There isn't. The logistic equation is commonly written in the form $ {dP\over dt}=rP\left(1-{P\over K}\right), \quad P(0)=P_0, $ and in the context of logistic population models,
- $P$ is population
- $t$ is time
- $r$ is the intrinsic growth rate
- $K$ is the carry capacity of the environment
- $P_0$ is the initial population
Because of their physical meaning, each is taken to be positive. The solution is $ P(t)={KP_0\over P_0+(K-P_0)e^{-rt}}={K\over 1+\left({K\over P_0}-1\right)e^{-rt}}. $ This latter formulation matches the first form of your solution, just with $b:=r$ and $a:=r/K$, and $c:={K\over P_0}-1$.
There is no mathematical nor physical reason why we must have $c>0$. A negative value for $c$ would just mean that the initial population happened to be greater than the carrying capacity.