Quantcast

Got Homework?

Connect with other students for help. It's a free community.

  • across
    MIT Grad Student
    Online now
  • laura*
    Helped 1,000 students
    Online now
  • Hero
    College Math Guru
    Online now

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

MathSofiya Group Title

Suppose that a population grows according to a logistic model with carrying capacity 6000 and k=0.0015 per year. Write a logistic differential equation for these data.

  • one year ago
  • one year ago

  • This Question is Closed
  1. TuringTest Group Title
    Best Response
    You've already chosen the best response.
    Medals 1

    this terminology is the only thing confusing me here...

    • one year ago
  2. TuringTest Group Title
    Best Response
    You've already chosen the best response.
    Medals 1

    I guess I don't know what a "logistic" DE is

    • one year ago
  3. experimentX Group Title
    Best Response
    You've already chosen the best response.
    Medals 1

    |dw:1343418531683:dw|

    • one year ago
  4. TuringTest Group Title
    Best Response
    You've already chosen the best response.
    Medals 1

    ok... that makes sense

    • one year ago
  5. experimentX Group Title
    Best Response
    You've already chosen the best response.
    Medals 1

    something like population growth or bacteria growth ... etc etc that depends on initial population

    • one year ago
  6. MathSofiya Group Title
    Best Response
    You've already chosen the best response.
    Medals 1

    \[y=6000e^{0.0015t}\]

    • one year ago
  7. MathSofiya Group Title
    Best Response
    You've already chosen the best response.
    Medals 1

    ?

    • one year ago
  8. experimentX Group Title
    Best Response
    You've already chosen the best response.
    Medals 1

    |dw:1343508816527:dw|

    • one year ago
  9. experimentX Group Title
    Best Response
    You've already chosen the best response.
    Medals 1

    these kids of equation, differentiate once .... remove these terms|dw:1343509003699:dw|

    • one year ago
  10. dpaInc Group Title
    Best Response
    You've already chosen the best response.
    Medals 0

    are those logistic functions? the 6000 is not your initial population, it's your carrying capacity.... the differential equation is: \(\large \frac{dy}{dt}=ky(6000-y) \)

    • one year ago
  11. MathSofiya Group Title
    Best Response
    You've already chosen the best response.
    Medals 1

    What's the thought process...how could I come up with this on my own

    • one year ago
  12. lgbasallote Group Title
    Best Response
    You've already chosen the best response.
    Medals 0

    ...so what's this difference from the usual differential equation for exponential growth?

    • one year ago
  13. MathSofiya Group Title
    Best Response
    You've already chosen the best response.
    Medals 1

    so we start with \[y=y_0e^kt\] and we take the derivative of this to find a rate of change?

    • one year ago
  14. TuringTest Group Title
    Best Response
    You've already chosen the best response.
    Medals 1

    no, start with the idea that the current rate of change is proportional to the current population\[\frac{dy}{dt}\propto y=ky\]separate the variables\[\frac{dy}y=kdt\]integrate both sides:\[\ln y=kt+C\]\[y(t)=Ce^{kt}\]by plugging in t=0 we find that C the original population\[y(0)=Ce^0=C\implies C=y_0\]so we get\[y=y_0e^{kt}\]

    • one year ago
  15. TuringTest Group Title
    Best Response
    You've already chosen the best response.
    Medals 1

    I just don't know all the terminology above lol, but I'm familiar with the problems

    • one year ago
  16. MathSofiya Group Title
    Best Response
    You've already chosen the best response.
    Medals 1

    how do we go from \[\ln y=kt+C\] to \[y(t)=Ce^{kt}\] I presume it was doing e^.... on both sides but why is the C multiplied to e?

    • one year ago
  17. TuringTest Group Title
    Best Response
    You've already chosen the best response.
    Medals 1

    \[e^{kt+C}=e^{kt}\cdot e^C\]and \(e^C\) is just another constant, so I just called it C again. I maybe should have numbered them for the sake of distinction, but that gets tedious in higher math\[e^{kt+C_1}=C_2e^{kt}\]

    • one year ago
  18. MathSofiya Group Title
    Best Response
    You've already chosen the best response.
    Medals 1

    Oh I see. Now that I have this equation \[y(t)=Ce^{kt}\] why am I taking the derivative again?

    • one year ago
  19. MathSofiya Group Title
    Best Response
    You've already chosen the best response.
    Medals 1

    or do I just plug in numbers at this point?

    • one year ago
  20. TuringTest Group Title
    Best Response
    You've already chosen the best response.
    Medals 1

    just plug it in usually, but carrying capacity... not sure of that is at time t=0 sounds like a maximum to me again the terminology is messing me up

    • one year ago
  21. MathSofiya Group Title
    Best Response
    You've already chosen the best response.
    Medals 1

    @dpalnc statement looks legit...(don't know why) \[\large \frac{dy}{dt}=ky(6000-y)\]

    • one year ago
  22. MathSofiya Group Title
    Best Response
    You've already chosen the best response.
    Medals 1

    check this out: \[\frac{dP}{dt}=kP(1-\frac{P}{K})\]

    • one year ago
  23. MathSofiya Group Title
    Best Response
    You've already chosen the best response.
    Medals 1

    Now I'm really confused

    • one year ago
  24. MathSofiya Group Title
    Best Response
    You've already chosen the best response.
    Medals 1

    I am soo sorry! My book calls the logistic equation little k and carrying capacity big K....

    • one year ago
  25. TuringTest Group Title
    Best Response
    You've already chosen the best response.
    Medals 1

    I'm just going to say what I'm sure of, which is the initial derivation

    • one year ago
  26. MathSofiya Group Title
    Best Response
    You've already chosen the best response.
    Medals 1

    I think I'm kinda done with this problem LOL. I suck at word problems. But your steps in the initial derivative makes sense though.

    • one year ago
  27. MathSofiya Group Title
    Best Response
    You've already chosen the best response.
    Medals 1

    It seems like my book just gave me this equation \[\frac{dP}{dt}=kP(1-\frac{P}{K})\] and just wants me to plug and chug numbers. ....fine with me!

    • one year ago
  28. TuringTest Group Title
    Best Response
    You've already chosen the best response.
    Medals 1

    I guess that's what I'd do as well :p

    • one year ago
  29. MathSofiya Group Title
    Best Response
    You've already chosen the best response.
    Medals 1

    Sounds like a plan :P Thanks Turing

    • one year ago
  30. TuringTest Group Title
    Best Response
    You've already chosen the best response.
    Medals 1

    that equation makes sense too because when P=K, dP/dt=0 i.e. when the population = max capacity it stops changing reasonable to me :)

    • one year ago
    • Attachments:

See more questions >>>

Your question is ready. Sign up for free to start getting answers.

spraguer (Moderator)
5 → View Detailed Profile

is replying to Can someone tell me what button the professor is hitting...

23

  • Teamwork 19 Teammate
  • Problem Solving 19 Hero
  • You have blocked this person.
  • ✔ You're a fan Checking fan status...

Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.