Impact of Time Delays and Environmental Noise on the Extinction of a Population Dynamics Model

In this paper, we examine a population model with carrying capacity, time delay, and sources of additive and multiplicative environmental noise. We find that time delay, noise sources and their correlation induce regime shifts and transitions between the population survival state and the extinction state. To explore the transition mechanism between these two states, we analyzed the shift time to extinction, or the delayed extinction time, of populations. The main finding is that the extinction transition time as a function of the noise intensity shows a maximum, indicating the existence of an appropriate noise intensity leading to a maximal delayed extinction. This nonmonotonic behavior, with a maximum, is a signature of the noise-enhanced stability phenomenon, observed in many physical and complex metastable systems. In particular, this maximum increases (or decreases) as the cross-correlation intensity or the delay time in the death process increases. Furthermore, the signal-to-noise ratio as a function of noise intensity shows a maximum, which is a signature of the stochastic resonance phenomenon in the population dynamics model investigated in the presence of time delay and environmental noise.