Survival dynamics in zombie post-apocalypse I'm new here. I have a world-building problem that I'm trying to translate into a simple mathematical model. I apologize in advance if I'm using the wrong terminology.
A world (2d plane) is filled with survivors of the apocalypse (infinite number of points on the vertices on a 2d cartesian grid).
After the apocalypse (at time 0), all survivors start to move with the same constant speed in the direction of the zombie shelter (a single point on the plane). If a survivor gets to the shelter, they there indefinitely.
While the survivors move, they have a constant probability of being eaten by zombies (disappear) before they reach the shelter. So the bigger the distance is between a survivor and the shelter, the less are the chances the survivor will reach the shelter.
I'd like to find out the frequency of arrivals to the shelter (and the number of survivors in the shelter), as a function of time.
 A: Since, the plane is infinite, we can take any point to be the zombie shelter. Let the zombie shelter be the origin.
Let the probability of survival for 1 km be $\chi$, and the further you are from the shelter the less your chances of survival, as you have mentioned in the comment. Then, Probability of survival at a distance $r$ km is given by: $${\chi}^{r}$$
Suppose the survivors are moving in with a speed $v$. Suppose there are $x$ survivors per unit area.
Within a thin ring of area $2\pi rdr$, there are $2\pi xrdr$ survivors. After time $t$, a portion of the survivors within a distance of $vt$, in keeping with the probability of survival ${\chi}^{r}$, will be in the shelter, so the number of people in the shelter at time t is given by $N(t)$: $$N(t)=\int_0^{vt}{\chi}^{r}2\pi xrdr=\frac{2πx(vtln(\chi)−1){\chi}^{vt}}{(ln(\chi))^2}$$ 
and differentiating it, the frequency of arrivals is: $$f(t)=\frac{2π\chi^{vt}vx(ln(\chi)vt−1)}{ln(\chi)}+\frac{2π\chi^{vt}vx}{ln(\chi)}=2π\chi^{vt}{v^2}xt$$
