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The logistic differential equation $y'=y(b-ay) \, \textrm{with}\, a\neq 0, b\neq 0$ has the non-trivial solution $y(t) = \frac{\frac{b}{a}}{1+e^{-bt}}, \quad (1)$ where $c$ is a constant.

My questions is:

We have that $\frac{y'}{y}= b-ay$. But how can I understand this intuitively. What does it give?

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    I am thinking what does $\frac{y'}{y}$ tell me.2012-11-13

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If $y(t)$ represents the size of a population, then $y'$ is the total growth rate for the entire population ("the number of new individuals produced during each unit of time"). Therefore $y'/y$ is the growth rate per capita ("the number of new individuals that each individual produces during each unit of time, on average").

In the logistic model, the growth rate per capita is assumed to decrease linearly with the size of the population (as seen in the right-hand side $b-ay$), for example because of competition about resources.

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    @Reader: I'm not sure that I understand the question. The quantity $K=b/a$ is called the *carrying capacity* of the environment; it's the largest population size that can be supported in the long run. If the initial population is smaller than $K$ it will grow and approach $K$ from below as $t \to \infty$, and if it's larger than $K$ it will decrease and approach $K$ from above.2012-11-14