{"id":276,"date":"2024-10-18T21:19:55","date_gmt":"2024-10-18T21:19:55","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/chapter\/logarithms-and-logistic-growth-fresh-take\/"},"modified":"2024-10-18T21:24:52","modified_gmt":"2024-10-18T21:24:52","slug":"logarithms-and-logistic-growth-fresh-take","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/chapter\/logarithms-and-logistic-growth-fresh-take\/","title":{"raw":"Logarithms and Logistic Growth: Fresh Take","rendered":"Logarithms and Logistic Growth: Fresh Take"},"content":{"raw":"\n<section class=\"textbox learningGoals\">\n<ul>\n\t<li>Use the properties of logarithms to solve exponential models for time<\/li>\n\t<li>Identify the carrying capacity in a logistic growth model<\/li>\n\t<li>Use a logistic growth model to predict growth<\/li>\n<\/ul>\n<\/section>\n<h2>Reversing an Exponent<\/h2>\n<div class=\"textbox shaded\"><strong>The Main Idea&nbsp;<\/strong> <br>\n<br>\n<p>Logarithms are mathematical tools that help us work with exponential models, particularly in solving for time or growth rates. They are the inverse of exponentials, meaning they 'undo' the exponential function.<\/p>\n<p><strong>Key Concepts:<\/strong><\/p>\n<ul>\n\t<li><strong>Common Logarithm<\/strong>: Denoted as [latex]log(x)[\/latex], it is based on the exponential [latex]10^x[\/latex] and is used to solve equations where the variable is an exponent.\n\n\n<ul>\n\t<li>This means the statement [latex]10^{a} = b[\/latex] is equivalent to the statement [latex]log(b) = a[\/latex]<\/li>\n\t<li>Exponent property: [latex]\\log\\left({{A}^{r}}\\right)=r\\log\\left(A\\right)[\/latex]<\/li>\n<\/ul>\n<\/li>\n\t<li><strong>Properties of Logarithms<\/strong>: These include product rule, quotient rule, power rule, and more, which are essential for simplifying and solving exponential equations.<\/li>\n<\/ul>\n\n\nTo solve exponential equations with logarithms:\n\n<ol>\n\t<li>Isolate the exponential. In other words, get it by itself on one side of the equation. This usually involves dividing by a number multiplying it.<\/li>\n\t<li>Take the log of both sides of the equation.<\/li>\n\t<li>Use the exponent property of logs to rewrite the exponential with the variable exponent multiplying the logarithm.<\/li>\n\t<li>Divide as needed to solve for the variable.<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/Z5myJ8dg_rM\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\n<p>You can view the&nbsp;<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Logarithms+_+Logarithms+_+Algebra+II+_+Khan+Academy.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cLogarithms | Logarithms | Algebra II | Khan Academy\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox watchIt\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=11328604&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=kqVpPSzkTYA&amp;video_target=tpm-plugin-yrfe6wc9-kqVpPSzkTYA\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe>\n<p>You can view the&nbsp;<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Logarithms+-+The+Easy+Way!.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cLogarithms - The Easy Way!\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox example\">Evaluate [latex]log(300)[\/latex].<br>\n[reveal-answer q=\"517989\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"517989\"]Using a calculator, [latex]log(300)[\/latex] is approximately&nbsp;[latex]2.477121[\/latex][\/hidden-answer]<\/section>\n<section class=\"textbox example\">Rewrite&nbsp;[latex]log(25)[\/latex] using the exponent property for logs.<br>\n[reveal-answer q=\"925419\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"925419\"][latex]log(25) = log(5^{2}) = 2log(5)[\/latex][\/hidden-answer]<\/section>\n<h2>Limits on Exponential Growth<\/h2>\n<div class=\"textbox shaded\"><strong>The Main Idea&nbsp;<\/strong> <br>\n<br>\n<p>In real-world situations, exponential growth often encounters limits, such as resource constraints or environmental factors. The logistic model adjusts the exponential growth formula to account for these limits, introducing the concept of carrying capacity.<\/p>\n<p><strong>Key Concepts:<\/strong><\/p>\n<ul>\n\t<li><strong>Carrying Capacity ([latex]K[\/latex])<\/strong>: The maximum population or quantity that an environment can sustain.<\/li>\n\t<li><strong>Adjusted Growth Rate<\/strong>: The growth rate in logistic models decreases as the population approaches the carrying capacity.<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox watchIt\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=11328605&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=0BSaMH4hINY&amp;video_target=tpm-plugin-6kfx69zg-0BSaMH4hINY\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe>\n<p>You can view the&nbsp;<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Exponential+Growth+a+Commonsense+Explanation.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cExponential Growth: a Commonsense Explanation.\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox watchIt\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=11328606&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=ozW7y-y6Ymw&amp;video_target=tpm-plugin-b27f9twt-ozW7y-y6Ymw\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe>\n<p>You can view the&nbsp;<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Ecological+Carrying+Capacity-Biology.