Introduction to the Modeling and Analysis of Complex Systems

5.6. ASYMPTOTIC BEHAVIOR OF DISCRETE-TIME LINEAR DYNAMICAL... 81You should try variable rescaling to eliminate as many parameters as possible fromyour model before conducting a mathematical analysis. You may be able to eliminateas many parameters as the variables in your model. To rescale variables, do thefollowing:1. Replace all variables with a non-zero constant times a new variable.2. Simplify the model equations.3. Find “convenient” choices for the constants that will make your equations assimple as possible.4. Define new parameters, as needed, to make the equations even simpler.Exercise 5.9x t =ax t−1 + bSimplify the following difference equation by variable rescaling:(5.30)Exercise 5.10 Simplify the following two-dimensional predator-prey differenceequation model by variable rescaling:(x t = x t−1 + rx t−1 1 − x ) ()t−11− 1 −x t−1 (5.31)K by t−1 + 1y t = y t−1 − dy t−1 + cx t−1 y t−1 (5.32)5.6 Asymptotic Behavior of Discrete-Time Linear DynamicalSystemsOne of the main objectives of rule-based modeling is to make predictions of the future.So, it is a natural question to ask where the system will eventually go in the (infinite) longrun. This is called the asymptotic behavior of the system when time is taken to infinity,which turns out to be fully predictable if the system is linear.Within the scope of discrete-time models, linear dynamical systems are systems whose