- What is a difference equation?
- Stair-step diagrams.
- Geometry of linear maps
- Stability tpyes of fixed points.
- Linearization theorem.
- Application: Why Newton's method converges
fast for good starting values?
- Shifting around:
Describe the fixed points and their stability types of the linear map
x1 -> a*x1 + b
for all values of the parameters a, b and initial conditions x0.
- As a parameter is varied:
Consider the map x1 -> a + x1 + 2.0*x1*x1 where a is a parameter.
Find three values of the parameter a for which the map has 0, 1, or 2
fixed points. Determine the stability types of these fixed points.
You should enter this map into Phaser as a Custom equation. Then using the
stair-step diagrams try to find the requested parameter values.
- Investing for Med School:
Suppose you want to make an investment that will pay for your child's
Medical School education. You figure you will need $72,000 when your child
starts Med School, 14 years from now. You can buy a long term certificate
of deposite, or CD, that pays a 6.4 per cent annual interest
compounded mothly. What size CD should you buy?
Beverton-Holt Stock-recruitment model:
xn+1 = rxn/[1 + ((r - 1)/k)xn]
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.
To understand the geometric meanings of the parameters,
fix k (say at 1.25) and vary r. Make a Gallery of growth curves using
the stair-step view of PHASER (study the Phaser Tutorials Lesson 6
and Lesson 13 to learn about the Gallery and making SlideShow).
You may want to take big window size
(-1 , 15; -1, 15) to see what happens as the population gets large.
Next, fix r = 1.35 and vary k. Describe biologically what you observe in these
Find the fixed points of the model as a function of the parameters.
For what ranges of the parameters the fixed points are biologically significant?
Determine the stability type of the fixed points, using the
Linearization Theorem. Can the positive fixed point
become unstable as the parameter r or k is increased?
- Write a summary of the possible dynamics of a population descibed by
the Beverton-Holt model and interpret your findings in biological terms.