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5.4: Logistic solution

  • Page ID
    25445
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    Recall that the carrying capacity is −r /s, also called K. So wherever −r /s appears, substitute K, as follows.

    \(N(t)\,=\frac{1}{(\frac{s}{r}\,+\frac{1}{N_0})\,e^{-rt}\,-\frac{s}{r}}\)

    \(\,=\frac{1}{(\,-\frac{1}{K}\,+\frac{1}{N_0})\,e^{-rt}\,+\frac{1}{K}}\)

    \(\,=\frac{K}{(\,-\,1\,+\frac{K}{N_0})\,e^{-rt}\,+\,1}\)

    \(\,=\frac{K}{(\frac{K\,-\,N_0}{N_0})\,e^{-rt}\,+\,1}\)

    This is the solution given in textbooks for logistic growth. There are slight variations in ways it is written, but they are equivalent.


    This page titled 5.4: Logistic solution is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Clarence Lehman, Shelby Loberg, & Adam Clark (University of Minnesota Libraries Publishing) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.