Introduction to the Modeling and Analysis of Complex Systems

244 CHAPTER 13. CONTINUOUS FIELD MODELS I: MODELING1. The population will never increase or decrease within the time scale being consideredin this model.2. The economy is activated by having more people in a region.3. The economy diminishes without people.In the meantime, their spatial dynamics (movements) are assumed as follows:4. Both the population and the economic activity diffuse gradually.5. People are attracted to the regions where economy is more active.Our job is to translate these assumptions into mathematical equations by filling in theboxes below.∂p∂t =∂m∂t =First, give it a try by yourself! Then go on to the following.Were you able to develop your own model? Let’s go over all the assumptions to seehow we can write each of them in mathematical form.Assumption 1 says there is no change in the population by default. So, this can bewritten simply as∂p∂t= 0. (13.19)Assumption 2 is about the increase of economy (m) caused by people (p). The simplestpossible expression to model this is∂m∂t= αp, (13.20)where α is a positive constant that determines the rate of production of the economy percapita.Assumption 3 tells that the economy should naturally decay if there is no influence frompeople. The simplest model of such decay would be that of exponential decay, which can