Introduction to the Modeling and Analysis of Complex Systems

5.5. VARIABLE RESCALING 79before jumping right into such paper-and-pencil mathematical work, I would like to showyou a very useful technique that can make your mathematical work much easier. It iscalled variable rescaling.Variable rescaling is a technique to eliminate parameters from your model withoutlosing generality. The basic idea is this: Variables that appear in your model representquantities that are measured in some kind of units, but those units can be arbitrarily chosenwithout changing the dynamics of the system being modeled. This must be true for allscientific quantities that have physical dimensions—switching from inches to centimetersshouldn’t cause any change in how physics works! This means that you have the freedomto choose any convenient unit for each variable, some of which may simplify your modelequations.Let’s see how variable rescaling works with an example. Here is the logistic growthmodel we discussed before:(x t = x t−1 + rx t−1 1 − x )t−1K(5.20)There is only one variable in this model, x, so there is only one unit we can change,i.e., the unit of the population counts. The first step of variable rescaling is to replaceeach of the variables with a new notation made of a non-zero constant and a new statevariable, like this:x → αx ′ (5.21)With this replacement, the model equation becomesαx ′ t = αx ′ t−1 + rαx ′ t−1( )1 − αx′ t−1. (5.22)KThe second step is to simplify the equation and then find a “convenient” choice forthe constant that will make your equation simpler. This is a rather open-ended processwith many different directions to go, so you will need to do some explorations to find outwhat kind of unit choices make your model simplest. For the logistic growth model, theequation can be further simplified, for example, like