- 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?
- When the Linearization does not suffice:
Using PHASER, determine if the origin is an unstable, stable,
or asymptotically stable fixed point of the following maps.
Make sure to include one screen image of stair-step diagram
for each map. Show that the origin is a fixed point and the derivative of each equation
at the origin is either 1 or -1. Now explain why the Linearization theorem
has nothing to offer when the derivative at a fixed point is 1 or -1.
Hint: Enter the MAP x1 -> a*x1 + b*x1*x1*x1 into PHASER. Now set
the parameter values appropriately to study any one of the maps above.
- x1 -> x1 - 1.15*x1*x1*x1
- x1 -> x1 + 1.15*x1*x1*x1
- x1 -> -x1 - 1.15*x1*x1*x1
- x1 -> -x1 + 1.15*x1*x1*x1
- As a parameter is varied:
Consider the map x1 -> a + x1 + 1.2*x1*x1 where a is a parameter.
Draw at least three stair step diagrams, say, for a = -0.16,
a = 0, and a = 0.16. Describe the changes in the number and stability
types of fixed points as the parameter a is changed pass 0.
How much can you borrow?:
Suppose you borrow x0 dollars from a bank at 2 per
cent per month interest,
and that you pay your bank 40 dollars per month.
Derive the difference equation that models your debt to the bank at the end of each month.
(Hint: the equation you need is a linear equation. So
use the Linear 1D in the Map library of PHASER
and set the parameters appropriately.)
If your original debt is 560 dollars, how long will it take you to pay the loan?
If your original debt is 3700 dollars, how long will it take you to pay the loan?
What is the maximum amount you can barrow?
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.