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anonymous

  • one year ago

how exactly does the standard form of the exponential function y[x] = k E^(r x) relate to this form of the logistic differential equation y'[t] == r y[t] (1 - y[t]/b) and it's counterpart y[t] = (b*E^(r*t + b*k))/(-1 + E^(r*t + b*k)) r is obvious k is not so obvious but probably the same between y[t] and y[x] but b???!!! b is a problem.

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  1. UsukiDoll
    • one year ago
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    omg I feel like I'm in Mathematical Biology again

  2. jagr2713
    • one year ago
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    @nincompoop @Australopithecus

  3. anonymous
    • one year ago
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    lol, its awful, Im in hell

  4. jagr2713
    • one year ago
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    Please come help

  5. anonymous
    • one year ago
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    3 days I been trying to work this out

  6. UsukiDoll
    • one year ago
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    where there are two parameters in the logistic differential equation... r is the growth rate k is carrying capacity. ewww I don't want to do this again!

  7. anonymous
    • one year ago
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    you any good with mathematica usuki?

  8. UsukiDoll
    • one year ago
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    too long ago.

  9. UsukiDoll
    • one year ago
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    but I know this logistic equation

  10. anonymous
    • one year ago
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    In all the examples they provide b.. but then in the problem.. they dont.. I do have a table of data though.. an exponential sample.

  11. anonymous
    • one year ago
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    is that required to work out b in the final equation?

  12. jagr2713
    • one year ago
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    @Michele_Laino

  13. anonymous
    • one year ago
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    my best guess is that the logistical curve is going to be an S .. at the middle of that S, in the 2nd derivative will be a 0 marking the point at which the concave changes.. this point is going to be at almost exactly 1/2 of b.. if there is proven to be a relationship between the corresponding exponential curve and the logistical curve then I can find b.

  14. anonymous
    • one year ago
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    like the slope of the derivatives, might have some relationship that is propotional to r.

  15. anonymous
    • one year ago
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    at that inflection point.

  16. Astrophysics
    • one year ago
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    What exactly is the problem, this seems like logistic differential equation, I mean like does k represent the carrying capacity, and r,k >0 constant?

  17. anonymous
    • one year ago
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    My problem is that I was given the top equation.. and asked to turn it into the bottom equation. after they showed me the middle equation.. and they're asking me to work it out.

  18. anonymous
    • one year ago
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    if we name the equations 1,2,3. I know how to go from equation 2 to 3 .. but not from equation 1 to 2 or 1 to 3.

  19. anonymous
    • one year ago
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    k as far as I can tell, is the value where x=0 y[0], I do have a value for it.. I have a data table for the exponential points.. and a formula that approximates those data points.

  20. anonymous
    • one year ago
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    the formula is in the standard exponential form [x] = k E^(r x)

  21. anonymous
    • one year ago
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    oops y[x] = k E^(r x)

  22. Astrophysics
    • one year ago
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    The counterpart to me seems to be integrated, but I'm not really used to this stuff yet (haven't done much ODE atm), I've done similar but for word problems haha..mhm. I would take an attempt at it, but I don't want to lead you to the wrong place so I'll let the big guns handle it, so to speak haha @ganeshie8 @SithsAndGiggles

  23. anonymous
    • one year ago
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    thank you astro for taking a look. appreciated.

  24. jagr2713
    • one year ago
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    Sorry about the inconvenience @hughfuve i dont want to make false promises but i will try and find someone to help :D @freckles

  25. UsukiDoll
    • one year ago
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    I've done logistic equation... but like I mentioned I don't want to go back to Mathematical Biology >_<

  26. anonymous
    • one year ago
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    well I appreciate all help.. no matter how it turns out..

  27. UsukiDoll
    • one year ago
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    y'[t] == r y[t] (1 - y[t]/b) there are two parameters r and b. r is the growth rate and b is the carrying capacity.

  28. UsukiDoll
    • one year ago
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    you can't assign values to parameters.. the derivative is a constant.

  29. anonymous
    • one year ago
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    so if b is the carrying capacity here.. could it just k from the first equation? The problem I have is when I tried that I got a very small S curve.. the data I have represents the US population, and according to a value of b=k, then the US population should not go over 80million.. as that is clearly not the reality.. I am concerned.

  30. UsukiDoll
    • one year ago
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    hmmm mine had ecology situations

  31. UsukiDoll
    • one year ago
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    like for bird populations

  32. anonymous
    • one year ago
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    lol, you studied at the height of environmental hysteria

  33. anonymous
    • one year ago
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    this course one comes from a time of immigration hysteria

  34. UsukiDoll
    • one year ago
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    my Mathematical Biology book is still in my backpack since final exam week. I'm not taking it out XD!

  35. anonymous
    • one year ago
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    touching it probably makes your fingers rash

  36. anonymous
    • one year ago
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    my book is going to give me shingles one day.. I just know it.

