Thanks Paul. I have a pedagogical challenge. Suppose one was explaining L-V and simulation to, say a functional programmer or software engineer, who didn't have any background in the natural sciences. We're at a party with them, and are trying to convey to them at a high level the spirit of these topics.
For now let's start with the deterministic case, which is a big subject in itself. How would the "mini talk" go?
I would start by saying something like L-V is a system of ODEs, which arise as the rate equations for a predator/prey population model based on the following premises XYZ. For simulating these kinds of deterministic systems the goal boils down to understanding the structure of the solutions, which are all the paths through the vector field defined by the equation. Stable equilibrium points show up as attractor points that solution paths converge to. Unstable equilibria are repelling points which the paths lead out of. Oscillatory solutions are bounded paths that never fall into an attractor points.
Now how can simulators help us to understand this structure of solution paths? For starters, they can identify the equilibrium and attractor points. Next step is that they can be used to make a map of the basins of attractions for the attractor points. It is can be useful to make a "color chart" where the color shows ABC. Regions of oscillatory paths may be conjectured if they remain bounded for a "long time".
The structure of the L-V solutions has the following general topology... To explore aspect ABC of these solutions in greater detail, simulation type DEF is useful.
An important question in population dynamics is what are the regions that lead to the extinction of some subset of the populations...