Introduction to the Modeling and Analysis of Complex Systems

4.6. BUILDING YOUR OWN MODEL EQUATIONS WITH MULTIPLE VARIABLES 55equation into a more well-known form:(x t = − a − 1 )Kx t−1 + a x t−1 (4.24)(= − r )K x t−1 + r + 1 x t−1 (4.25)(= x t−1 + rx t−1 1 − x )t−1(4.26)KThis formula has two terms on its right hand side: the current population size (x t−1 ) andthe number of newborns (rx t−1 (· · · )). If x is much smaller than K, the value inside theparentheses gets closer to 1, and thus the model is approximated byx t ≈ x t−1 + rx t−1 . (4.27)This means that r times the current population is added to the population at each timestep, resulting in exponential growth. But when x comes close to K, inside the parenthesesapproaches 0, so there will be no net growth.Exercise 4.10 Create a mathematical model of population growth in which thegrowth ratio is highest at a certain optimal population size, but it goes down as thepopulation deviates from the optimal size. Then simulate its behavior and see howits behavior differs from that of the logistic growth model.4.6 Building Your Own Model Equations with MultipleVariablesWe can take one more step to increase the complexity of the model building, by includingmore than one variable. Following the theme of population growth, let’s consider ecologicalinteractions between two species. A typical scenario would be the predator-preyinteraction. Let’s stick to the population-level description of the system so each speciescan be described by one variable (say, x and y).The first thing you should consider is each variable’s inherent dynamics, i.e., whatwould happen if there were no influences coming from other variables. If there is alwaysplenty of food available for the prey, we can assume the following:• Prey grows if there are no predators.