Find the rate at which two regions mingle. I am considering a problem in population biology, but I need help with a numerical calculation.
Suppose the plane is densely populated with animals at a density of $\delta_0$ animals per unit area, except for in a bounded region $\Omega$, in which there are none.
Suppose the animals move according to Brownian motion, and that they may wander into $\Omega$ through a small opening $\ell$ in the boundary of $\Omega$. See my diagram:

How does the population density of $\Omega$ change over time? I know that as $t\to\infty$, the density of $\Omega$ approaches $\delta_0$. I suspect that the rate of convergence to $\delta_0$ is determined by the length of $\ell$, and also by the parameters of the random motion.
My question is: what is an expression for the density of $\Omega$ at a given time $t\in\Bbb R$?
 A: For $l$ sufficiently smaller compared to the circumference of $\Omega$, you could try this simple approach. 
Assume that after the "animals" enter the $\Omega$ area, they instantaneously distribute evenly in $\Omega$. Let $x, y$ denotes the density of the animal outside and inside of $\Omega$, respectively. Then what we want to find is: $\frac{dy}{dt}$.
Because the plane is densely populated with the animals at a fixed density of $\delta_0$, you can just consider the area immediately surrounding the opening of length $l$.
Since the animal is moving via diffusion and we assume that once entering $\Omega$, the animals distribute evenly, then the amount of animal entering the region $\Omega$ is proportional to the diffusive rate times the difference in density of the two regions. This gives:
$$
\frac{dy}{dt} = c(x-y)l = cl(\delta_0 - y).
$$
As you can see, for $t$ approaches $\infty$, $y$ also approaches $\delta_0$.
I do not know how realistic you would like your system to be, but this a simple way of modeling this system.
