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The following differential equation gives another model similar to the logistic model.

$\frac{dP}{dt}=kP(lnM-lnP)$;$k,M$ constants

Assume all quantities are positive, and $M>P.M$ still represents the carrying capacity.

a$.$ Show that with this model, the curve has an inflection point when $P=\frac{M}{e}.$ So$,$ the rate of growth slows down "sooner" than it would using the logistic growth model. Note: to do this, you need to show that the second derivative of $P_{-}$w$.$r$.$t$.\_t$ changes sign when $P=\frac{M}{e}.$