TITLE: Survivor distribution following a zombie outbreak
QUESTION [14 upvotes]: Suppose there is initially a population of $A$ humans and $B$ zombies. You are sitting on a nearby hill with a shotgun; however, you're not a very good shot, so each time you pull the trigger, one of the members of the population is hit uniformly at random.

If you hit a zombie, the zombie dies.
If you hit a human, the zombies pile onto the poor wounded human and turn them into a new zombie.

Undeterred by your lack of skill, you continue to take shots into the population until there are no more zombies. At this point, what can be said about the distribution of the number of human survivors?
For example, we can try to compute the probability $q(A,B)$ that there are no survivors at the end of the process. This quantity should satisfy the recurrence $$q(A,B) = \frac{A}{A+B} q(A-1, B+1) + \frac{B}{A+B} q(A, B-1)$$ with the initial conditions $q(0,k) = 1$ for $k\ge 0$ and $q(k,0) = 0$ for $k>0$. Is there a simple formula for $q(A,B)$ or reasonable asymptotics? What can be said about the probability $q(k,A,B)$ that there are $k$ survivors for $1\le k \le A$?
Also, looking at some data seems to suggest that we always have a "reverse unimodal" property $$q(0,A,B) > q(1,A,B) > \cdots < q(A-1, A, B) < q(A, A, B)$$ with $q(0,A, B) > q(A, A, B)$, i.e. it is likeliest to have zero survivors. Is it possible to show this in general?

REPLY [2 votes]: Too long for a comment. I think I can prove the following rather technical claim about the behavior of the zombie apocalypse started with large population.
Fix $\alpha, \beta > 0$. Then starting from the initial condition $(\lfloor \alpha n \rfloor, \lfloor \beta n \rfloor)$, denote by $(A_n, B_n)$ the pair of numbers of survivors/zombies after the $n$-th shot. If $T$ is the first time $n$ at which $B_n = 0$, then we consider the path $(Z_t)_{0 \leq t \leq T/n}$ defined by
$$ Z_t = \tfrac{1 - (nt - \lfloor nt \rfloor)}{n}(A_{\lfloor nt \rfloor}, B_{\lfloor nt \rfloor}) + \tfrac{nt - \lfloor nt \rfloor}{n}(A_{\lfloor nt \rfloor+1}, B_{\lfloor nt \rfloor+1}). $$
In other words, $(Z_t)$ is the piecewise linear path joining points $\frac{1}{n}(A_k, B_k)$'s and the speed is chosen so that each line segment from $\frac{1}{n}(A_k, B_k)$ to $\frac{1}{n}(A_{k+1}, B_{k+1})$ is traced at a uniform speed within the time interval $[k/n, (k+1)/n]$. Then

Technical Claim. As $n\to\infty$, $(Z_t)_{0 \leq t \leq T/n}$ converges in distribution to a deterministic curve $\gamma = (\gamma(t))_{0 \leq t \leq 2\alpha+\beta}$. Moreover, $\gamma(t) = (x(t), y(t))$ satisfies
  $$ x' = -\frac{x}{x+y}, \qquad y' = \frac{x-y}{x+y}, \qquad (x(0), y(0)) = (\alpha, \beta). $$
  Here, two paths on $\mathbb{R}^2$ are close if their graphs in $\mathbb{R}^3$ are close in Hausdorff distance.

Loosely speaking, the piecewise linear path joining $(A_n, B_n)$'s are close to the curve
$$ \frac{y}{x} + \log x = \text{constant}. $$
The following plot demonstrates this phenomenon in the case of $(a, b) = (1000, 500)$. The black line represents the curve $y/x + \log x = \text{const}$, where the constant is chosen so that it passes through $(a, b)$. Also, the colored zigzag curves represent 10 simulations of the history of zombie apocalypse.

Now, doing the same simulation with $(a, b) = (10000, 5000)$,

We clearly see concentration behavior emerges.