Discrete Time Simulation

Posted 27 Mar 2004 by hermanThe topic is introduced using the famous Lotka-Volterra Equations.

The Lotka-Volterra equations, described here, give an explanation of how predators and prey interact in a biological system. The equations are differential equations governing the rate of population growth (or decline) in two species. This page has a nice, intuitive rationale for the terms in the equations. While the equilibrium solution for these equations can be found directly, simulation is more versatile because we can modify the equations and easily see the result by simulation. This page has more details about the equations and suggests exploring the results by simulation. Rather than using a simulation package, you can simulate the populations of foxes and rabbits using a Python program that I wrote: volterra.py. I'll illustrate this in class and show how changing the parameters gives different results. I found some slides here that give some insight into how things change as the simulation parameters are modified.

A somewhat lengthy paper shows how ideas from the Lotka-Volterra equations can be generalized to very complex systems (glance at the diagram on pages 14-15, and you get the idea). In class we'll discuss how one might make the equations more realistic an explore the consequences using simulation.

Sim World, Simulations of Networks?It's not hard to imagine how a simulation based on equations could be used for guiding the way virtual realities and related games (Sims World) could be implemented. Simulation is also very useful for testing hypotheses about systems before they are built: we would like to know if a system will have unexpected behaviors before spending millions to implement it! Economists, businesses, chip-manufacturers, insurance companies -- are but a few examples of organizations where simulation has become a crucial tool. In the networking arena, simulators such as NS are very widely used to evaluate the performance of network protocols. We will find that many of these simulations go beyond differential equations and discrete time.