Problem Set: The Logistic Equation

For the following problems, consider the logistic equation in the form $P\prime =CP-{P}^{2}$ . Draw the directional field and find the stability of the equilibria.

C = 3

$C=3$

2. $C=0$

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A direction field with arrows pointing to the right. The arrows are horizontal along the x axis. The arrows point down above the x-axis and below the x axis. The further away the arrows are from the x axis, the more vertical the lines become.

P = 0

$P=0$ semi-stable

C = - 3

$C=-3$

4. Solve the logistic equation for $C=10$ and an initial condition of $P\left(0\right)=2$ .

Show Solution

P = \frac{10 e^{10 x}}{e^{10 x} + 4}

$P=\frac{10{e}^{10x}}{{e}^{10x}+4}$

5. Solve the logistic equation for

C = - 10

$C=-10$ and an initial condition of

P (0) = 2

$P\left(0\right)=2$ .

6. A population of deer inside a park has a carrying capacity of $200$ and a growth rate of $2\text{%}$ . If the initial population is $50$ deer, what is the population of deer at any given time?

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P (t) = \frac{10000 e^{0.02 t}}{150 + 50 e^{0.02 t}}

$P\left(t\right)=\frac{10000{e}^{0.02t}}{150+50{e}^{0.02t}}$

7. A population of frogs in a pond has a growth rate of

5 %

$5\text{%}$ . If the initial population is

1000

$1000$ frogs and the carrying capacity is

6000

$6000$ , what is the population of frogs at any given time?

8. [T] Bacteria grow at a rate of $20\text{%}$ per hour in a petri dish. If there is initially one bacterium and a carrying capacity of $1$ million cells, how long does it take to reach $500,000$ cells?

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69

$69$ hours

5

$5$ minutes

9. [T] Rabbits in a park have an initial population of

10

$10$ and grow at a rate of

4 %

$4\text{%}$ per year. If the carrying capacity is

500

$500$ , at what time does the population reach

100

$100$ rabbits?

10. [T] Two monkeys are placed on an island. After $5$ years, there are $8$ monkeys, and the estimated carrying capacity is $25$ monkeys. When does the population of monkeys reach $16$ monkeys?

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8

$8$ years

11

$11$ months

11. [T] A butterfly sanctuary is built that can hold

2000

$2000$ butterflies, and

400

$400$ butterflies are initially moved in. If after

2

$2$ months there are now

800

$800$ butterflies, when does the population get to

1500

$1500$ butterflies?

The following problems consider the logistic equation with an added term for depletion, either through death or emigration.

12. [T] The population of trout in a pond is given by $P\prime =0.4P\left(1-\frac{P}{10000}\right)-400$ , where $400$ trout are caught per year. Use your calculator or computer software to draw a directional field and draw a few sample solutions. What do you expect for the behavior?

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13. In the preceding problem, what are the stabilities of the equilibria

0 < P_{1} < P_{2} ?

$0<{P}_{1}<{P}_{2}?$

14. [T] For the preceding problem, use software to generate a directional field for the value $f=400$ . What are the stabilities of the equilibria?

Show Solution

A direction field with arrows pointing down and to the right. Around y = 4,000, the arrows are more horizontal. The further the arrows are from this line, the more vertical the arrows become.

P_{1}

${P}_{1}$ semi-stable

15. [T] For the preceding problems, use software to generate a directional field for the value

f = 600

$f=600$ . What are the stabilities of the equilibria?

16. [T] For the preceding problems, consider the case where a certain number of fish are added to the pond, or $f=-200$ . What are the nonnegative equilibria and their stabilities?

Show Solution

A direction field with arrows pointing up for P < 10,000 and arrows pointing down for P > 10,000.

P_{2} > 0

${P}_{2}>0$ stable

It is more likely that the amount of fishing is governed by the current number of fish present, so instead of a constant number of fish being caught, the rate is proportional to the current number of fish present, with proportionality constant $k$ , as

$P\prime =0.4P\left(1-\frac{P}{10000}\right)-kP$ .

17. [T] For the previous fishing problem, draw a directional field assuming

k = 0.1

$k=0.1$ . Draw some solutions that exhibit this behavior. What are the equilibria and what are their stabilities?

18. [T] Use software or a calculator to draw directional fields for $k=0.4$ . What are the nonnegative equilibria and their stabilities?

Show Solution

A direction field with arrows pointing to the right at P = 0. Below 0, the arrows point down and to the right. Above 0, the arrows point down and to the right. The further away from 0, the more vertical the arrows become.

P_{1} = 0

${P}_{1}=0$ is semi-stable

19. [T] Use software or a calculator to draw directional fields for

k = 0.6

$k=0.6$ . What are the equilibria and their stabilities?

20. Solve this equation, assuming a value of $k=0.05$ and an initial condition of $2000$ fish.

Show Solution

y = \frac{- 20}{4 \times 10^{- 6} - 0.002 e^{0.01 t}}

$y=\frac{-20}{4\times {10}^{-6}-0.002{e}^{0.01t}}$

21. Solve this equation, assuming a value of

k = 0.05

$k=0.05$ and an initial condition of

5000

$5000$ fish.

The following problems add in a minimal threshold value for the species to survive, $T$ , which changes the differential equation to $P\prime \left(t\right)=rP\left(1-\frac{P}{K}\right)\left(1-\frac{T}{P}\right)$ .

22. Draw the directional field of the threshold logistic equation, assuming $K=10,r=0.1,T=2$ . When does the population survive? When does it go extinct?

