Originally, I had implemented this as just a continuous time Markov chain; state (P,V) turns into The only problem with this is that we've lost a little bit of the biology; the contacts between predator and prey now come in two entirely different flavors: ones where prey disappear, and ones where new predators are created. It would be nice to couple these processes.

One possible solution is to assign each predator a state, namely, how many times it has fed; and once it hits the magic number, it's allowed to reproduce. While biologically warranted, perhaps, it's pretty clumsy on the mathematical side.

I tried something a little different. I say that encounters happen at rate S*P*V. Each encounter diminishes the prey count by one. It further has a chance of adding to the number of predator, with probability gamma. Here, I choose gamma so that the expected number of unsuccessful encounters before a new predator is produced is the same as that mandated by the continuous-time rate alpha.
Feel free to look at the dynamics of the original CTMC model.