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. 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). Despite its complicated form, this model has simple dynamics. We assume that both parameters take on non-negative values. Enter this model into PHASER.
   1. To understand the geometric meanings of the parameters, Fix k (say at 1) and vary r to make a Gallery of growth curves using the stair-step view of PHASER. You may want to take big window size (-1 , 17; -1, 17) to see what happens as the population gets large. Next, fix r = 1.6 and vary k. Describe biologically what you observe in these two sequences. Note: In this model varying the parameters is not dangerous (unlike the logistic model).

   2. Find the fixed points of the model as a function of the parameters. For what ranges of the parameters they are biologically significant?

   3. Determine the stability type of the fixed points. Can the positive fixed point become unstable as the parameter r or k is increased? Can either of the fixed points undergo period-doubling bifurcation?

2. Draw the bifurcation diagram of the Beverton-Holt Stock-recruitment model
   • Fix k=1 , and draw the bifurcation diagram for r. To get useful picture, set initial condition to 0.3, starttime=232, 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.

3. Write a summary of the possible dynamics of a population descibed by the Beverton-Holt model and interpret your findings in biological terms.

4. Decay constant for C-14: The half life of radioactive 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 120 and 210. Does the decay constant depend on the initial amount?