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# 5.4: Logistic solution


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.

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