I am having trouble finding information regarding the inclusion of random variables in differential equations. For example I have a simple model describing the growth of some population $X$ over time: $$ \dot{X} = X (r - \alpha X) $$
In this deterministic form we can easily solve to get the equilibrial and explicit solutions with regards to time
My question comes when I want to express random variations in parameter $\alpha$, allowing it be represented as a random variable. This means that $X$ also becomes a random variable. Can we derive equations to describe the change in this distribution and its moments over time? Note that here i am not taking about a time varying stochastic process (like brownian walks) but rather static variation in the parameters. Additionally can we include these moments in the differential equation above so that they depend on the moments of the distribution?
$$ \dot{X} = X (r - \alpha X - E[X]) $$