{"id":684,"date":"2021-05-10T18:59:28","date_gmt":"2021-05-10T18:59:28","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=684"},"modified":"2021-11-17T02:48:28","modified_gmt":"2021-11-17T02:48:28","slug":"summary-of-the-logistic-equation","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/summary-of-the-logistic-equation\/","title":{"raw":"Summary of the Logistic Equation","rendered":"Summary of the Logistic Equation"},"content":{"raw":"<section id=\"fs-id1170572311324\" class=\"key-concepts\" data-depth=\"1\">\r\n<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul id=\"fs-id1170572560614\" data-bullet-style=\"bullet\">\r\n \t<li>When studying population functions, different assumptions\u2014such as exponential growth, logistic growth, or threshold population\u2014lead to different rates of growth.<\/li>\r\n \t<li>The logistic differential equation incorporates the concept of a carrying capacity. This value is a limiting value on the population for any given environment.<\/li>\r\n \t<li>The logistic differential equation can be solved for any positive growth rate, initial population, and carrying capacity.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/section><section id=\"fs-id1170571598181\" class=\"key-equations\" data-depth=\"1\">\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1170572622480\" data-bullet-style=\"bullet\">\r\n \t<li><strong data-effect=\"bold\">Logistic differential equation and initial-value problem<\/strong><span data-type=\"newline\">\r\n<\/span>\r\n[latex]\\frac{dP}{dt}=rP\\left(1-\\frac{P}{K}\\right),P\\left(0\\right)={P}_{0}[\/latex]<\/li>\r\n \t<li><strong data-effect=\"bold\">Solution to the logistic differential equation\/initial-value problem<\/strong><span data-type=\"newline\">\r\n<\/span>\r\n[latex]P\\left(t\\right)=\\frac{{P}_{0}K{e}^{rt}}{\\left(K-{P}_{0}\\right)+{P}_{0}{e}^{rt}}[\/latex]<\/li>\r\n \t<li><strong data-effect=\"bold\">Threshold population model<\/strong><span data-type=\"newline\">\r\n<\/span>\r\n[latex]\\frac{dP}{dt}=\\text{-}rP\\left(1-\\frac{P}{K}\\right)\\left(1-\\frac{P}{T}\\right)[\/latex]<\/li>\r\n<\/ul>\r\n<\/section><section id=\"fs-id1170572245934\" class=\"section-exercises\" data-depth=\"1\"><\/section>\r\n<div data-type=\"glossary\">\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1170572147813\">\r\n \t<dt>carrying capacity<\/dt>\r\n \t<dd id=\"fs-id1170572351566\">the maximum population of an organism that the environment can sustain indefinitely<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572351570\">\r\n \t<dt>growth rate<\/dt>\r\n \t<dd id=\"fs-id1170572351576\">the constant [latex]r&gt;0[\/latex] in the exponential growth function [latex]P\\left(t\\right)={P}_{0}{e}^{rt}[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572290475\">\r\n \t<dt>initial population<\/dt>\r\n \t<dd id=\"fs-id1170572290480\">the population at time [latex]t=0[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170571688565\">\r\n \t<dt>logistic differential equation<\/dt>\r\n \t<dd id=\"fs-id1170572624812\">a differential equation that incorporates the carrying capacity [latex]K[\/latex] and growth rate [latex]r[\/latex] into a population model<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572379509\">\r\n \t<dt>phase line<\/dt>\r\n \t<dd id=\"fs-id1170572379514\">a visual representation of the behavior of solutions to an autonomous differential equation subject to various initial conditions<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572379519\">\r\n \t<dt>threshold population<\/dt>\r\n \t<dd id=\"fs-id1170571638272\">the minimum population that is necessary for a species to survive<\/dd>\r\n<\/dl>\r\n<\/div>","rendered":"<section id=\"fs-id1170572311324\" class=\"key-concepts\" data-depth=\"1\">\n<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul id=\"fs-id1170572560614\" data-bullet-style=\"bullet\">\n<li>When studying population functions, different assumptions\u2014such as exponential growth, logistic growth, or threshold population\u2014lead to different rates of growth.<\/li>\n<li>The logistic differential equation incorporates the concept of a carrying capacity. This value is a limiting value on the population for any given environment.