Understanding a derivative in a biology article

Question

From this biology article (5th page, right column)

They have...

$$1-P_t ≈ 1-(1+s)P_{t+1}+\frac{p_{t+1}^2}{2}$$

and they conclude that...

$$\frac{dP}{dt} ≈ -sP + \frac{P^2}{2}$$

I don't really understand this link. How did they do that?

Answer 1

I think that $1$ is small when compared to $t$, so they read $P_{t+1}$ as $P_t + dP/dt$. The first equation then becomes $$1 - P \approx 1 - (1 + s)(P + dP/dt) + \tfrac{1}{2}(P + dP/dt)^2.$$ Solving this for $dP/dt$ and ignoring the term $(dP/dt)^2$ gives their result.