constant death rate 0.02 (deaths per animal per month)

and with birth rate proportional to P.

And with...

... we have...

... which is suitably arranged as either...

... or else...

... according to taste.

Just in case a picture helps...

... where (key in spoiler) ...

Spoiler:

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Don't integrate - balloontegrate!

Balloon Calculus; standard integrals, derivatives and methods

Balloon Calculus Drawing with LaTeX and Asymptote!