Cobweb Diagrams

The main purpose of this section is for you to discover and prove a theorem about discrete linear dynamcial systems -- that is, systems of the form

p_n = m p_n-1 + b

where m and b are constants.

In general, the change in a discrete dynamical system is described by an equation of the form

p_n = f(p_n-1)

In a linear discrete dynamical system the function on the right side of this equation is linear.

f(p) = m p + b

We are interested in equilibrium points of discrete dynamical systems -- that is, points that satisfy the equation

f(p) = p

These points are called equilibrium points because if one term is at an equilibrium point then every subsequent term stays at the same point. In this situation we say that the system is in equilibrium.

The theorem we are looking for is important for its own sake and and for its practical implications. It is also a nice theorem to find and prove because it is accessible with only high school algebra and it can be looked at both geometrically and algebraically.

We begin with another theorem.

Theorem:
If m is not 1 then there is a unique equilibrium point
Missing equation
This can be seen graphically from the graph at the right. The black line is the diagonal line g(p) = p and the red line is the function f(p) = m p + b. Recall that an equilibrium point is a point at which these two lines intersect. If m is not 1 then these two lines are not parallel and they intersect at one point. This point is marked in blue on the graph.

Proof:

We can prove this theorem algebraically by solving the equation

p = f(p)

as shown below.

In an earlier section we discussed supply and demand models for the fluctuation of prices. In one of those models we worked with the supply and demand functions

S(p) = 1000 p - 400

D(p) = 1000 - 500 p

and the dynamical system

p_n = p_n-1 + k (D(p_n-1) - S(p_n-1))

where the positive constant k depends on the kind of marketplace we are studying. In a volatile marketplace where prices respond quickly to an imbalance in supply and demand, k is relatively large, but in a more conservative marketplace where people are slow to change, k is relatively small.

Verify that the dynamical system above is a linear dynamical system. Compute its equilibrium point using the theorem above and verify that it is the price at which supply and demand are equal.
Describe the behavior of this model for various different initial prices when k=.0001.
Describe the behavior of this model for various different initial prices when k=.0012.
Describe the behavior of this model for various different initial prices when k=.0014.
Summarize the behavior of this dynamical system for different values of k.
answer

As you saw above, linear discrete dynamical systems can behave in some surprizing ways. Sometimes a sequence obtained from a linear discrete dynamical system approaches its equilibrium point directly; sometimes it bounces around the equilibrium point and the bounces get smaller so that the sequence approaches the equilibrium point; sometimes it bounces around the equilibrium point but the bounces get larger so that the sequence does not approach the equilibrium point. Your goal in this section is to formulate and prove a theorem that determines when each of these three kinds of behavior occurs. We begin with a useful graphical tool.

Cobweb diagrams

In this section we construct the cobweb diagram for a dynamical system of the form

p_n = f(p_{n - 1}).

In our first example the function f(p) will be a linear function

f(p) = m p + b

so that

p_n = m p_{n - 1} + b

We begin our construction of such a cobweb diagram with the graph shown at the right.

We mark the initial point p₁ on the "x-axis" as shown in blue.

Then we draw a blue vertical line at this point going up to the red graph of the function f(p). The blue line hits the red graph at the point
(p₁, p₂)
since
p₂ = f(p₁).

Next we draw a horizontal line from the point
(p₁, p₂)
to the black diagonal line at the point
(p₂, p₂)

Then we draw a vertical line to the red graph of the function f(p).

This line hits the graph at the point
(p₂, p₃)
since
p₃ = f(p₂).

We continue in the same way, repeating the same two steps over and over.

Drawing a line horizontally from the red graph of the function f(p) to the black diagonal line.

Then drawing a vertical line from the black diagonal line to the red graph of the function f(p).

Notice that in this case the "spider web" stair steps up to the equilibrium point.

Draw a cobweb diagram for the following linear discrete dynamical system
p_n = 0.80 p_n-1 + 100, p₁ = 100.
Describe the long term behavior of this model.
Draw a cobweb diagram for the following linear discrete dynamical system
p_n = -0.80 p_n-1 + 100, p₁ = 100.
Describe the long term behavior of this model.
Draw a cobweb diagram for the following linear discrete dynamical system
p_n = 1.5 p_n-1 + 100, p₁ = 100.
Describe the long term behavior of this model.
Draw a cobweb diagram for the following linear discrete dynamical system
p_n = -1.5 p_n-1 + 100, p₁ = 100.
Describe the long term behavior of this model.
Formulate and prove a theorem that enables you to predict the long term behavior of the dynamical system
p_n = m p_n-1 + b, p₁ = a.
based on the values of the constants m, b, and a.
We can draw cobweb diagrams for other discrete dynamical systems. The six pictures below each show the graph of a function f(p) in red and the usual diagonal line in black. We are interested in the models
p_n = f(p_n-1)
with various different initial conditions. Print these graphs and use them to draw cobweb diagrams with several different initial conditions. Describe the long term behavior of these models on the basis of the cobweb diagrams that you drew.

[Next section -- Logistic Models and Cobweb Diagrams]

We begin our construction of such a cobweb diagram with the graph shown at the right. We mark the initial point p₁ on the "x-axis" as shown in blue. Then we draw a blue vertical line at this point going up to the red graph of the function f(p). The blue line hits the red graph at the point (p₁, p₂) since p₂ = f(p₁).
Next we draw a horizontal line from the point (p₁, p₂) to the black diagonal line at the point (p₂, p₂) Then we draw a vertical line to the red graph of the function f(p). This line hits the graph at the point (p₂, p₃) since p₃ = f(p₂).
We continue in the same way, repeating the same two steps over and over. Drawing a line horizontally from the red graph of the function f(p) to the black diagonal line. Then drawing a vertical line from the black diagonal line to the red graph of the function f(p). Notice that in this case the "spider web" stair steps up to the equilibrium point.