Introduction to the Modeling and Analysis of Complex Systems

128 CHAPTER 7. CONTINUOUS-TIME MODELS II: ANALYSISExercise 7.15 Consider the logistic growth model (r > 0, K > 0):dx(1dt = rx − x )K(7.70)Conduct a linear stability analysis to determine whether this model is stable or notat each of its equilibrium points x eq = 0, K.Exercise 7.16 Consider the following differential equations that describe the interactionbetween two species called commensalism (species x benefits from thepresence of species y but doesn’t influence y):dxdt = −x + rxy − x2 (7.71)dy= y(1 − y)dt(7.72)x ≥ 0, y ≥ 0, r > 1 (7.73)1. Find all the equilibrium points.2. Calculate the Jacobian matrix at the equilibrium point where x > 0 and y > 0.3. Calculate the eigenvalues of the matrix obtained above.4. Based on the result, classify the equilibrium point into one of the following:Stable point, unstable point, saddle point, stable spiral focus, unstable spiralfocus, or neutral center.Exercise 7.17Consider the differential equations of the SIR model:dS= −aSIdt(7.74)dI= aSI − bIdt(7.75)dR= bIdt(7.76)As you see in the equations above, R doesn’t influence the behaviors of S and I,so you can safely ignore the third equation to make the model two-dimensional.Do the following: