Introduction to the Modeling and Analysis of Complex Systems

52 CHAPTER 4. DISCRETE-TIME MODELS I: MODELINGMathematics is a language that is used to describe the world. Just like that there is nosingle correct way to describe an idea in English, there is no single correct way to build amathematical model equation either. It is highly dependent on your own personal literacy,creativity, and expressiveness in the language of mathematics. Ultimately, you just needto keep “reading” and “writing” math every day, in order to get better in mathematicalmodel building.Having said that, there are some practical tips I can offer to help you build your ownmodel equations. Here they are:Practical tips for mathematical model building1. If you aren’t sure where to start, just grab an existing model and tweak it.2. Implement each model assumption one by one. Don’t try to reach the finalmodel in one jump.3. To implement a new assumption, first identify which part of the model equationrepresents the quantity you are about to change, replace it with an unknownfunction, and then design the function.4. Whenever possible, adopt the simplest mathematical form.5. Once your equation is complete, check if it behaves as you desired. It is oftenhelpful to test its behavior with extreme values assigned to variables and/orparameters.Let me illustrate each of those tips by going through an example. Consider buildinganother population growth model that can show not just exponential growth but alsoconvergence to a certain population limit. Any ideas about where to start?As the first tip suggests, you could use an existing model that is similar to what youwant to model, and then modify it for your needs. Since this example is about populationgrowth, we already know one such model: the exponential growth model. So let’s startthere:x t = ax t−1 (4.19)This model is very simple. It consists of just two components: growth ratio a and populationsize x t−1 .The second tip says you should take a step-by-step approach. So let’s think aboutwhat we additionally need to implement in this model. Our new model should show thefollowing two behaviors: