CSC 210

Week 5



  1. Bacterial growth experiment: A culture of 8 bacteria are placed in a test tube, and their numbers is counted daily. At first, when their numbers was small they grew at a rate of %185 per day. After a few days, their numbers stabilized at about 410.

  2. Predator-prey model with logistic growth: Recall that in class we considered the Predator-prey system ODE from Phaser
    x1' = x1(a - bx2 - mx1)
    x2' = x2(-c + dx1 -nx2)
    Set a=2, b=1.1, c=1.5, d=0.5, m=0, and n=0. With these parameter values we assume that in the absence of the other, each species grows or dies exponentially. Take several (at least two) initial conditions and describe the behavior of the populations in biological terms.
    Now, increase the parameter m from 0 to 2.5 while keeping the other parameters fixed. That is, assume that the prey grows according to the logistic model. Describe the behavior of the populations as the parameter m is varied.

  3. Error vs. step size in Euler: We mentioned in class that the global error bound in Euler's algorithm is proportional to the step size. Now, in PHASER solve the initial-value problem x' = x, x(0) = 1 to compute x(1) = e = 2.7182818284590452354, using Euler's algorithm with six different step sizes. Using your favorite plotting program, plot the errors against the step sizes. Do you get a linear relationship? Be careful of the scales on your graph.

  4. Impossible computation: Consider the "explosion" problem x'= x^2, x(0)=1, and try to compute x(1) using several different algorithms with various step sizes.

  5. Two steps of Improved Euler(2): Consider the initial value problem x1' = x, x(0)=1. With step size h = 0.5, compute, by hand using paper and pencil, two steps of Improved Euler (2) algorithm to obtain the approximate value of x(1). You can look up the formula for Improved Euler (2) in Phaser Help; make sure you show all the intermediate numbers Ks. Now compare your answer with the one you get from Phaser.

  6. Reading Euler: Try to read the original paper of Euler listed above.