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Species extinction is a core process that affects the diversity of life on Earth. Competition between species in a population is considered by ecological niche-based theories as a key factor leading to different severity of species extinctions. There are population dynamics models that describe a simple and easily understandable mechanism for resource competition. However, these models can not efficiently characterize and quantify new emergent extinctions in a large population appearing due to environmental forcing. To address this issue we develop a stochastic physics-inspired approach to analyze how environmental forcing influences the severity of species extinctions in such models. This approach is based on the large deviations theory of stochastic processes (the Freidlin-Wentzell theory). We show that there are three possible fundamentally different scenarios of extinctions, which we call catastrophic extinctions, asymmetric ones, and extinctions with exponentially small probabilities. The realization of those scenarios depends on environmental noise properties and the boundaries of niches, which define the domain, where species survive. Furthermore, we describe a hysteresis effect in species extinction showing that fluctuations can lead to dramatic consequences even if an averaged resource supply is sufficient to support population survival. Our stochastic physics-inspired approach generalizes niche theory by accounting environmental forcing and will be useful to find, by available data, which environmental perturbations may induce extinctions.