The original problem asks me to solve the following model for the whale population: $$\frac{dP}{dt} = k(M-P)(P-m)$$ Where $P$ is the population at time $t$, $M$ is the carrying capacity, $m$ is the survival limit, and $k$ is a positive constant.
I feel like I may have the right answer, but with the amount of algebra that came along with solving it I would like some clarification.
I won't show every step I took, as typing all of it out would take a long time, but I will show some of them with some explanations so that I can see if I am along the right track.
After separating and integrating, I got that $$\ln|P-m| - \ln|M-P|=(M-m)kt+C$$ Plugging in the initial condition of $P_0$ at time $t_0$ to solve for the constant gave me $$\ln\left|\frac{P-m}{M-P}\right|=(M-m)kt+\ln\left|\frac{P_0-m}{M-P_0}\right| - (M-m)kt_0$$
Now all that is left is to solve for $P$. After removing the logs by taking $e$ to the power of both sides we get $$\frac{(P-m)(M-P_0)}{(M-P)(P_0-m)} = e^{k(M-m)(t-t_0)}$$
And then after expanding both sides out to isolate $P$ I eventually get a final answer of $$P=\frac{Mm-P_0m+(MP_0-Mm)e^{k(M-m)(t-t_0)}}{M-P_0+(P_0-m)e^{k(M-m)(t-t_0)}}$$
Is what I have correct? I'm sure this kind of problem has been solved enough times that there is some sort of layout already somewhere but I could not find it with any google searches so any help would be greatly appreciated.
I feel like what I have makes sense, because if we take the limit as $t$ goes to $\infty$, we do get that $P$ goes to $M$, which is how it should be for this kind of model.
