Topics:

• Application: Population dynamics, LOGISTIC MAP x1 -> a*x1*(1.0 - x1) continued.
• Bifurcation diagrams.
• Time series (X vs Time) plots; how to recognize stable and oscillating population from data.
• Discrete Predator-prey model.
• Introduction to continuous models: Radioactive decay.
• Lecture notes by fellow students

Assignment:

1. Draw the bifurcation diagram of the Beverton-Holt Stock-recruitment model
   x_n+1 = rx_n/[1 + ((r - 1)/k)x_n]
   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 previous week for further info on this model.
   • Fix k=1 , and draw the bifurcation diagram for r. To get useful picture, set initial condition to 0.3, starttime=222, stoptime=444. Window size (0.5, 3) (-1, 3). When you fix k=1 in the parameters panel, and assign the parameter r to be changed to the x-axis in the Bifurcation Diagram panel at the top of the tree, the current value of r is ignored and r is changed from x-min to x-max while k=1 is held fixed.
   • Draw another bifurcation while k =2 and r varied as above.
   • Interpret the diagrams in terms of the population's fate.
   • Next, draw three bifurcation diagrams by, fixing r = 0.7 and varying k, fixing r = 1 and varying k, fixing r =2 and varying k.
   • Interpret your bifurcation diagrams biologically.

2. 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 x1 vs. time window, and also in the Phase portrait view.

3. Decay constant for C-14: The half life of radiactive carbon C14 is known to be appproximately 5700 years. Using Phaser, determine the decay constant k in the decay differential equation x1' = k*x1. Do the calculations with two different initial conditions, say 100 and 200. Does the decay constant depend on the initial amount?