How long before the prey can escape?

Question

I've (sort of) come across the following problem in my research. The actual scenario is a little abstract to explain, so I'm rephrasing the problem in terms of a predator/prey scenario. I'm tagging this as a soft question because:

• I'm not even sure if the problem is analytically tractable, and I'm interested to know what others think about this;
• I'm not familiar with this kind of mathematics (which I'm assuming has mostly to do with cellular automata and probability?), and so even hints as to which techniques might be useful would be helpful;
• The formulation is not set in stone, and I'm open to modifications that make the problem more tractable.

A prey (red) is being guarded by 8 predators (blue) on a square lattice (left diagram). The predators' discipline slowly wanes away, and they start pacing around. If at any time the prey is not being guarded by at least one predator, it finds itself free to escape by flying away (right diagram).

I am interested in the expected time for which the predators are able to keep the prey from escaping.

I propose the following rules for the predators' movement:

1. The prey remains stationary.
2. At each time step $k$, each predator randomly chooses 1 out of its 8 neighboring squares to move into (regardless of these squares' occupancy).
3. For each predator whose chosen square has not also been chosen by another predator, and is not the prey's square, the predator moves into its chosen square.
4. Repeat Step 3 until no predator can make a further move.
5. $k \leftarrow k+1$ and go to Step 1 if the prey is not free, or terminate if it is.

Answer 1

You allowed for some changes in the rules, so here are new rules. I hope they do not stray too far from your intent.

(1) To start, we put the prey at $(0, 0)$, and the $8$ predators at the $8$ grid points adjacent to the prey.

(2) At each step, each of the 8 predatora moves to one of the $8$ points nearest to it (or stays put) each with probability $1/9.$

(2a) We do not care if a grid point is occupied by multiple actors (prey or predators). That is relative uncommon and not likely to have much effect on the result. (2b) We do not restrict movement to points in the $9 \times 9$ grid. Mostly, few predators are going far anyhow. (2c) The main REASON for (a) and (b) above is ease in programming. The main JUSTIFICATIONS are that intuitively I don't think these departures from your rules are going to make much difference, and I don't understand what you mean by your Rule #4. (Box: "All models are wrong, but some are useful.")

(3) The process stops as soon as all eight of the grid points adjacent to the prey (your grey points) are empty, allowing the prey to escape.

Let $T$ be the number of steps until the prey escapes. Following @Henry's suggestion that simulation is the best way to answer this, here is R code implementing the altered rules, along with (presumably explanatory) notations:

 m = 100000;  t = numeric(m)   # 100,000 performances of expt
   for (i in 1:m) {
   x = c(1,1,0,-1,-1,-1, 0, 1)  # starting x-coords of 8 predators
   y = c(0,1,1,1,0,-1,-1, -1)   # their starting y-coordinates
   cnt = 0;  far = 1            # init count & min predator dist from prey
     while(far < 2) {
     dx = sample(-1:1, 8, rep=T) #incremental movements
     dy = sample(-1:1, 8, rep=T) #  of predators
     x = x + dx;  y = y + dy     # their new locations
     far = min(sqrt(x^2+y^2))    # their min dist from prey
     cnt = cnt+1 }
   t[i] = cnt }
 mean(t);  sd(t)
 ## 12.13223
 ## 6.258674
 summary(t)
 ##    Min. 1st Qu.  Median   Mean 3rd Qu.    Max. 
 ##   1.00    8.00   11.00   12.13   15.00   61.00 

The approximation gives $E(T) \approx 12.1$ and $SD(T) \approx 6.3.$ The graph below shows an example of one of the 100,000 runs in which it took 19 steps for the prey to escape, and a histogram of all 100,000 simulated values of $T$. (If the option at each step not to move were omitted the mean would be a little smaller.)