Introduction to the Modeling and Analysis of Complex Systems

Chapter 7Continuous-Time Models II: Analysis7.1 Finding Equilibrium PointsFinding equilibrium points of a continuous-time model dx/dt = G(x) can be done in thesame way as for a discrete-time model, i.e., by replacing all x’s with x eq ’s (again, note thatthese could be vectors). This actually makes the left hand side zero, because x eq is nolonger a dynamical variable but just a static constant. Therefore, things come down to justsolving the following equation0 = G(x eq ) (7.1)with regard to x eq . For example, consider the following logistic growth model:dx(1dt = rx − x )K(7.2)Replacing all the x’s with x eq ’s, we obtain(0 = rx eq 1 − x )eq(7.3)Kx eq = 0, K (7.4)It turns out that the result is the same as that of its discrete-time counterpart (see Eq. (5.6)).Exercise 7.1Find the equilibrium points of the following model:dxdt = x2 − rx + 1 (7.5)111