4.4 The logistic equation  (Page 5/12)

$C=3$

$C=0$

$C=\mathrm{-3}$

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

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

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?

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 $\mathrm{500,000}$ cells?

[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?

[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?

[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?

[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 $f=\mathrm{-200}.$ What are the nonnegative equilibria and their stabilities?