Answer
False
Work Step by Step
The concavity can be expressed as:
$$\frac{d^2 P}{dt^2}[f(t)]$$
Take the derivative of the logistic model
$$\frac{d}{dt}\frac{dP}{dt}=\frac{d}{dt}[Pr(1-\frac{P}{C})]$$
At some point the derivative is 0: $\frac{d^2 P}{dt^2}=0$
Hence, the concavity does not depend on the initial population or the carrying capacity.