更改

* 理论奠基：逻辑斯蒂函数由比利时数学家皮埃尔·弗朗索瓦·韦尔霍斯特在1838年至1847年间发表的三篇论文中系统性地提出并发展。他在阿道夫·凯特勒的指导下，将其作为调整指数增长模型后的人口增长模型。韦尔霍斯特在1838年的论文中首次提出了逻辑斯蒂方程的微分形式，其中包含种群平衡点对应环境承载能力<ref>Verhulst, Pierre Francois (1838). Notice sur la loi que la population suit dans son accroissement. Correspondance mathématique et Physigue, publiée par A. Quetelet, 10, 113-120.</ref>。1845年，他正式将方程的解曲线命名为“逻辑斯蒂曲线”<ref>Verhulst, Pierre Francois (1845). Recherches mathématiques sur la loi d'accroissement dela population. Nouveaur Mémoires de l'Académie Royale des Sciences, des Lettres et desBeaux-Arts de Belgique, 18,1-38.</ref>。1847年，他的第三篇论文调整了对比利时人口增长模型中的修正项<ref>Verhulst, Pierre Francois (1847). Deuxième Mémoire sur la loi d'accroissement de la popula.tion. Nouveaur Mémoires de l'Académie Rouale des Sciences, des Lettres et des Beaur-Artsde Belgique, 20,1-32.</ref>。

* 理论奠基：逻辑斯蒂函数由比利时数学家皮埃尔·弗朗索瓦·韦尔霍斯特在1838年至1847年间发表的三篇论文中系统性地提出并发展。他在阿道夫·凯特勒的指导下，将其作为调整指数增长模型后的人口增长模型。韦尔霍斯特在1838年的论文中首次提出了逻辑斯蒂方程的微分形式，其中包含种群平衡点对应环境承载能力<ref>Verhulst, Pierre Francois (1838). Notice sur la loi que la population suit dans son accroissement. Correspondance mathématique et Physigue, publiée par A. Quetelet, 10, 113-120.</ref>。1845年，他正式将方程的解曲线命名为“逻辑斯蒂曲线”<ref>Verhulst, Pierre Francois (1845). Recherches mathématiques sur la loi d'accroissement dela population. Nouveaur Mémoires de l'Académie Royale des Sciences, des Lettres et desBeaux-Arts de Belgique, 18,1-38.</ref>。1847年，他的第三篇论文调整了对比利时人口增长模型中的修正项<ref>Verhulst, Pierre Francois (1847). Deuxième Mémoire sur la loi d'accroissement de la popula.tion. Nouveaur Mémoires de l'Académie Rouale des Sciences, des Lettres et des Beaur-Artsde Belgique, 20,1-32.</ref>。

* 重新发现与推广：韦尔霍斯特的工作在当时并未引起广泛关注。直到1920年，美国生物学家雷蒙德·珀尔和洛厄尔·里德在研究果蝇种群增长和人口问题时独立重新发现了这一方程，并提出了现在广为流传的形式。他们的工作使得该方程在生物学界广为人知，因此该方程有时也被称为韦尔霍斯特-珀尔阻碍方程。

* 重新发现与推广：韦尔霍斯特的工作在当时并未引起广泛关注。直到1920年，美国生物学家雷蒙德·珀尔和洛厄尔·里德在研究果蝇种群增长和人口问题时独立重新发现了这一方程，并提出了现在广为流传的形式。他们的工作使得该方程在生物学界广为人知，因此该方程有时也被称为韦尔霍斯特-珀尔阻碍方程。

* 混沌理论发展：随着20世纪系统科学和混沌理论的发展，该方程的离散形式——[[逻辑斯谛映射]]被深入研究。罗伯特·梅在1976年发表于《自然》杂志的论文《表现非常复杂的动力学的简单数学模型》中<ref>May R. 1976. Simple mathematical models with very complicated dynamics. Nature 261, 459–467. <nowiki>https://doi.org/10.1038/261459a0</nowiki></ref>，系统性地阐述了这一简单非线性方程如何产生复杂的混沌行为，从而使该模型成为混沌研究的经典范例。

* 混沌理论发展：随着20世纪系统科学和混沌理论的发展，该方程的离散形式——[[逻辑斯谛映射]]被深入研究。罗伯特·梅在1976年发表于《自然》杂志的论文《表现非常复杂的动力学的简单数学模型》中<ref>May R. 1976. Simple mathematical models with very complicated dynamics. Nature 261, 459–467. <nowiki>https://doi.org/</nowiki>[https://doi.org/10.1038/261459a0 10.1038/261459a0]<span class="sciapp-btn" title="DOI: 10.1038/261459a0" paper_id="10.1038/261459a0" paper_id_type="DOI"><svg xmlns="http://www.w3.org/2000/svg" class="sciapp-svgicon" viewBox="0 0 32 32"><image href="data:image/png;base64,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" width="32" height="32"></image></svg></span></ref>，系统性地阐述了这一简单非线性方程如何产生复杂的混沌行为，从而使该模型成为混沌研究的经典范例。

*  

*  

===拐点===

===拐点===

逻辑斯谛曲线的拐点出现在 <math>N = \frac{K}{2}</math>处，此时种群增长率达到最大值<math>\frac{rK}{4}</math> 。这一性质在实际应用中具有重要意义，例如在资源管理和流行病控制中，拐点可以帮助确定干预措施的最佳时机。

逻辑斯谛曲线的拐点出现在 <math>N = \frac{K}{2}</math>处，此时种群增长率达到最大值<math>\frac{rK}{4}</math> 。这一性质在实际应用中具有重要意义，例如在资源管理和流行病控制中，拐点可以帮助确定干预措施的最佳时机。

[[文件:Logistic curve.jpg|替代=|居中|缩略图|680x680像素|图1. 简单逻辑斯谛曲线的示意图。增长率函数在同一图上以钟形曲线表示。引自<ref>Kucharavy, D., Roland, DG. (2015). Application of logistic growth curve. Procedia engineering, 131, 280-290. https://doi.org/10.1016/j.proeng.2015.12.390<span class="scicrx-ico" title="DOI: 10.1016/j.proeng.2015.12.390" paper_id="10.1016/j.proeng.2015.12.390" paper_id_type="DOI"></span><span class="sciapp-btn" title="DOI: 10.1016/j.proeng.2015.12.390" paper_id="10.1016/j.proeng.2015.12.390" paper_id_type="DOI"><svg xmlns="http://www.w3.org/2000/svg" class="sciapp-svgicon" viewBox="0 0 32 32"><image href="data:image/png;base64,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" width="32" height="32"></image></svg></span></ref>]]

[[文件:Logistic curve.jpg|替代=|居中|缩略图|680x680像素|图1. 简单逻辑斯谛曲线的示意图。增长率函数在同一图上以钟形曲线表示。引自<ref>Kucharavy, D., Roland, DG. (2015). Application of logistic growth curve. Procedia engineering, 131, 280-290. https://doi.org/10.1016/j.proeng.2015.12.390</ref>]]

==离散形式的逻辑斯谛方程==

==离散形式的逻辑斯谛方程==