更改

===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S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., & De Guio, R. (2015). Application of logistic growth curve. Procedia engineering, 131, 280-290. <nowiki>https://doi.org/10.1016/j.proeng.2015.12.390</nowiki><span class="es-webpage-collect main" es-id-type="DOI" es-id="10.1016/j.proeng.2015.12.390" es-collect-button-id="1" cslsource="crossref"><svg xmlns="http://www.w3.org/2000/svg" version="1.1" x="0px" y="0px" viewBox="0 0 1000 1000" enable-background="new 0 0 1000 1000" space="preserve" height="1em" style="cursor: pointer; padding: 0.05em; border-radius: 0.5em; background: transparent;">

[[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., & De Guio, R. (2015). Application of logistic growth curve. Procedia engineering, 131, 280-290. <nowiki>https://doi.org/10.1016/j.proeng.2015.12.390</nowiki><span class="es-webpage-collect main" es-id-type="DOI" es-id="10.1016/j.proeng.2015.12.390" es-collect-button-id="1" cslsource="crossref"><svg xmlns="http://www.w3.org/2000/svg" version="1.1" x="0px" y="0px" viewBox="0 0 1000 1000" enable-background="new 0 0 1000 1000" space="preserve" height="1em" style="cursor: pointer; padding: 0.05em; border-radius: 0.5em; background: transparent;">

         <g><path d="M852.3,135.6l-47.7,48.3c157.6,178.7,157.4,451,0.1,631l48.8,49.5C1035.9,672.1,1035.5,327.2,852.3,135.6z M740.9,248.6l-49.3,49.9c96.1,115.5,96.1,285.8,0.5,402.1l48.8,49.6C863.4,606.1,863.6,391.5,740.9,248.6z M195.4,183.9l-47.7-48.3c-183.2,191.6-183.6,536.5-1.2,728.8l48.8-49.5C37.9,635,37.8,362.6,195.4,183.9z M259.1,248.6C136.4,391.5,136.6,606.1,259,750.3l48.8-49.6c-95.6-116.4-95.6-286.6,0.4-402.1L259.1,248.6z M499.8,340.4c-87.3,0-158.1,71.8-158.1,160.4c0,88.5,70.8,160.4,158.1,160.4c87.3,0,158.1-71.8,158.1-160.4C657.8,412.2,587,340.4,499.8,340.4z" style="fill: rgb(32, 85, 138);"></path></g>

         <g><path d="M852.3,135.6l-47.7,48.3c157.6,178.7,157.4,451,0.1,631l48.8,49.5C1035.9,672.1,1035.5,327.2,852.3,135.6z M740.9,248.6l-49.3,49.9c96.1,115.5,96.1,285.8,0.5,402.1l48.8,49.6C863.4,606.1,863.6,391.5,740.9,248.6z M195.4,183.9l-47.7-48.3c-183.2,191.6-183.6,536.5-1.2,728.8l48.8-49.5C37.9,635,37.8,362.6,195.4,183.9z M259.1,248.6C136.4,391.5,136.6,606.1,259,750.3l48.8-49.6c-95.6-116.4-95.6-286.6,0.4-402.1L259.1,248.6z M499.8,340.4c-87.3,0-158.1,71.8-158.1,160.4c0,88.5,70.8,160.4,158.1,160.4c87.3,0,158.1-71.8,158.1-160.4C657.8,412.2,587,340.4,499.8,340.4z" style="fill: rgb(32, 85, 138);"></path></g>

         </svg><span class="es-inline-ifqb" style="display: none; margin: 0px 0px 0px 0.2em; font-size: 1em; vertical-align: text-bottom; text-indent: 0px; text-decoration: none; cursor: pointer;"><span class="es-imf" style="background: rgb(0, 113, 203); color: white; padding: 0px 0.2em; display: none;" value="NA">IF: NA </span><span class="es-jcr" style="padding: 0px 0.2em; color: black; display: none;" value="NA">NA </span><span class="es-cas" style="display: none; padding: 0px 0.2em; color: black;" value="NA">NA</span></span></span></ref>]]

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==离散形式的逻辑斯谛方程==

==离散形式的逻辑斯谛方程==