Math 19. Lecture 9Equilibrium in Two Component SystemsT. JudsonFall 20041 Uniqueness of Solution sA two-component linear system is a system of differential equationsdxdt= ax + by,dydt= cx + dy.Given initial conditions, (x(0), y(0)) = (x0, y0), the system has a uniquesolution and is completely predictive. We can also write this system in matrixform asx0(t) = Ax(t),whereA =a bc d, x(t) =x(t)y(t), and x0(t) =x0(t)y0(t).An equilibrium solution to the system where x(t) = (x(t), y(t)) is a constantvector.2 DeterminantsThe system x0(t) = Ax(t) has an equilibrium solution at (0, 0) if it has onlythe solution x = y = 0. Another way of viewing this fact is to observe that1the two linesax + by = 0cx + dy = 0are not parallel if and only if ad − bc 6= 0. We define t he determinant of Ato bedet(A) = ad − bc.3 Stability CriterionThe constant solution 0 is said to be stable when all trajectories that startin some region with 0 inside move closer to 0 as t → ∞. Otherwise, 0 isunstable. The system x0= Ax is stable if a nd only iftr(A) < 0det(A) > 0.4 An Equatio n for x( t)Let us examine 2 × 2 linear systems more closely. Letx0(t)y0(t)=a bc dx(t)y(t),2where x(0) = x0and y(0) = y0.4.1 An Uncoupled SystemLet us first a ssume that b = c = 0. Then t he solution to the systemx0= axy0= dyisx = x0eaty = y0edt.This system is stable if both a and d are negative. This occurs exactly whendet(A) > 0 and tr(A) < 0.4.2 The General CaseFor the general case, we will letx0=x(0)y(0)and w0=(a − d)x(0)/2 + by(0)cx(0) + (d − a)y(0)/2and∆ =14tr(A)2− det(A).We have exactly three types of solutions.1• Case 1: ∆ > 0.x(t) =12etr(A)t/2e√∆t(x0+ ∆−1/2w0) + e−√∆t(x0− ∆−1/2w0).• Case 2: ∆ = 0.x(t) = etr(A)t/2(x0+ tw0)).• Case 3: ∆ < 0.x(t) =12etr(A)t/2cos|∆|1/2tx0+ |∆|−1/2sin|∆|1/2tw0.1These solutions can be derived using linear algebra. See Math 21b or Math 106.3In each case, you can get an unstable solution if x0is chosen poorly and theconditionstr(A) < 0det(A) > 0are violated.Homework• Chapter 8. Exercises 1, 2, 3, 4, 5, 7; pp. 138–139.Readings and References• C. Taubes. Modeling Differential Eq uations in Biology. Prentice Hall,Upper Saddle River, NJ, 2001. Chapter 8.• “Better Protection for the Ozone Layer,” pp.
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