Introduction to the Modeling and Analysis of Complex Systems

62 CHAPTER 5. DISCRETE-TIME MODELS II: ANALYSISReplacing all the x’s with x eq , we obtain the following:(x eq = x eq + rx eq 1 − x )eq(5.4)( K0 = rx eq 1 − x )eq(5.5)Kx eq = 0, K (5.6)The result shows that the population will not change if there are no organisms (x eq = 0)or if the population size reaches the carrying capacity of the environment (x eq = K). Bothmake perfect sense.Exercise 5.1Obtain the equilibrium point(s) of the following difference equation:x t = 2x t−1 − x 2 t−1 (5.7)Exercise 5.2 Obtain the equilibrium point(s) of the following two-dimensional differenceequation model:x t = x t−1 y t−1 (5.8)y t = y t−1 (x t−1 − 1) (5.9)Exercise 5.3Obtain the equilibrium point(s) of the following difference equation:x t = x t−1 − x 2 t−2 + 1 (5.10)Note that this is a second-order difference equation, so you will need to first convertit into a first-order form and then find the equilibrium point(s).5.2 Phase Space Visualization of Continuous-StateDiscrete-Time ModelsOnce you find where the equilibrium points of the system are, the next natural step ofanalysis would be to draw the entire picture of its phase space (if the system is two orthree dimensional).