4.4 The logistic equation  (Page 5/12)

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Key concepts

• When studying population functions, different assumptions—such as exponential growth, logistic growth, or threshold population—lead to different rates of growth.
• The logistic differential equation incorporates the concept of a carrying capacity. This value is a limiting value on the population for any given environment.
• The logistic differential equation can be solved for any positive growth rate, initial population, and carrying capacity.

Key equations

• Logistic differential equation and initial-value problem
$\frac{dP}{dt}=rP\left(1-\frac{P}{K}\right),\phantom{\rule{1em}{0ex}}P\left(0\right)={P}_{0}$
• Solution to the logistic differential equation/initial-value problem
$P\left(t\right)=\frac{{P}_{0}K{e}^{rt}}{\left(K-{P}_{0}\right)+{P}_{0}{e}^{rt}}$
• Threshold population model
$\frac{dP}{dt}=\text{−}rP\left(1-\frac{P}{K}\right)\left(1-\frac{P}{T}\right)$

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.

$C=3$

$C=0$

$P=0$ semi-stable

$C=-3$

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

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

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

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?

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

A population of frogs in a pond has a growth rate of $5\text{%}.$ If the initial population is $1000$ frogs and the carrying capacity is $6000,$ what is the population of frogs at any given time?

[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?

$69$ hours $5$ minutes

[T] Rabbits in a park have an initial population of $10$ and grow at a rate of $4\text{%}$ per year. If the carrying capacity is $500,$ at what time does the population reach $100$ rabbits?

[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?

$7$ years $2$ months

[T] A butterfly sanctuary is built that can hold $2000$ butterflies, and $400$ butterflies are initially moved in. If after $2$ months there are now $800$ butterflies, when does the population get to $1500$ butterflies?

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

[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?

In the preceding problem, what are the stabilities of the equilibria $0<{P}_{1}<{P}_{2}?$

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

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

[T] For the preceding problems, use software to generate a directional field for the value $f=600.$ What are the stabilities of the equilibria?

[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?

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

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