Environmental Stochasticity Driving the Extinction of Top Predators in a Food Chain Chemostat Model

Understanding the process of extinction in natural populations is crucial for the preservation of ecosystem stability and biodiversity, both theoretically and practically. The risk of extinction in these populations is often influenced by environmental stochasticity, which has a significant impact on birth and mortality rates. In this study, we propose a tri-trophic food chain model that incorporates random disturbances in the environment, represented by a chemostat, which is an ideal mathematical model for simulating diverse ecosystems. In the absence of noise, the model exhibits two types of bistability, indicating that the stochastic system has two distinct paths to extinction: from a stationary state or from an oscillatory state. For each type, we determine the tipping value of environmental stochasticity that leads to the extinction of top predators by constructing confidence regions for the corresponding coexisting attractor. Furthermore, we observe a high skewness and heavy-tailed distribution of extinction times for intermediate and high levels of environmental stochasticity, consistent with empirical data. To analyze extinction times, we employ the Lévy distribution, a statistical model that describes power-law tail distributions. Our findings demonstrate that, for a fixed dilution rate, increasing environmental stochasticity reduces the average extinction time, thereby accelerating species extinction. Additionally, for a certain level of stochasticity, the average extinction time decreases with the magnitude of the dilution rate due to the heavy-tailed nature of the extinction time distribution.