Introduction to the Modeling and Analysis of Complex Systems

4.5. BUILDING YOUR OWN MODEL EQUATION 53• Exponential growth• Convergence to a certain population limitWe should check the first one first. The original model already shows exponential growthby itself, so this is already done. So we move on to the second one. Obviously, the originalmodel doesn’t show such convergence, so this is what we will need to implement in themodel.The third tip says you need to focus on a specific component to be revised. There aremany options here. You could revise a, x t−1 , or you could even add another term to theright hand side. But in this particular case, the convergence to a certain limit means thatthe growth ratio a should go to 1 (i.e., no net growth). So, we can focus on the a part, andreplace it by an unknown function of the population size f(x t−1 ). The model equation nowlooks like this:x t = f(x t−1 )x t−1 (4.20)Now your task just got simpler: just to design function f(x). Think about constraints it hasto satisfy. f(x) should be close to the original constant a when the population is small,i.e., when there are enough environmental resources, to show exponential growth. In themeantime, f(x) should approach 1 when the population approaches a carrying capacity ofthe environment (let’s call it K for now). Mathematically speaking, these constraints meanthat the function f(x) needs to go through the following two points: (x, f(x)) = (0, a) and(K, 1).And this is where the fourth tip comes in. If you have no additional information aboutwhat the model should look like, you should choose the simplest possible form that satisfiesthe requirements. In this particular case, a straight line that connects the two pointsabove is the simplest one (Fig. 4.5), which is given byf(x) = − a − 1 x + a. (4.21)KYou can plug this form into the original equation, to complete a new mathematical equation:(x t = − a − 1 )Kx t−1 + a x t−1 (4.22)Now it seems your model building is complete. Following the fifth tip, let’s check ifthe new model behaves the way you intended. As the tip suggests, testing with extremevalues often helps find out possible issues in the model. What happens when x t−1 = 0? In