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When both parameters are set to $1$, the differential equation for logistic population growth is $$x'=x(1-x)$$

Suppose the population is also harvested at the constant rate $h$. The differential equation then apparently becomes $$x'=x(1-x)-h$$

Here's what I don't understand: If the population is harvested at the constant rate $h$, it means that $x$ is reduced by $h$ periodically. Why would this also mean that $x'$ is reduced by $h$?

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The model isn't that you harvest periodically, it's that you harvest continuously all the time, $h$ units of population per unit time.

So in an infinitesimal time interval $dt$, the natural population growth is $x(1-x)\, dt$, and you harvest $h\, dt$, which results in the population change $dx = x(1-x) \, dt - h \, dt$.

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