  37. UsukiDoll
    • one year ago
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    well it's not a discrete model (that's for difference equations) it's a continuous model.. what I 'm saying may be a different point of view for the logistic equation

  38. anonymous
    • one year ago
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    yes this is one of the first problems of my chapter introducing differential equations. what an intro.. geeez.

  39. UsukiDoll
    • one year ago
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    pfffffffft you should be doing integrating factor as the first topic

  40. UsukiDoll
    • one year ago
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    forgot to mention that the logistic equation is separable -_-!

  41. Astrophysics
    • one year ago
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    Does that look right Hugh?

  42. Astrophysics
    • one year ago
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    |dw:1435638707408:dw|

  43. anonymous
    • one year ago
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    looks good to me.

  44. anonymous
    • one year ago
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    I am on chapter 6.. integration isn't until chapter 10. I have a hunch that perhaps this course is not for teaching calculus, but instead for teaching mathematica to those who already know calculus.

  45. UsukiDoll
    • one year ago
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    your curriculum is backwards -_-

  46. UsukiDoll
    • one year ago
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    Populations tend to get larger until there is no longer enough food or space to support so many individuals. This type of growth is called logistic population growth What Is Logistic Population Growth? A group of individuals of the same species living in the same area is called a population. The measurement of how the size of a population changes over time is called the population growth rate, and it depends upon the population size, birth rate and death rate. As long as there are enough resources available, there will be an increase in the number of individuals in a population over time, or a positive growth rate. However, most populations cannot continue to grow forever because they will eventually run out of water, food, sunlight, space or other resources. As these resources begin to run out, population growth will start to slow down. When the growth rate of a population decreases as the number of individuals increases, this is called logistic population growth.

  47. anonymous
    • one year ago
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    I have the textbook calculus concepts and contexts by james stewart.. I just found a section in there on this formula.. Its almost word for word

  48. anonymous
    • one year ago
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    doing some reading

  49. UsukiDoll
    • one year ago
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    James Stewart? I had a Single Variable Calculus Book from that author, but it was only Calculus I and II.

  50. UsukiDoll
    • one year ago
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    logistic equation is a 100% ODE though

  51. Astrophysics
    • one year ago
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    Is this in james ste 7e?!

  52. anonymous
    • one year ago
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    I got the larson hostetler, edwards video seriies too.. maybe theres something in there.

  53. anonymous
    • one year ago
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    7e?

  54. Astrophysics
    • one year ago
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    james stewart 7th edition

  55. anonymous
    • one year ago
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    oh no.. its 2nd edition

  56. UsukiDoll
    • one year ago
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    OH THAT IS OLD !

  57. anonymous
    • one year ago
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    2001, it was on our bookshelf.. my g/f used it way back when

  58. UsukiDoll
    • one year ago
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    :/

  59. anonymous
    • one year ago
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    lol, no wonder Im struggling.. I need some new stuff..

  60. anonymous
    • one year ago
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    I have pearsons college mathematics 13th edition if that helps

  61. anonymous
    • one year ago
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    but they dont say much on this topic

  62. anonymous
    • one year ago
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    actually this looks promising http://reference.wolfram.com/language/tutorial/DSolveOverview.html

  63. anonymous
    • one year ago
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    might have found a possible solution, just have to work it backwards. if y[x] = logarithmic S curve f[x] = corresponding exponential curve with same r then y''[n]==0 solve for n finds the mid point of the S curve. Then take that n of midpoint of S curve and f[n]+f[0] = value for b

  64. anonymous
    • one year ago
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    well well y[n]/2 =f[n]+f[0]

  65. anonymous
    • one year ago
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    might have found a possible solution, just have to work it backwards. if y[x] = logarithmic S curve f[x] = corresponding exponential curve with same r then y''[n]==0 solve for n finds the mid point of the S curve. Then take that n of midpoint of S curve and f[n]+f[0] = value for b well well y[n]/2 =f[n]+f[0] so (y[n]/2)-f[0] =f[n]

  66. UsukiDoll
    • one year ago
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    You are right about the logistic equation producing scenarios that can't happen. Like the one for your 80 million people in the US. Obviously to sustain 80 million, you have to force people to not have plenty of kids. It's like the logistic equation is saying ok your world must sustain exactly 80 million people or less and if it goes 1 over, you'll have a shortage of supplies. Same with the birds. Either 100 birds or less...otherwise if there are 1000 birds there will be less resources and eventually the birds will die.

  67. anonymous
    • one year ago
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    so does that mean then that b is arbitrary? and you can set it to whatever you want? say if you determine that you are going to put some cap on on growth and force it?

  68. UsukiDoll
    • one year ago
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    you can try to force it .. but in reality that's not possible... humans are going to mate anyways... same with birds.

  69. UsukiDoll
    • one year ago
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    @Michele_Laino is here!

  70. UsukiDoll
    • one year ago
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    I was told that Mathematical Biology is harder ... it actually takes Calculus III, IV, ODE, and Linear Algebra... puts it in a blender and messes everything up.

