# Math Insight

### The idea of stability of equilibria for discrete systems

Math 1241, Fall 2020
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Due date: Oct. 2, 2020, 11:59 p.m.
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Total points: 3
1. The below graph shows the function $f$ for the dynamical system $m_{t+1} = f(m_t)$.

What are the equilibria of the dynamical system? (They are integers; enter them in increasing order.)
$E=$

For each equilibrium, cobweb (by hand or with optional applet) starting with initial conditions just above and below the equilibrium to determine its stability. Specify the stability of each equilibrium in the same order as above. Enter stable if an equilibrium is stable or unstable if it is unstable. Separate answers by commas.

Stability of equilibria:

Feedback from applet
Points on diagonal:
Points on function:

For example, if there are three equilibria and they are, in order, stable, unstable, and unstable, then you should enter stable, unstable, unstable in the answer blank.

2. The below graph shows the function $g$ for the dynamical system $q_{t+1} = g(q_t)$.

What are the equilibria of the dynamical system? (They are integers; enter in increasing order.)
$E =$

For each equilibrium, cobweb starting with initial conditions just above and below the equilibrium to determine if the equilibrium is stable or unstable. Specify the stability of each equilibrium in the same order as above. Enter stable if an equilibrium is stable or unstable if it is unstable. Separate answers by commas.

Stability of equilibria:

Feedback from applet
Points on diagonal:
Points on function:

3. The below graph shows the function $f$ for the dynamical system $r_{n+1} = f(r_n)$.

What are the equilibria of the dynamical system? (They are integers; enter in increasing order.)
$E =$

For each equilibrium, cobweb starting with initial conditions just above and below the equilibrium to determine if the equilibrium is stable or unstable. Specify the stability of each equilibrium in the same order as above. Enter stable if an equilibrium is stable or unstable if it is unstable. Separate answers by commas.

Stability of equilibria:

Feedback from applet
Points on diagonal:
Points on function:

4. The below graph shows the function $g$ for the dynamical system $h_{n+1} = g(h_n)$.

What are the equilibria of the dynamical system? (They are integers; enter in increasing order.)
$E =$

For each equilibrium, cobweb starting with initial conditions just above and below the equilibrium to determine if the equilibrium is stable or unstable. Specify the stability of each equilibrium in the same order as above. Enter stable if an equilibrium is stable or unstable if it is unstable. Separate answers by commas.

Stability of equilibria:

5. The below graph shows the function $f$ and the diagonal for the dynamical system \begin{align*} p_{n+1} &= f(p_n)\\ p_0 &= p_0. \end{align*}
1. What are the equilibria of the dynamical system? (They are integers; enter in increasing order.)
$E=$
2. For each equilibrium, cobweb starting with initial conditions just above and below the equilibrium to determine if the equilibrium is stable or unstable. Specify the stability of each equilibrium in the same order as above. Enter stable if an equilibrium is stable or unstable if it is unstable. Separate answers by commas.

Stability of equilibria:

3. The basin of attraction of a stable equilibrium is the set of initial conditions $p_0$ for which the solution $p_n$ tends to the equilibrium for large time $n$. For this example, there is one stable equilibrium, it is $E =$
. For that stable equilibrium, determine its basin of attraction.

The basin of attraction is all values $p_0$ satisfying

$< p_0 <$

6. The population of fish $f_t$ in a lake in year $t$ evolve according to the dynamical system \begin{align*} f_{t+1}=g(f_{t}), \quad \text{for $t=0,1,2,3, \ldots$} \end{align*} where $g$ is graphed by the thick curve. The thin line is the diagonal $f_{t+1}=f_t$.
1. Identify all equilibria of the system. Estimate their values in increasing order.
$E =$

Getting to nearest 100 is fine. They are all close to multiples of 50.

2. Determine the stability of the equilibria. (On an exam, you'd want to be sure to justify your answer by explaining the method you used.) Only values $f_t \ge 0$ make sense, so for one equilibrium, you need to check just on one side. Enter stability in same order as the equilibria.

Stability of equilibria:

3. (Basins of attraction) For what values of the initial population size $f_0$ will the fish population disappear? This tricky. Be sure to check what happens for very large $f_0$.
The population will disappear if $f_0 <$
or if $f_0 >$

(Getting to nearest 100 is fine.)

For what values of $f_0$ will the population stay at a relatively large number (we'll call it a healthy population.)
We will have a healthy population of fish if
$< f_0 <$

When we have this healthy population, approximately how many fish are there?