



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 initialvalue 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/initialvalue problem
$P\left(t\right)=\frac{{P}_{0}K{e}^{rt}}{\left(K{P}_{0}\right)+{P}_{0}{e}^{rt}}$

Threshold population model
$\frac{dP}{dt}=\text{\u2212}rP\left(1\frac{P}{K}\right)\left(1\frac{P}{T}\right)$
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.
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?
$P\left(t\right)=\frac{10000{e}^{0.02t}}{150+50{e}^{0.02t}}$
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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?
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[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
$\mathrm{500,000}$ cells?
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[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?
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[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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[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?
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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?
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[T] For the preceding problem, use software to generate a directional field for the value
$f=400.$ What are the stabilities of the equilibria?
${P}_{1}$ semistable
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[T] For the preceding problems, consider the case where a certain number of fish are added to the pond, or
$f=\mathrm{200}.$ What are the nonnegative equilibria and their stabilities?
${P}_{2}>0$ stable
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Questions & Answers
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Graphene has a hexagonal structure
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what's the easiest and fastest way to the synthesize AgNP?
I start with an easy one. carbon nanotubes woven into a long filament like a string
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AMJAD
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At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
how did you get the value of 2000N.What calculations are needed to arrive at it
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Leaves accumulate on the forest floor at a rate of 2 g/cm2/yr and also decompose at a rate of 90% per year. Write a differential equation governing the number of grams of leaf litter per square centimeter of forest floor, assuming at time 0 there is no leaf litter on the ground. Does this amount approach a steady value? What is that value?
You have a cup of coffee at temperature 70°C, which you let cool 10 minutes before you pour in the same amount of milk at 1°C as in the preceding problem. How does the temperature compare to the previous cup after 10 minutes?
Abdul
Source:
OpenStax, Calculus volume 2. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11965/1.2
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