Back of the envelope

Consider the diffusion equation for settled systems in the free-streaming limit (which I think should be appropriate)

\frac{\partial \rho}{\partial t} = k \frac{\partial ^2 \rho}{\partial x^2} \rightarrow \frac{\partial \rho}{\partial t} = v \frac{\partial \rho}{\partial x} where v is the free streaming velocity roughly corresponding to the thermal velocity. (For flux-limited radiation diffusion, this is just the speed of light).

We can then combine that with a logistic style source term that accounts for local probe activity

S=\frac{1}{T_p}\rho(1 - \frac{\rho}{\rho_{max}})

We then normalize the density of settled systems by the density of total systems

\eta=\frac{\rho}{\rho_{max}}

which gives us

\frac{\partial \eta}{\partial t} = -v \frac{\partial \eta}{\partial x} + \frac{1}{T_p} \eta (1-\eta)

We then look for self-similar solutions of the form

\eta(\xi) where \xi = x - \mathcal{V} t

Upon substituting that in to the continuity equation we arrive at

-\mathcal{V}\frac{\partial \eta}{\partial \xi} = -v\frac{\partial \eta}{\partial \xi} + \frac{1}{T_p} \eta \left ( 1 - \eta \right )

or

\frac{\partial \eta}{\partial \xi} = -\frac{\eta (1-\eta)}{T_p (\mathcal{V}-v)}

which is just a logistic curve with length scale L=T_p \left (\mathcal{V}-v \right) where v is the thermal speed and \mathcal{V} is the speed of the front (maximum speed from maxwellian).

Simulation result

Earth Params
v_rms	30 km/s
v_probe	30 km/s
d_probe	10 pc
launch_period	1e4 yr
rho	1e5/(50 pc)3

Ran a simulation with periodic BC's and 10⁴ systems, but particles enter from the right unsettled and enter from the left settled. Systems on the left half start settled. Also shifted the random velocities by the expected front velocity ~3 v_rms due to the maxwellian distribution. (Click on the picture below for a movie). Settled systems are in red, probes in blue, and unsettled systems are not shown.

Then fit the number of settled systems at the 'steady state' to a logistic curve

\rho = \frac{\rho_{\max}}{1+\exp^{(x-x0)/L}}

This gives L=8.6 pc

where if we calculate T_p (\mathcal{V}-v) we get 0.6 pc

The travel time to neighboring stars is a factor of 3 longer than the launch period - and the travel time to the maximum probe range is another factor of 10 on top of that, so .6 pc is definitely a lower bound for the logistic growth rate. There is also potentially a selection bias in that the fastest moving systems (those out in front) will see more systems traveling quickly in the other direction which can also explain the larger then expected value for L

And for reference - here is the same setup, but with no velocity fluctuations