Homework 3 Math 375 Uhlenbeck/Buck Due October 9 1. Find the equilibria of the following system which models disease: dS/dt = (-aI + b(1 - S/K))S dI/dt = -cI +a SI. What is your answer if a = .1, b = .05, c= 1 and K = .02? What does each number a, b, c and K represent in the model, if S is the susceptible population and I is the infected? 2. Write the equation (1) on page 128 as a vector and matrix equation. Just to practice for the computer, use B_t = x(1), B_s = x(2) and C = x(3). 3. Write out the equation dX/dt = Q X F as three equations, if X = [A] (as in the conservation biology U .3 5 -.6 article) and Q is the matrix 0 1 2 . 0 .3 -4 4. Compute dh/dx and dh/dy when a) h(x,y) = x - x^2 - 2 xy b) h(x,y) = xy^3 -2 c) h(x,y) = x cos(y) d) h(x,y) = exp(x)/y -3y = e^x/y - 3y 5. Decide whether 0 is a sink, source or saddle for the equation dx/dt = -2x + 3y dy/dt = x - 6y