Logistic function?

Solve for the logistic function with initial condition = 14, limit to growth = 42 and passing through (1, 28).

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  • Ian H
    Lv 7
    1 month ago

    With finite resources an initial population Po grows nearly exponentially at first, but the population P eventually approaches a sort of saturation level, known as the carrying capacity K. This is the population level at which the birth and death rates of a species precisely match, resulting in a stable population over time.

    Such a logistics growth model is derived by assuming dP/dt= rP(1 - P/K)

    Details of the derivation can be found at

    http://www.math.northwestern.edu/~mlerma/courses/m...

    The solution takes the form

    P = K/[1 + Ae^-rt] where A = (K - Po )/Po, ....(1)

    In this example K = 42, Po = 14, so, A = 28/14 = 2 and

    P = 42/[1 + 2e^-rt] and we can find r using P = 28 when t = 1

    28 = 42/[1 + 2e^-r] yielding r = ln 4 and -r = ln(1/4)

    e^-rt = (1/4)^t

    P(t) = 42 /[1 + 2*(1/4)^t] or if you prefer

    P(t) = 42 /[1 + 2*2^(-2t)]

    https://www.wolframalpha.com/input/?i=P%28t%29+%3D...

    Note: We could have started with the form given by Slowfinger

    P(t) = K/[1 + Ab^t) which uses b = e^-r = 1/4

  • 1 month ago

    General form

    f(x) = c / (1 + ab^x)

    limit to growth is 42 => c=42

    initial condition is 14 meaning f(0)=14

    in above equation, substitute c=42, f(x)=14 and x=0

    14 = f(0) = 42 / (1 + ab^0) = 42 / (1+a)

    divide by 14

    1 = 3 / (1+a)

    solve for a

    1+a = 3/1 = 3

    a = 3 - 1 = 2

    b is still unknown. Find it from condition f(1)=28

    28 = 42 / (1 + 2*b^1)

    28 = 42 / (1 + 2b)

    solve for b

    1+2b = 42/28

    1+2b = 3/2

    2b = 3/2 - 1 = 1/2

    b = 1/4

    Finally, we have a logistic equation

    f(x) = 42 / (1 + 2(1/4)^x)

    Attachment image
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