Math 5270 - Modeling with Dynamical Systems
Homework Assignments

Spring 2000
Prof. Peckham



  • Midterm 1 Wednesday March 22, 2000. See Midterm Topic List
  • Preliminary Project Topic Description. Due Friday, March 24, 2000. Describe in a half page or less a brief outline of a project that you plan on doing. Indicate roughly where you expect to spend your effort: literature search, model development, analyzing a model, data collection, writing up results, ... .
  • Logistic Model, Part 2. Due Monday, 4/10/2000.
    1. (Due date pushed back to 4/17/2000, but no late ones will be accepted after the beginning of class on 4/26 since we will review all the logistic models in class on 4/17.) 2D Differential Equation. We developed several models with two age classes, but I will choose the following one to analyze:

      x(t)=juvenile population at time t (unable to reproduce)
      y(t)=adult population at time t (able to reproduce)

      Assuming the populations behave logistically, we came up with the follwing systems of differential equations:

      dx/dt = B y - D1 x - D2 x^2 - D3 x y - G x
      dy/dt = G x - D4 y - D5 y^2 - D6 x y

      Explain the significance of each term in the system. Assume all parameters are positive. Show that by rescaling both x and y we can only eliminate one parameter. Therefore, choose the same rescaling factor for both and show that the rescaling factor can be chosen to allow the system to be rewritten as

      dx/dt = b y - d1 x - d1 x^2 - d3 x y - g x
      dy/dt = g x - d4 y - d5 y^2 - d6 x y

      • What is the relationship between all corresponding parameters before and after rescaling? For simplicity, assume the parameters d3 and d6 are zero.
      • Determine nullclines, and how they vary with the parameters. (Why did I decide to make d3 and d6 zero?) Use the nullclines to help locate all equilibrium points and how they depend on the parameters. For which parameter values is there a positive carrying capacity?
      • Determine the stability of each equilibrium point by linearization, and determine how the stability depends on the parameters.
      • Could this model ever allow a positive population to exist without limiting to a constant value (a ``carrying capacity")? If so, for what parameter values? (Hint: Look at the determinant and trace of the Jacobian matrix used in the linearization.)
      • Corroborate your analysis with a numerical phase portrait (for example using Differential Systems software). Label any stable and unstable manifolds of saddles. (You may draw these in by hand if you wish.) Use the phase portrait to describe the long term behavior of the population and how it depends on the starting populations.


    2. (Work on this problem will not be accepted after the beginning of class on Wednesday April 26, 2000.) Delayed logistic map (discrete dynamical system). Combination of 1D discrete dynamical system and the 1D delay differential equation. Start with the 1D discrete map: x_(n+1) = a x_n - b (x_n)^2, but replace one of the x_n's in (x_n)^2 with x_(n-1), analogous to the 1D delay differential equation. Rescale to make a new discrete map: x_(n+1) = a x_n - a x_n x_(n-1). Convert this second order difference equation into a system of two first order difference equations:

      y_(n+1)=x_n
      x_(n+1)=a x_n (1-y_n)

      Locate all nonnegative fixed points and how they depend on the parameter a. Determine the stability of each equilibrium point by linearization, and determine how the stability depends on a. Could this model ever allow a positive population to exist without limiting to a constant value (a ``carrying capacity")? If so, for what parameter values? (Hint: Locate a bifurcation where the positive fixed point changes from attracting to repelling.)


  • Homotopy between two predator-prey models to locate a bifurcation. Due Monday May 1, 2000.

    We saw in class that two different versions of a predator-prey model exhibited different behavior:

    dR/dt=5R-R^2 - 4RF
    dF/dt=-5F+4RF

    had an attracting positive population for both predator and prey.

    dR/dt=R-R^2 - 4RF/(1+4R)
    dF/dt=-.5F+4RF/(1+4R)

    had a repelling positive population for both predator and prey.




    This page is maintained by Bruce Peckham (bpeckham@d.umn.edu) and was last modified on Thursday, 20-Apr-2000 17:48:16 CDT.