Introduction to the Modeling and Analysis of Complex Systems

80 CHAPTER 5. DISCRETE-TIME MODELS II: ANALYSISx ′ t = x ′ t−1 + rx ′ t−1(= x ′ t−1 1 + r= x ′ t−1( )1 − αx′ t−1K( ))1 − αx′ t−1K)(1 + r − rαx′ t−1K= (1 + r)x ′ t−1(1 − rαx′ t−1K(1 + r)(5.23)(5.24)(5.25)). (5.26)Here, the most convenient choice of α would be α = K(1 + r)/r, with which the equationabove becomesx ′ t = (1 + r)x ′ t−1(1 − x ′ t−1). (5.27)Furthermore, you can always define new parameters to make equations even simpler.Here, you can define a new parameter r ′ = 1 + r, with which you obtain the following finalequation:x ′ t = r ′ x ′ t−1(1 − x ′ t−1) (5.28)Note that this is not the only way of rescaling variables; there are other ways to simplify themodel. Nonetheless, you might be surprised to see how simple the model can become.The dynamics of the rescaled model are still exactly the same as before, i.e., the originalmodel and the rescaled model have the same mathematical properties. We can learn afew more things from this result. First, α = K(1 + r)/r tells us what is the meaningfulunit for you to use in measuring the population in this context. Second, even though theoriginal model appeared to have two parameters (r and K), this model essentially hasonly one parameter, r ′ , so exploring values of r ′ should give you all possible dynamicsof the model (i.e., there is no need to explore in the r-K parameter space). In general,if your model has k variables, you may be able to eliminate up to k parameters from themodel by variable rescaling (but not always).By the way, this simplified version of the logistic growth model obtained above,x t = rx t−1 (1 − x t−1 ), (5.29)has a designated name; it is called the logistic map. It is arguably the most extensivelystudied 1-D nonlinear iterative map. This will be discussed in more detail in Chapter 8.Here is a summary of variable rescaling: