Chap 41 Exercises

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Exercise 1 The following R/mosaic statements will define a dynamical function named $f()$ for the differential equation \[\partial_t x = f(x)\ .\] Your task is to use R/mosaic statements to locate all of the fixed points in the interval $-10 \leq x \leq 10$. For each fixed point find the stability (stable or unstable) and find the parameter in the exponential solution $e^{at}$ near the fixed point. (Hints: Zeros() and D().)

f <- doodle_fun(~ x, seed=385)

Exercise 2 The graphs below show two dynamical functions, $f(x)$ and $g(x)$, in the two differential equations $\partial_t x = f(x)$ and $\partial_t x = g(x)$.

Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
ℹ Please use `linewidth` instead.

For function $f(x)$,
1. locate all of the fixed points in the domain $-10 \leq x \leq 10$ and determine whether each one is stable or not
2. find the state value $x$ where the state is growing the fastest.
3. find the state value $x$ where the state is decreasing the fastest.
Answer the questions in (a) but for $g(x)$.

Exercise 3 The dynamical function in the graph is a sixth-order polynomial. It has several fixed points, labelled “a”, “b”, “c”, and so on.

The numbers i, ii, iii, etc. mark some initial conditions.

For each of the initial conditions marked in the graph, say whether the long-term behavior of the trajectory from that point will lead to one of the fixed points (and which one) or to $-\infty$ or $\infty$. (Hint: That the function is a sixth-order polynomial should tell you something about the functions behavior beyond the domain shown in the graph.)

Exercise 4 Consider the first-order differential equation

\[\partial_t V = g(V)\] with a fixed point at $V^\star$.

For each of the following mathematical objects, say whether the object is a function or a number.

$V$
$g()$
$\partial_t V$
$V^\star$

Exercise 5 Here is a generic, linear differential equation:

\[\partial_t x = a x + b\ .\] 1. How many fixed points are in this system?

For one of the fixed points you identified in (1), what is the value for the location $x^\star$ in terms of the parameters $a$ and $b$?
For one of the fixed points, translate the equation to be in terms of $y$ in \[\partial_t y = ???\ .\]
Write down the symbolic solution to the equation in (3). (Hint: it will involve the parameter $a$ but not $b$.)
Translate the solution in (4) back into terms of $x$. (Hint: The solution will involve both parameters $a$ and $b$.)

Exercise 6 Use differentiation to verify that \[y(t) = 4 e^{2 t} + 7\] is a solution to the differential equation \[\partial_t y = 2 y - 14\ .\]

Exercise 7

For the dynamical system graphed above, sketch a plausible time series solution starting from initial condition $x_0=-2.5$.

Exercise 8

Suppose that $p(t)$ is the population density of squirrels over time. Draw the dynamical function $s(p)$ in the differential equation \[\partial_t p = s(p)\] that will produce a population that grows form a very small density of 4 squirrels per hectare to a carrying capacity of 50 squirrels per hectare. Mark the units on your graph.
Challenge: Arrange the units on the output of $s(p)$ so that it will take roughly 3 months to reach the carrying capacity.

Exercise 9 Coffee at an initial temperature of $T=$ 80 degrees C is in a closed thermos at a (very cold) construction site where the ambient temperature is -20 C. Suppose, with the thermos closed, it takes 4 hours for the coffee to fall to a tepid 30 degrees C. The thermos cools in accord with Newton’s Law of Cooling.

Draw a quantitatively reasonable graph of the dynamical function consistent with these facts in the differential equation \[\partial_t T = f(T)\ .\]

Exercise 10 A first-order dynamical system $\partial_t x = g(x)$ can have any number of fixed points, depending on the shape of $g(x)$.

A. Draw a graph of a $g(x)$ that has seven fixed points.

B. Explain why, when there are multiple fixed points, a stable fixed point will have a neighboring fixed point that is unstable, and vice versa.

Exercise 11 In the cows eating grass model, we proposed the model \[ \partial_t v = \frac{v}{3} \left(1-\frac{v}{25}\right)\] for the growth of grass in the field, where $v$ is in tons of biomass and $t$

A. According to the model, what is the carrying capacity of the field?

B. The field is most productive when the grass is growing at the highest rate possible. At what level of vegetation biomass is the biomass increasing the most rapidly?

C. At that most productive level of biomass, what is the rate of growth of the vegetation? (Make sure to give units.)

Exercise 13 Consider the differential equation \[\partial_t x = \sin(x)^2 + x + 1\ .\]

Where is the fixed point?
Is the fixed point stable?

Exercise 14 The interactive app displayed below implements a version of the pasture-cow dynamics with different parameters than the one in the main body of the text.

The app draws the various functions involved:

The intrinsic logistic model growth dynamics with no cows.
The total consumption by the herd of cows.
The net growth, which is the difference between (1) and (2).

And, you can play the role of farmer by changing the herd size.

Using the app, answer these two questions:

How many cows can you put in the pasture before the addition of even a single additional cow will cause collapse of the ecosystem?
Once a collapse has occurred, how many cows will you need to remove in order for the pasture to show growth toward a healthy equilibrium?

Exercise 15 MC_xref("fig-cow-consumption") shows a model of a cow’s consumption of vegetation as a sigmoid function of the amount of vegetation. In addition to the mathematics of the function, it is helpful to be able to translate the shape of the function into terms that make sense in the context of the model.

At what level of available biomass is the cow most hungry at the end of the day? (Hint: Think about how you would measure how hungry a cow is at the end of the day.)

```{mcq}
#| label: cow1
#| show_hints: true
1. 2 tons [ correct hint: You're right The consumption function is steepest here, meaning that if more grass were available the cow would eat the biggest proportion of it. ]
2. 5 tons [ hint: The level of hunger can be measured by the slope of the consumption curve. If a cow would eat a lot more if the biomass were available, she must be hungry. ]
3. 10 tons [ hint: The level of hunger can be measured by the slope of the consumption curve. If a cow would eat a lot more if the biomass were available, she must be hungry. ]
4. 15 tons [ hint: Even if more grass were available, the cow wouldn't each much of it. So, the cow is not very hungry. ]
```

Exercise 16 A debit card is a way to make withdrawals from a bank account. Funds in the account earn interest. In the economic environment of 2021, the interest rate is practically zero. But to make it possible to see the growth on a graph, we will stipulate it to be 4% per year. But if the account is in debt, then the debt grows at a much faster rate, perhaps 15% per year.

Suppose the amount in the bank account is $m$ (in dollars) and time is $t$ (in years). If $m$ is positive, you are earning interest. If negative, you are in debt and the debt will grow.

Which of the following graphs reflects the dynamics of the system?

On the graph paper below, sketch out the modifications of the bank-card dynamics described. Identify any fixed points by marking them on the graph and say if they are stable or unstable.

) )

In addition to the normal bank-card dynamics, your employer makes a steady retirement deposit into your account at the rate of $100 per year.
Your darling aunt looks at your account balance. If you have less than $300 in your account, she will add money daily at a rate of $100 per year. Otherwise, she does nothing.

Exercise 17 You and your older cousin are on a road trip. She rented a car that has not only “cruise control” (a technology from the 1970s that holds a constant speed without driver intervention) but also “car following,” that keeps the car a set distance $D$ behind the car in front. Your cousin, knowing that you are a MOSAIC Calculus reader, asks you how the “car following” system works.

First, define some terms. You tell your cousin to denote the current, instantaneous distance from the car ahead as $\xi$. You point out that when $\xi - D > 0$, you’re a greater distance than $D$ from the car ahead and therefore the velocity should be increased so you catch up. On the other hand, when you’re too close ($\xi < D$) you should decrease your velocity. The car-following system automates this, the relationship being $\partial_t \xi = b (\xi - D)$.

Your cousin has never encountered a name like $\xi$ so you decide to simplify. “Let $x = \xi - D$, so $x$ is positive if you are too far behind, zero when you’re at the right distance, and negative if you are too far ahead.” Conveniently $\partial_t x = \partial_t \xi$, so you can write down the automatic relationship in terms of $x$: $\partial_t x = b x$ with $b$ a positive number, such as 1.

Where is the fixed point of the dynamics $\partial_t\, x = b x$?

```{mcq}
#| label: following1-1
#| show_hints: true
1. $x=D$ [ hint: No, it should be $\xi=D$, which means that $x=0$. ]
2. $x=0$ [ correct hint: Good job!  ]
3. $\xi = 0$ [ hint: This corresponds to zero distance between you and the car in front of you. Not what we want! ]
4. There is no fixed point. 
```

Is the fixed point of $\partial_t\, x = b x$ (with $b > 0$) stable?

```{mcq}
#| label: following1-2
#| show_hints: true
1. Yes, that is  why we can leave it to the automatic system. [ hint: True, we would want the automatic system to show stable fixed point behavior, but that is  not what's happening here. ]
2. No, the solution grows exponentially as $e^{bt}$. [ correct hint: right-o  ]
3. Yes, the solution decays exponentially as $e^{bt}$ [ hint: But since $b > 0$, this wouldn't be exponential decay but growth. ]
```

Something’s wrong with the system you’ve sketched out. You want a stable following distance, doing the right thing if the car ahead speeds up or slows down, but your system is unstable.

Which of the following rules will have a stable fixed point at $x=0$?

```{mcq}
#| label: following1-3
#| show_hints: true
1. $\partial_t\, x = -b x$ (with $b > 0$) [ correct hint: Good job!  ]
2. $\partial_t\, x = b^2 x$ [ hint: The coefficient on $x$ is still positive, so this system is unstable. ]
3. $\partial_t\, x = b x^2$ [ hint: This system is stable for disturbances that make $x < 0$, but **unstable** for disturbances that make $x > 0$. ]
4. $\partial_t\, x = 0 x$ [ hint: In this system, $x$ does not change. ]
```

The word “feedback” is used to describe systems where the change in the state is a function of the state. “Negative feedback” is when the change is negative when the state is positive, as in $\partial_t\, x = -b\, x$. Negative feedback is desirable when you want to keep things stable. “Positive feedback” is when the change is the same sign as the state, as in $\partial_t\, x = b\, x$ and makes things unstable.

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