Abstract We establish the existence and uniqueness of global (in time) positive strong solutions for a generalized population dynamics equation with environmental noise, while the global existence fails for the deterministic equation. Particularly, we prove the global existence of positive strong solutions for the following stochastic differential equation d X t = ( θ X t m 0 + k X t m ) d t + e X t m + 1 2 φ ( X t ) d W t , t > 0 , X t > 0 , m > m 0 ⩾ 1 , X 0 = x > 0 , with θ , k , e ∈ R being constants and φ ( r ) = r ϑ or | log ( r ) | ϑ ( ϑ > 0 ) , and we also show that the index ϑ > 0 is sharp in the sense that if ϑ = 0 , one can choose certain proper constants θ , k and e such that the solution X t will explode in a finite time almost surely.

中文翻译：

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带环境噪声的广义种群动力学方程

摘要 我们建立了具有环境噪声的广义种群动力学方程的全局（在时间上）正强解的存在性和唯一性，而确定性方程的全局存在性不成立。特别地，我们证明了以下随机微分方程的正强解的全局存在性 d X t = ( θ X tm 0 + k X tm ) dt + e X tm + 1 2 φ ( X t ) d W t , t > 0 , X t > 0 , m > m 0 ⩾ 1 , X 0 = x > 0 ，其中 θ , k , e ∈ R 为常数且 φ ( r ) = r ϑ 或 | 日志 ( r ) | ϑ ( ϑ > 0 ) ，并且我们还表明指数 ϑ > 0 是尖锐的，如果 ϑ = 0 ，人们可以选择某些适当的常数 θ , k 和 e 使得解 X t 将在有限范围内爆炸时间几乎可以肯定。