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23. For the preceding problem, solve the logistic threshold equation, assuming the initial condition

P (0) = P_{0}

$P\left(0\right)={P}_{0}$ .

24. Bengal tigers in a conservation park have a carrying capacity of $100$ and need a minimum of $10$ to survive. If they grow in population at a rate of $1\text{%}$ per year, with an initial population of $15$ tigers, solve for the number of tigers present.

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P (t) = \frac{850 + 500 e^{0.009 t}}{85 + 5 e^{0.009 t}}

$P\left(t\right)=\frac{850+500{e}^{0.009t}}{85+5{e}^{0.009t}}$

25. A forest containing ring-tailed lemurs in Madagascar has the potential to support

5000

$5000$ individuals, and the lemur population grows at a rate of

5 %

$5\text{%}$ per year. A minimum of

500

$500$ individuals is needed for the lemurs to survive. Given an initial population of

600

$600$ lemurs, solve for the population of lemurs.

26. The population of mountain lions in Northern Arizona has an estimated carrying capacity of $250$ and grows at a rate of $0.25\text{%}$ per year and there must be $25$ for the population to survive. With an initial population of $30$ mountain lions, how many years will it take to get the mountain lions off the endangered species list (at least $100$ ?)

Show Solution

13

$13$ years months

The following questions consider the Gompertz equation, a modification for logistic growth, which is often used for modeling cancer growth, specifically the number of tumor cells.

27. The Gompertz equation is given by

P (t)' = α ln (\frac{K}{P (t)}) P (t)

$P\left(t\right)\prime =\alpha \text{ln}\left(\frac{K}{P\left(t\right)}\right)P\left(t\right)$ . Draw the directional fields for this equation assuming all parameters are positive, and given that

K = 1

$K=1$ .

28. Assume that for a population, $K=1000$ and $\alpha =0.05$ . Draw the directional field associated with this differential equation and draw a few solutions. What is the behavior of the population?

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29. Solve the Gompertz equation for generic

α

$\alpha$ and

K

$K$ and

P (0) = P_{0}

$P\left(0\right)={P}_{0}$ .

30. [T] The Gompertz equation has been used to model tumor growth in the human body. Starting from one tumor cell on day $1$ and assuming $\alpha =0.1$ and a carrying capacity of $10$ million cells, how long does it take to reach “detection” stage at $5$ million cells?

Show Solution

31.465

$31.465$ days

31. [T] It is estimated that the world human population reached

3

$3$ billion people in

1959

$1959$ and

6

$6$ billion in

1999

$1999$ . Assuming a carrying capacity of

16

$16$ billion humans, write and solve the differential equation for logistic growth, and determine what year the population reached

7

$7$ billion.

32. [T] It is estimated that the world human population reached $3$ billion people in $1959$ and $6$ billion in $1999$ . Assuming a carrying capacity of $16$ billion humans, write and solve the differential equation for Gompertz growth, and determine what year the population reached $7$ billion. Was logistic growth or Gompertz growth more accurate, considering world population reached $7$ billion on October $31,2011?$

Show Solution

September

2008

$2008$

33. Show that the population grows fastest when it reaches half the carrying capacity for the logistic equation

P' = r P (1 - \frac{P}{K})

$P\prime =rP\left(1-\frac{P}{K}\right)$ .

34. When does population increase the fastest in the threshold logistic equation $P\prime \left(t\right)=rP\left(1-\frac{P}{K}\right)\left(1-\frac{T}{P}\right)?$

Show Solution

\frac{K + T}{2}

$\frac{K+T}{2}$

35. When does population increase the fastest for the Gompertz equation

P (t)' = α ln (\frac{K}{P (t)}) P (t) ?

$P\left(t\right)\prime =\alpha \text{ln}\left(\frac{K}{P\left(t\right)}\right)P\left(t\right)?$

Below is a table of the populations of whooping cranes in the wild from $1940\text{ to }2000$ . The population rebounded from near extinction after conservation efforts began. The following problems consider applying population models to fit the data. Assume a carrying capacity of $10,000$ cranes. Fit the data assuming years since $1940$ (so your initial population at time $0$ would be $22$ cranes).

*Source:* https://www.savingcranes.org/images/stories/site_images/conservation/whooping_crane/pdfs/historic_wc_numbers.pdf
Year (years since conservation began)	Whooping Crane Population
$1940\left(0\right)$	$22$
$1950\left(10\right)$	$31$
$1960\left(20\right)$	$36$
$1970\left(30\right)$	$57$
$1980\left(40\right)$	$91$
$1990\left(50\right)$	$159$
$2000\left(60\right)$	$256$

36. Find the equation and parameter $r$ that best fit the data for the logistic equation.

Show Solution

r = 0.0405

$r=0.0405$

37. Find the equation and parameters

r

$r$ and

T

$T$ that best fit the data for the threshold logistic equation.

38. Find the equation and parameter $\alpha$ that best fit the data for the Gompertz equation.

Show Solution

α = 0.0081

$\alpha =0.0081$

39. Graph all three solutions and the data on the same graph. Which model appears to be most accurate?

40. Using the three equations found in the previous problems, estimate the population in $2010$ (year $70$ after conservation). The real population measured at that time was $437$ . Which model is most accurate?

Show Solution

Logistic:

361

$361$ , Threshold:

436

$436$ , Gompertz:

309

$309$ .

Differential Equations: Supplemental Content

Problem Set: The Logistic Equation

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