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

**1.**[latex]C=3[/latex]

**2. **[latex]C=0[/latex]

**3.**[latex]C=-3[/latex]

**4. **Solve the logistic equation for [latex]C=10[/latex] and an initial condition of [latex]P\left(0\right)=2[/latex].

**5.**Solve the logistic equation for [latex]C=-10[/latex] and an initial condition of [latex]P\left(0\right)=2[/latex].

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

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

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

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

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

**11. [T]**A butterfly sanctuary is built that can hold [latex]2000[/latex] butterflies, and [latex]400[/latex] butterflies are initially moved in. If after [latex]2[/latex] months there are now [latex]800[/latex] butterflies, when does the population get to [latex]1500[/latex] 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 [latex]P\prime =0.4P\left(1-\frac{P}{10000}\right)-400[/latex], where [latex]400[/latex] 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?

**13.**In the preceding problem, what are the stabilities of the equilibria [latex]0<{P}_{1}<{P}_{2}?[/latex]

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

**15. [T]**For the preceding problems, use software to generate a directional field for the value [latex]f=600[/latex]. 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 [latex]f=-200[/latex]. What are the nonnegative equilibria and their stabilities?

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 [latex]k[/latex], as

[latex]P\prime =0.4P\left(1-\frac{P}{10000}\right)-kP[/latex].

**17. [T]**For the previous fishing problem, draw a directional field assuming [latex]k=0.1[/latex]. 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 [latex]k=0.4[/latex]. What are the nonnegative equilibria and their stabilities?

**19. [T]**Use software or a calculator to draw directional fields for [latex]k=0.6[/latex]. What are the equilibria and their stabilities?

**20. **Solve this equation, assuming a value of [latex]k=0.05[/latex] and an initial condition of [latex]2000[/latex] fish.

**21.**Solve this equation, assuming a value of [latex]k=0.05[/latex] and an initial condition of [latex]5000[/latex] fish.

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

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

**23.**For the preceding problem, solve the logistic threshold equation, assuming the initial condition [latex]P\left(0\right)={P}_{0}[/latex].

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

**25.**A forest containing ring-tailed lemurs in Madagascar has the potential to support [latex]5000[/latex] individuals, and the lemur population grows at a rate of [latex]5\text{%}[/latex] per year. A minimum of [latex]500[/latex] individuals is needed for the lemurs to survive. Given an initial population of [latex]600[/latex] lemurs, solve for the population of lemurs.

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

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 [latex]P\left(t\right)\prime =\alpha \text{ln}\left(\frac{K}{P\left(t\right)}\right)P\left(t\right)[/latex]. Draw the directional fields for this equation assuming all parameters are positive, and given that [latex]K=1[/latex].

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

**29.**Solve the Gompertz equation for generic [latex]\alpha [/latex] and [latex]K[/latex] and [latex]P\left(0\right)={P}_{0}[/latex].

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

**31. [T]**It is estimated that the world human population reached [latex]3[/latex] billion people in [latex]1959[/latex] and [latex]6[/latex] billion in [latex]1999[/latex]. Assuming a carrying capacity of [latex]16[/latex] billion humans, write and solve the differential equation for logistic growth, and determine what year the population reached [latex]7[/latex] billion.

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

**33.**Show that the population grows fastest when it reaches half the carrying capacity for the logistic equation [latex]P\prime =rP\left(1-\frac{P}{K}\right)[/latex].

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

**35.**When does population increase the fastest for the Gompertz equation [latex]P\left(t\right)\prime =\alpha \text{ln}\left(\frac{K}{P\left(t\right)}\right)P\left(t\right)?[/latex]

Below is a table of the populations of whooping cranes in the wild from [latex]1940\text{ to }2000[/latex]. 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 [latex]10,000[/latex] cranes. Fit the data assuming years since [latex]1940[/latex] (so your initial population at time [latex]0[/latex] would be [latex]22[/latex] cranes).

Year (years since conservation began) | Whooping Crane Population |
---|---|

[latex]1940\left(0\right)[/latex] | [latex]22[/latex] |

[latex]1950\left(10\right)[/latex] | [latex]31[/latex] |

[latex]1960\left(20\right)[/latex] | [latex]36[/latex] |

[latex]1970\left(30\right)[/latex] | [latex]57[/latex] |

[latex]1980\left(40\right)[/latex] | [latex]91[/latex] |

[latex]1990\left(50\right)[/latex] | [latex]159[/latex] |

[latex]2000\left(60\right)[/latex] | [latex]256[/latex] |

**36. **Find the equation and parameter [latex]r[/latex] that best fit the data for the logistic equation.

**37.**Find the equation and parameters [latex]r[/latex] and [latex]T[/latex] that best fit the data for the threshold logistic equation.

**38. **Find the equation and parameter [latex]\alpha [/latex] that best fit the data for the Gompertz equation.

**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 [latex]2010[/latex] (year [latex]70[/latex] after conservation). The real population measured at that time was [latex]437[/latex]. Which model is most accurate?