  The logistic equation has many applications and is used frequently in
different fields, such as biology, medicine, and economics. In this paper, we
study the stability of a single-species logistic model with a general
distribution delay kernel and an inflow of nutritional resources at a constant
rate. In particular, we provide precise conditions for the linear stability of
the positive equilibrium and the occurrence of Hopf bifurcation. We apply the
results to three delay distribution kernels: Uniform, Dirac-delta, and gamma
distributions. Without an inflow, we show that the positive equilibrium is
stable for a relatively small delay and then loses its stability through the
Hopf bifurcation when the mean delay ${\tau_m}$ increases with the three
distributions. In the presence of an inflow, the model dynamics depend on the
delay distribution kernel. In the uniform and Dirac-delta distributions cases,
we find that the dynamics are similar to the absence of a nutrient influx. In
contrast, the dynamics depend on the delay order p when considering the gamma
distribution. For $p=1$, the positive equilibrium is always stable. While for
$p=2$ and $p=3$, we find stability switching of the positive equilibrium
resulting from the increase of the value of ${\tau_m}$, where the positive
equilibrium is stable for a relatively short period; then, it loses stability
via Hopf bifurcation as ${\tau_m}$ increases; after then, it stabilizes again
with an increase in ${\tau_m}$. The main difference between the delay orders
$p=2$ and $p=3$ is that for relatively large ${\tau_m}$ and intrinsic growth
rate, the positive equilibrium can be stable when $p=2$, but it will be
unstable when $p=3$.
