Topics:

• Discrete population models and their simulations.
• Linear model: x1 -> r*x1
• LOGISTIC MAP x1 -> r*x1*(1.0 - x1) Review article from NATURE.
• Stable equilibrium, stable oscillations, period-doubling bifurcations, chaotic (non-periodic) solutions, sensitive dependence on initial conditions.

Assignment:

1. Fixed points of the Logistic MAP: Find all fixed points of the Logistic map as a function of the parameter r ( answer: x = 0 and x = 1 - 1/r) For which values of r these fixed points are biologically relevant? Using the linearization theorem, determine the values of the parameter r for which the fixed point at the origin is asymptotically stable or unstable. Next do the same for the second fixed point. Note that for r = 3, the positive fixed point has derivative -1 and thus the Linearization Theorem does not apply. Using PHASER determine if this fixed point is asymptotically stable or unstable for r = 3.

2. Sensitive dependence on initial conditions: In the Logistic model, first, set r = 3.83 and take two initial population densities 0.2 and 0.200001. As time goes on, what are the eventual fates of these solutions. Next, change the growth rate to r = 3.86. What are the fates of the two solutions with the initial sizes above? Notice how close the initial population densities are. How many generations does it take when no digits of the two solutions agree? What are the biological implications of these computer experiments?

3. Period-doubling everywhere: In the Logistic MAP, set the parameter r = 3.83. Fix Start time= 1000, Stop Time=1500 and initial condition 0.2. Draw the Stair-Step diagram to see period-3 solution. CHeck the number in Xi-Values view to make sure that the numbers repeat every fourth iteration. Now, change r carefully to make this solution to bifurcate to a period-6 solution. Change r a bit more to obtain a period-12 solution. Submit your parameter values, Xi values, and the Stair step diagrams.

4. Read as much of the Nature article linked above as you can. Extract a statement of your choice, or formulate a problem, and illustrate it using PHASER.

5. 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.
   1. To understand the geometric meanings of the parameters, Fix k (say at 1) and vary r make a sequence of growth curves using the stair-step view of PHASER. You may want to take big window size (-1 , 15; -1, 15) to see what happens as the population gets large. Next, fix r = 1.5 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?

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