  71. UsukiDoll
    • one year ago
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    plus I had to use Matlab for assignments and a project... I still think my project about Killer Zombies saved my grade.

  72. anonymous
    • one year ago
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    maybe I got thrown off because the examples show exponential plots vs logarithmic plots that are proportional to each other in respect to the value for b. They created a log plot, then used that to make the exponential plot. But now I am thinking that maybe the two plots are always proportional in this way no matter what the value for b.. and b is arbitrary, and it depends on what controls you want to place on the model. So Im looking for a shadow

  73. anonymous
    • one year ago
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    crap.. I think it is the case

  74. Michele_Laino
    • one year ago
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    if we try this substitution: \[y\left( t \right) = \frac{1}{{z\left( t \right)}}\] where z is the new variable, then our starting equation, namely equation #2 can be rewritten as follows: \[\frac{{dz}}{{dt}} + rz = \frac{r}{b}\]

  75. anonymous
    • one year ago
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    Hi michele.. welcome :)

  76. UsukiDoll
    • one year ago
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    now it looks like the integrating factor in the form of \[\frac{dy}{dx} =p(x)y=q(x)\]

  77. Michele_Laino
    • one year ago
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    Hi :) @hughfuve

  78. UsukiDoll
    • one year ago
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    NUGH NOT MY NIGHT! \[\frac{dy}{dx} +p(x)y=q(x)\]

  79. UsukiDoll
    • one year ago
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    I can't do math in the heat D:!

  80. Michele_Laino
    • one year ago
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    the solutions to that ODE are: \[\Large z\left( t \right) = {e^{ - rt}}\left( {c + \int {\frac{r}{b}{e^{rt}}dt} } \right)\]

  81. UsukiDoll
    • one year ago
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    so you do need integrating factor right?

  82. Michele_Laino
    • one year ago
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    right! so we have: \[\Large z\left( t \right) = c{e^{ - rt}} + \frac{1}{b}\]

  83. Michele_Laino
    • one year ago
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    now we have to return to our old variable y(t): \[\Large y\left( t \right) = \frac{b}{{bc{e^{ - rt}} + 1}}\] where c is an arbitrary real constant

  84. UsukiDoll
    • one year ago
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    @hughfuve didn't you mention earlier that integrating factor was not until real later in your book?

  85. UsukiDoll
    • one year ago
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    "hughfuve Best Response Medals 1 I am on chapter 6.. integration isn't until chapter 10. I have a hunch that perhaps this course is not for teaching calculus, but instead for teaching mathematica to those who already know calculus."

  86. anonymous
    • one year ago
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    yes, but that's okay.. I can still use the Iintegrate[] mathematica command in this problem at least..

  87. UsukiDoll
    • one year ago
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    I would complain to your instructor if I were you. How can you do this problem without integrating factor?

  88. anonymous
    • one year ago
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    Its okay.. Im learning for now.. this will at least put me in a place where I can put this one to rest..

  89. UsukiDoll
    • one year ago
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    .___. last time I checked all of differential equations come first...then the Mathematical Biology comes second

  90. Michele_Laino
    • one year ago
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    now we can multiply both numerator and denominator by \[\Large {{e^{rt}}}\] so we get: \[\Large y\left( t \right) = \frac{{b{e^{rt}}}}{{bc + {e^{rt}}}}\]

  91. Michele_Laino
    • one year ago
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    now we have to know the initial conditions, since the formula above gives us infinite solutions

  92. anonymous
    • one year ago
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    so the exponential function I started with.. r = 0.01300265909581666; k = E^(4.382968909053888); yExp[x] = k E^(r x) I can basically pull the variables here as the starting conditions and then b is whatever we want to set the limit of the function too?

  93. anonymous
    • one year ago
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    I assume c =k r = r b = whatever

  94. anonymous
    • one year ago
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    and I will have an exponential version and a logarithmic version that start out the same, but end up going in different directions.

  95. anonymous
    • one year ago
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    hence the idea that.. logistical growth is controlled growth ?

  96. Michele_Laino
    • one year ago
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    the coefficient b is called "carrying capacity"

  97. UsukiDoll
    • one year ago
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    I got that one too

  98. anonymous
    • one year ago
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    ah gotcha, thanks.. the language helps

  99. Michele_Laino
    • one year ago
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    it is suffice to use this substitution: \[\Large \frac{1}{c} = {e^{bk}}\]

  100. Michele_Laino
    • one year ago
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    I have integrated that ODE using the separation of variable method, and I got this solutions: \[\Large y\left( t \right) = \frac{{b{c_1}{e^{rt}}}}{{1 + {c_1}{e^{rt}}}}\] where c_1 is the integration constant. Now it is suffice apply this substitution: \[\Large {c_1} = {e^{bk}}\]

  101. UsukiDoll
    • one year ago
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    yes that's correct... though these days I prefer integrating factor... I know it's more steps but taking the antiderivative is so much easier than integrating on both sides with separation of variables and end up getting stuck on the left side.

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