Stability Analysis of A Single-Species Model with Distributed Delay

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 $p=3$.