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.