Stochastic analysis of survival and sensitivity in a competition model influenced by toxins under a fluctuating environment

This paper proposed a stochastic toxin-dependent competition model to investigate the impact of environmental noise on species interaction dynamics. First, a survival analysis was conducted to establish the sufficient conditions for population extinction and persistence. Second, we proved the existence of a unique ergodic stationary distribution. Finally, the spatial arrangement of random states near the deterministic attractor was investigated using the stochastic sensitivity functions technique. This analytical approach facilitates constructing confidence ellipses and estimating critical noise intensity corresponding to the onset of transition. Both theoretical and numerical findings demonstrated that significant levels of noise experienced by one species lead to its extinction while promoting persistence in its competitor; conversely, negligible levels of noise did not alter the original competition outcomes in the deterministic model. However, when both species encounter moderate levels of noise, various modifications can occur in competition outcomes. These findings have significant implications for preserving ecosystem diversity.