Resumen de Exploring the Impact of Distributed Delay and Environmental Fluctuation on Food Chain Stability in a Chemostat Model
Miaomiao Gao, Xiao Chen, Daqing Jiang
The food chain chemostat model is a mathematical model employed to evaluate the stability of food chains and the responsiveness of ecosystems. Based on the relationships and interactions among various species within the food chain, the model discloses changes in population size and the adaptability of ecosystems to environmental changes by analyzing the flow of energy and matter. In this study, we propose the predator-prey chemostat model incorporating log-normal Ornstein-Uhlenbeck process and distributed delay, and study dynamics of the stochastic system. Initially, we establish existence and uniqueness of the global positive solution. Next, threshold conditions for the extinction in two scenarios are derived: one where both predator and prey become extinct, and another where predator goes extinct while the prey survives.
Additionally, the stationary distribution indicates the long-term coexistence of prey and predator. By constructing suitable Lyapunov functions, we get the sufficient condition for existence of the stationary distribution. Then the explicit form of the density function near the positive equilibrium of the corresponding deterministic system is provided. Finally, we use numerical simulations to study the influence of mean delay, volatility intensity, and reversion speed on biological population.
