Introduction to the Modeling and Analysis of Complex Systems

58 CHAPTER 4. DISCRETE-TIME MODELS I: MODELINGThe death rate of the prey should be 0 if there are no predators, while it should approach1 (= 100% mortality rate!) if there are too many predators. There are a numberof mathematical formulas that behave this way. A simple example would be the followinghyperbolic functiond x (y) = 1 − 1by + 1 , (4.32)where b determines how quickly d x increases as y increases.The growth rate of the predators should be 0 if there are no prey, while it can go up indefinitelyas the prey population increases. Therefore, the simplest possible mathematicalform could ber y (x) = cx, (4.33)where c determines how quickly r y increases as x increases.Let’s put these functions back into the equations. We obtain the following:(1 −(x t = x t−1 + rx t−1 1 − x )t−1−K1by t−1 + 1)x t−1 (4.34)y t = y t−1 − dy t−1 + cx t−1 y t−1 (4.35)Exercise 4.11 Test the equations above by assuming extreme values for x and y,and make sure the model behaves as we intended.Exercise 4.12 Implement a simulation code for the equations above, and observethe model behavior for various parameter settings.Figure 4.8 shows a sample simulation result with r = b = d = c = 1, K = 5 and x 0 =y 0 = 1. The system shows an oscillatory behavior, but as its phase space visualization(Fig. 4.8 right) indicates, it is not a harmonic oscillation seen in linear systems, but it is anonlinear oscillation with distorted orbits.The model we have created above is actually a variation of the Lotka-Volterra model,which describes various forms of predator-prey interactions. The Lotka-Volterra model isprobably one of the most famous mathematical models of nonlinear dynamical systemsthat involves multiple variables.