We consider a Richards growth model (modified logistic model) driven by correlated multiplicative and additive colored noises, and investigate the effects of noises on the eventual distribution of population size with the help of steady-state analysis. An approximative Fokker–Planck equation is first derived for the stochastic model. By performing detailed theoretical analysis and numerical simulation for the steady-state solution of the Fokker–Planck equation, i.e., stationary probability distribution (SPD) of the stochastic model, we find that the correlated noises have complex effects on the statistical property of the stochastic model. Specifically, the phenomenological bifurcation may be caused by the noises. The position of extrema of the SPD depends on the model parameter and the characters of noises in different ways.

中文翻译：

---

由乘性和加性有色噪声驱动的理查兹增长模型：稳态分析

我们考虑由相关的乘性和加性有色噪声驱动的 Richards 增长模型（改进的逻辑模型），并在稳态分析的帮助下研究噪声对人口规模最终分布的影响。首先推导出随机模型的近似 Fokker-Planck 方程。通过对福克-普朗克方程的稳态解，即随机模型的平稳概率分布（SPD）进行详细的理论分析和数值模拟，我们发现相关噪声对随机模型的统计特性具有复杂的影响。模型。具体来说，现象学分叉可能是由噪声引起的。SPD的极值位置取决于模型参数和噪声特性的不同方式。