Introduction to the Modeling and Analysis of Complex Systems

114 CHAPTER 7. CONTINUOUS-TIME MODELS II: ANALYSIS2.52.01.51.00.50.00.0 0.5 1.0 1.5 2.0 2.5Figure 7.1: Phase space drawn with Code 7.1.as “walls” that separate the phase space into multiple contiguous regions. Inside eachregion, the signs of the time derivatives never change (if they did, they would be caughtin a nullcline), so just sampling one point in each region gives you a rough picture of howthe phase space looks.Let’s learn how this analytical process works with the following Lotka-Volterra model:dx= ax − bxydt(7.13)dy= −cy + dxydt(7.14)x ≥ 0, y ≥ 0, a > 0, b > 0, c > 0, d > 0 (7.15)First, find the nullclines. This is a two-dimensional system with two time derivatives, sothere must be two sets of nullclines; one set is derived from dx/dt = 0, and another set isderived from dy/dt = 0. They can be obtained by solving each of the following equations:0 = ax − bxy (7.16)0 = −cy + dxy (7.17)