# Week 3

### Topics:

• Application: Population dynamics, LOGISTIC MAP x1 -> a*x1*(1.0 - x1) continued.
• LOGISTIC MAP x1 -> a*x1*(1.0 - x1) Review article from NATURE.
• Ricker's original paper
• Bifurcation diagrams.
• Time series (X vs Time) plots; how to recognize stable and oscillating population from data.
• Discrete Predator-prey model.

### Assignment:

1. Logistic vs. Logistic: Note that in the Equation -> MAP Library of Phaser, there are two versions of the logistic MAP:
Logistic MAP: x1->a*x1*(1 - x1)
Logistic III MAP: x1 -> a*x1 - a*x1*x1
These two maps are mathematically identical (just open up the parantheses). On the computer, however, they can behave differently.
1. Set a = 2.7 and compute 70 iterates of the initial condition 0.23 first using Logistic MAP and next using Logistic III MAP. In the Xi Values view, look at the two sets of numbers. Do the numbers look the same at each iterate?
2. Now set a = 4.0, compute 70 iterations of the same initial condition with both versions of the Logistic. Do the numbers look the same? If not, at what iterate the values have no digits in common? What do you make of this puzzling, and disturbing, computational experiment? For a possible hint, read about floating-point arithmetic; see, for example, Wikipedia article.

2. Sensitive dependence on initial conditions: In the Logistic model, first set a = 2.66. Now take two initial population sizes and determine the population densities after 70 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 70 generations. Repeat the same experiment with the growth rate a = 3.951. What are the biological implications of these experiments?

3. Verifying calculations in the scientific literature: Read as much of the Nature article linked above as you can. This could be difficult reading for you, but do not be discouraged. In particular, examine Table 3 on page 464 regarding the periodic orbits of the Logistic map. Try to verify the first three numbers on the 4th row: period 5(a) orbit appears for a = 3.7382 and disappears at a = 3.7411. Is the claim in the paper true? Support your answer by drawing stair-step diagrams, Xi-values, for at least three parameter values: two near the boundaries, and one inside.
Notice that there other parameter windows for which there are period 5 orbits. If you like, you can explore the rows 5(b) and 5(c).

4. 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 link above to the original paper of Ricker.
• Construct the bifurcation diagram of the Ricker MAP by fixing k = 1, and varying r.
• By examing your bifurcation diagram, locate a value of r for which the map has a periodic orbit of period 3. Draw a stair-step diagram of your period-3 orbit. What are the values of the population size on this orbit?