Project 1 (Due March 29)

(Problem 68 in Section 8.1 of Thomas/Finney)
The population of certain species of bees are described by the logistic equation

$$X_{n+1} = r X_n ( 1 - X_n) \hspace{0.2in} 0<r<4,\hspace{0.2in} 0<X_n<1$$

where X_n is the population of the bees after n seasons and r is a parameter which describes the effective growth rate of the population (i.e. the difference between birth rate and the death rate). In the above equation the population X_n is assumed to be given in millions; so X_n=0.3 denotes a population of 0.3 million. The maximum population that the environment can support is 1 million, so 0<X_n<1.

The aim of the project is to understand the long-term behaviour of the population of the bees, for different values of the parameter r. We would find that if the effective growth rate r is small then the bee population eventually dies out and if r is large, then the long term behaviour of the bee population becomes chaotic.

Throughout this exercise choose the initial population X₀ such that 0<X₀<1, say X₀=0.3.

1.

Choose r=3/4. Calculate and plot the population for the first hundred seasons. Does the population appear to converge? What do you guess is the limit of the population sequence? Does this final limit seem to depend on your choice of X₀?

2.

Choose several values of r in the interval 1<r<3 and repeat the procedures given in the previous part. Describe the behaviour of the sequences you observe in your plots.

3.

Now examine the values of r near the endpoints of the interval 3<r<3.45. The transition value r=3 is called a bifurcation value and the new behaviour of the sequence in the interval is called an attracting 2-cycle.

4.

Next explore the behaviour for r values near the endpoints of each of the intervals 3.45<r<3.54 and 3.54<r< 3.55. Plot the first 200 terms of the sequences. Describe in your own words the behaviour observed in your plots for each interval. Among how many values does the sequence appear to oscillate for each interval? The values r=3.45 and r=3.54 are also called bifurcation points since the behaviour of the sequence changes as r crosses over those values.

5.

The situation gets even more interesting for 3.55<r<3.57. As r varies between this range we get more complicated attracting cycles. Choose r=3.5695 and plot 300 points. Describe the behaviour.

6.

Let us see what happens when r>3.57. Choose r=3.65 and calculate and plot the first 300 terms of the sequence. Observe how the population wanders around in an unpredicatable, chaotic fashion. You cannot guess the value of X_n+1 from the value of X_n.

7.

For r=3.65 choose two starting values of X₀ that are close together, say, X₀=0.3 and X₀=0.301. Calculate and plot the first 300 values of the sequences determined by each starting value. Compare the behavior observed in your plots (It is best to plot them in the same graph). How far out do you go before the corresponding terms of your two sequences appear to depart from each other? Repeat the exploration for r=3.75. Notice how different the plots look depending on your choice of initial population X₀. This sensitive dependence to initial condition is one of the most important characteristics of chaotic behavior.