CSC 210

Week 3


Topics:


Assignment:

  1. Sensitive dependence on initial conditions: In the Logistic model, first set a = 2.35. Now take two initial population sizes and determine the population densities after 100 generations. Suppose that you make 0.0003 percent error in the determination of the initial population sizes. Determine the relative percentage error in the population densities after 100 generations. Repeat the same experiment with the growth rate a = 4.0. What are the biological implications of these experiments?

  2. Ricker model:
    xn+1 = xn * exp(r*(1- xn/k))
    This is a biologically important fisheries model containing two parameters r (growth rate) and k (carrying capacity). We assume that both parameters take on non-negative values. See the following link for further info on this model. Ricker MAP and also the link above to the original paper of Ricker.

  3. Bifurcation diagram for Discrete Predator-Prey: Download the following phaser Project file discrete_pp_bifurcation.ppf by just clicking on it ( or by right-click and save it to our computer. Now load this file into Phaser). You should see the following bifurcation diagram:

    In this diagram, parameter b is fixed and a is varied. The horizontal axis is the parameter a and the vertical axis is the x1 variable (prey). Notice that the prey population is periodic with period 6 in a small window of the bifurcation diagram. Pick an a value in this window and for this value of the parameter a, plot the prey and predator in the Xi vs. Time window, and also in the Phase Portrait view. Is the predator population is periodic also? Interpret your pictures in biological terms.

  4. Nicholson-Bailey model: The following pair of difference equations is a famous model that describes the intereaction of host-parasitoid populations (one insect feeds on another):
    Hn+1 = k Hn e( -a Pn )
    Pn+1 = c Hn [1 - e( -a Pn ) ]

    Variables
    Hn : Density of host at generation n
    Hn+1 : Density of host at generation n+1
    Pn : Density of parasitoid at generation n
    Pn+1 : Density of parasitoid at generation n+1

    Parameters
    k : Reproductive rate of host
    a : Searching efficiency constant of parasitoid
    c : Average number of viable eggs deposited by parasitoid on a single host

    You can read more about this model at Phaser Web site .

    In 1941, Debach and Smith in their laboratory experiment started with 36 housefly, Musca domestica, and 18 of its pupal parasite Nasonia vitripennis and followed the populations for seven generations. They arranged the fecundity rate of the host to be 2 (k = 2, c = 1), and determined the searching efficiency constant to be a = 0.045. Please read their original paper at the link below:

    DE BACH, P. and SMITH, H.S. [1941]. "Are Population Oscillations Inherent in the Host-Parasite Relation?" Ecology, 22, 363-369. JSTOR URL

    In particular examine their data on Table I on page 367 and Figure 1 on page 368 of their article. Their Figure 1 looks like the following Phaser view: Download the following phaser Project file debach_smith.ppf by just clicking on it ( or by right-click and save it to our computer. Now load this file into Phaser).

    What is the experimental count of the host and parasitoid in the second and fifth generations in the paper of DeBach and Smith?
    What does the Phaser simulation of the Nicholson-Bailey model predict for these counts?
    What are the relative errors in the model predictions?