Stochastic disturbances and delays drive coexistence and extinction in competing populations

• A stochastic competition model with S-type delays is developed. • Global positivity and stability are proved via Lyapunov and Markov semigroup theory. • Disturbance intensity and time delays drive coexistence, fluctuation, or extinction. • Secondary species and environment affect weak species under small disturbances. The coexistence and extinction of competing populations are influenced not only by primary competing species but also by environmental stochastic disturbances, time delays, and secondary competing species. Based on the Lotka-Volterra model, this study establishes a stochastic population competition model with S-type distributed delays, incorporating the influence factor q of secondary species and the environment. Intrinsic and interspecific disturbances are represented using Ornstein-Uhlenbeck processes and white noise. Theoretical analysis and numerical simulations were conducted as follows: (1) A Lyapunov function was constructed to prove the existence and uniqueness of the global positive solution of the model, and conditions for global asymptotic stability were derived based on Markov semigroup theory. (2) Under low-intensity stochastic disturbances, the population with stronger competitiveness increased in size, while the weaker population tended toward extinction. When there was no difference in competitive strength, both populations could coexist in the long term; Under medium-intensity disturbances, coexistence and extinction exhibited complex patterns; Under high-intensity disturbances, randomness became the dominant factor, leading to rapid population declines, abnormal fluctuations, and eventual extinction. Nevertheless, competitiveness and time delays further complicated population dynamics; Time delays alleviated population fluctuations and promoted system stability under low-intensity disturbances. However, under higher disturbances, they initially amplified fluctuations and introduced periodic dynamics but ultimately helped stabilize the system; The influence factor q moderated population fluctuations and improved the survival conditions of weaker populations under low-intensity disturbances but showed diminished effects under medium- and high-intensity disturbances. This study enhances the understanding of coexistence and extinction mechanisms in competing populations, offering significant insights into ecosystem management and conservation strategies.