(t + \Delta t) = y \,|\, X(t) = x) = \begin{cases} a x \Delta t & \text{if $y=x+1$}\\ 1-(a+b)x\Delta t & \text{if $y=x$}\\ b x\Delta t & \text{if $y=x-1$}\\ 0 & \text{otherwise} \end{cases} \end{align*}
Under what conditions is the deterministic model a poor approximation to the stochastic model? What features of the stochastic model are completely missed by the deterministic model?
From some preliminary assessments it appears that new colonization events of empty meadows occur when butterflies travel between meadows (rather than traveling in from outside). Butterflies move at a rate of 0.4 per year.
Let $p$ be the fraction of meadows that are currently occupied by butterflies.