<\/li>\n<li>The logistic differential equation can be solved for any positive growth rate, initial population, and carrying capacity.<\/li>\n<\/ul>\n<\/div>\n<\/section>\n<section id=\"fs-id1170571598181\" class=\"key-equations\" data-depth=\"1\">\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1170572622480\" data-bullet-style=\"bullet\">\n<li><strong data-effect=\"bold\">Logistic differential equation and initial-value problem<\/strong><span data-type=\"newline\"><br \/>\n<\/span><br \/>\n[latex]\\frac{dP}{dt}=rP\\left(1-\\frac{P}{K}\\right),P\\left(0\\right)={P}_{0}[\/latex]<\/li>\n<li><strong data-effect=\"bold\">Solution to the logistic differential equation\/initial-value problem<\/strong><span data-type=\"newline\"><br \/>\n<\/span><br \/>\n[latex]P\\left(t\\right)=\\frac{{P}_{0}K{e}^{rt}}{\\left(K-{P}_{0}\\right)+{P}_{0}{e}^{rt}}[\/latex]<\/li>\n<li><strong data-effect=\"bold\">Threshold population model<\/strong><span data-type=\"newline\"><br \/>\n<\/span><br \/>\n[latex]\\frac{dP}{dt}=\\text{-}rP\\left(1-\\frac{P}{K}\\right)\\left(1-\\frac{P}{T}\\right)[\/latex]<\/li>\n<\/ul>\n<\/section>\n<section id=\"fs-id1170572245934\" class=\"section-exercises\" data-depth=\"1\"><\/section>\n<div data-type=\"glossary\">\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1170572147813\">\n<dt>carrying capacity<\/dt>\n<dd id=\"fs-id1170572351566\">the maximum population of an organism that the environment can sustain indefinitely<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572351570\">\n<dt>growth rate<\/dt>\n<dd id=\"fs-id1170572351576\">the constant [latex]r>0[\/latex] in the exponential growth function [latex]P\\left(t\\right)={P}_{0}{e}^{rt}[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572290475\">\n<dt>initial population<\/dt>\n<dd id=\"fs-id1170572290480\">the population at time [latex]t=0[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1170571688565\">\n<dt>logistic differential equation<\/dt>\n<dd id=\"fs-id1170572624812\">a differential equation that incorporates the carrying capacity [latex]K[\/latex] and growth rate [latex]r[\/latex] into a population model<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572379509\">\n<dt>phase line<\/dt>\n<dd id=\"fs-id1170572379514\">a visual representation of the behavior of solutions to an autonomous differential equation subject to various initial conditions<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572379519\">\n<dt>threshold population<\/dt>\n<dd id=\"fs-id1170571638272\">the minimum population that is necessary for a species to survive<\/dd>\n<\/dl>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-684\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Calculus Volume 2. <strong>Authored by<\/strong>: Gilbert Strang, Edwin (Jed) Herman. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction\">https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\">CC BY-NC-SA: Attribution-NonCommercial-ShareAlike<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":416434,"menu_order":16,"template":"","meta":{"_candela_citation":"{\"1\":{\"type\":\"cc\",\"description\":\"Calculus Volume 2\",\"author\":\"Gilbert Strang, Edwin (Jed) 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https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction\"}}","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-684","chapter","type-chapter","status-publish","hentry"],"part":159,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/684","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/users\/416434"}],"version-history":[{"count":4,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/684\/revisions"}],"predecessor-version":[{"id":1043,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/684\/revisions\/1043"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/159"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/684\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=684"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=684"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=684"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=684"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}