CSC 210

Week 5



  1. Harvest to extinction: Suppose that the size x(t) of a population is governed by the differential equation x' = 1.25 x - 0.16 x^2 - 0.45. This is the logistic model with constant harvesting. Using Phaser answer the following questions. Default Algorithm Dormand-Prince 5(4) and the Step Size 0.01 are fine to use.
    If the initial population size is x(0) = 4.1, find the time at which the population doubles.
    If the initial population size is x(0) = 0.4, find the time at which the population is exterminated.

  2. Predator-Prey model: Consider the generalized predator-prey (as it appears in Phaser ODE Library)
    x1' = x1*(a - b*x2 - m*x1)
    x2' = x2(-c + d*x1 - n*x2),
    where the parameters take on nonnegtive values.

  3. Another form of Logistic: Consider the continuous analog of the logistic model using the ODE:
    x' = r x (1 - x/k)
    where x(t) is the population size at time t, r (growth rate) and k (carrying capacity) are positive parameters. This form of the Logistic ODE is preferred by ecologists.

  4. Gompertz model of cancer growth: The differential equation
    x' = a*(exp(-b*t))*x
    is used to describe the growth of a tumor, where x(t) is a measure of its size (e.g. weight or number of cells), and a and b are parameters specific to a particular tumor. To get started, let us take a = 3 and b = 2, and x(0) = 5; the solution in the XivsTime view is shown in the image below.

    Load the image into your PHASER by clicking on the picture. This picture is not to scale. Adjust the Window size to get a true aspect ratio and draw the Direction Field. Using the Direction Field and solution curves answer the following questions.
    Explain what this model predicts for the tumor growth. Experiment with several different initial tumor sizes. How does the future size of tumor depend on its initial size?
    Experiment with several values of the parameters a and b. Explain the roles of these parameters in the tumor growth.

    Consult the famous article "Dynamics of Tumor Growth" by A.K. Laird in British Journal of Cancer (1964) 18, 490-502. doi:10.1038/bjc.1964.55
    Verify that the formula for Gompertz curve x = x_0 exp((a/b)(1 - exp(-bt)) that she gives in this article satisfies Gompertz's differential equation above.