1. MA3264 Mathematical Modelling Lecture 13 Practice Problems Homework 4 Answers

2. Practice Problems 1. Consider the predator prey equations a. Find equilibrium points 2. Find the max value of b. Sketch phase plane c. Find curves on which the trajectories move d. Find average values of x and y e. What does fishing for krill x do to x + whales y? a. In the positive quadrant (PQ) b. In each closed rectangular region in the PQ c. Show that K(x,y) is a concave function

3. Homework 4 Answers Equilibrium values are x = 0, x = 1, and x = -a/b if –a/b is in (0,1) Phase line vectors are pos/neg iff a+bx is always pos./neg. If positive then 0 is unstable and 1 is asymptotically stable 1. Consider the population growth model on the interval [0,1]. Determine all equilibrium values as a function of the real parameters a and b and determine their stability. If negative then 1 is unstable and 0 is asymptotically stable If a > 0 and –a/b is in (0,1) then 0 and 1 are unstable and –a/b is asymptotically stable If a < 0 and –a/b is in (0,1) then 0 and 1 are asymptotically stable and –a/b is unstable

4. Homework 4 Answers displacement S(t) of a harmonic oscillator (object on a spring). a. Convert this equation to a system of first order equations. b. Determine the most general solution (in a closed form). c. Determine the parameters of the general solution in terms of and 2. Consider the second order differential equation that models the

5. Homework 4 Answers 3. Consider the system of differential equations Determine the functionsuch that the solution trajectories are parabolas.

6. Homework 4 Answers 5. Do problem 2 on page 424 in the textbook. 6. Do problem 5 a,b,c,d,e,f on pages 436-437 in the textbook. 4. Do problem 9 a,b,c,d on page 417 in the textbook.