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEcological Carrying Capacity-Biology\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Logistic Growth<\/h2>\n<div class=\"textbox shaded\"><strong>The Main Idea&nbsp;<\/strong> <br>\n<br>\n<p>The logistic model provides a framework for understanding growth in confined environments, such as population growth in a limited habitat or product adoption in a market.<\/p>\n<p>The logistic growth model can be represented as [latex]P_n=P_{n-1}+r\\left(1-\\dfrac{P_{n-1}}{K}\\right)P_{n-1}[\/latex], where [latex]P_n[\/latex] is the population at time [latex]n[\/latex], [latex]r[\/latex] is the growth rate, and [latex]K[\/latex] is the carrying capacity.<\/p>\n<p>This model can be applied to various scenarios, like predicting population of a specific type of animal in an environment or the spread of a technology in a market.<\/p>\n<\/div>\n<section class=\"textbox watchIt\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=11328607&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=zZLRUPAjBaM&amp;video_target=tpm-plugin-3f3mo6ui-zZLRUPAjBaM\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe>\n<p>You can view the&nbsp;<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Exponential+vs+Logistic+Growth.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cExponential vs Logistic Growth\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox example\">On an island that can support a population of&nbsp;[latex]1000[\/latex] lizards, there is currently a population of&nbsp;[latex]600[\/latex]. These lizards have a lot of offspring and not a lot of natural predators, so have very high growth rate, around&nbsp;[latex]150\\%[\/latex]. Calculating out the next couple generations\n\n\n<p style=\"text-align: center;\">[latex]{{P}_{1}}={{P}_{0}}+1.50\\left(1-\\frac{{{P}_{0}}}{1000}\\right){{P}_{0}}=600+1.50\\left(1-\\frac{600}{1000}\\right)600=960[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]{{P}_{2}}={{P}_{1}}+1.50\\left(1-\\frac{{{P}_{1}}}{1000}\\right){{P}_{1}}=960+1.50\\left(1-\\frac{960}{1000}\\right)960=1018[\/latex]<\/p>\n<p>Interestingly, even though the factor that limits the growth rate slowed the growth a lot, the population still overshot the carrying capacity. We would expect the population to decline the next year.<\/p>\n<p style=\"text-align: center;\">[latex]{{P}_{3}}={{P}_{2}}+1.50\\left(1-\\frac{{{P}_{3}}}{1000}\\right){{P}_{3}}=1018+1.50\\left(1-\\frac{1018}{1000}\\right)1018=991[\/latex]<\/p>\n<p>Calculating out a few more years and plotting the results, we see the population wavers above and below the carrying capacity, but eventually settles down, leaving a steady population near the carrying capacity.<\/p>\n<center><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/12\/21175945\/Screen-Shot-2016-12-21-at-12.59.24-PM.png\"><img class=\"aligncenter wp-image-910\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/12\/21175945\/Screen-Shot-2016-12-21-at-12.59.24-PM.png\" alt=\"Graph. Vertical measures Population, in increments of 200 from 0 to 1200. Horizontal measures Years, in increments of 1 from 0 to 10. Year 0 shows population of 600, jumping to ~1000 in year 1, a little over 1000 in year 2, and staying close to 1000 in every subsequent year. \" width=\"350\" height=\"277\"><\/a><\/center><\/section>\n<section class=\"textbox example\">On a neighboring island to the one from the previous example, there is another population of lizards, but the growth rate is even higher \u2013 about&nbsp;[latex]205\\%[\/latex].Calculating out several generations and plotting the results, we get a surprise: the population seems to be oscillating between two values, a pattern called a 2-cycle.<center><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/12\/21184924\/Screen-Shot-2016-12-21-at-1.47.24-PM.png\"><img class=\"aligncenter wp-image-911\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/12\/21184924\/Screen-Shot-2016-12-21-at-1.47.24-PM.png\" alt=\"Graph. Vertical measures Population, in increments of 200 from 0 to 1200. Horizontal measures Years, in increments of 1 from 0 to 10. Year 0 shows population of 600, jumping to ~1100 in year 1, down to ~900 in year 2, and vacillating from 1100 to 800 in alternating years through the rest of the graph.\" width=\"350\" height=\"276\"><\/a><\/center>While it would be tempting to treat this only as a strange side effect of mathematics, this has actually been observed in nature. Researchers from the University of California observed a stable 2-cycle in a lizard population in California.[footnote]<a>http:\/\/users.rcn.com\/jkimball.ma.ultranet\/BiologyPages\/P\/Populations2.html<\/a>[\/footnote]Taking this even further, we get more and more extreme behaviors as the growth rate increases higher. It is possible to get stable 4-cycles, 8-cycles, and higher. Quickly, though, the behavior approaches chaos (remember the movie <em>Jurassic Park<\/em>?).<center><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/12\/21185438\/Screen-Shot-2016-12-21-at-1.50.01-PM.png\"><img class=\"aligncenter wp-image-913 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/12\/21185438\/Screen-Shot-2016-12-21-at-1.50.01-PM.png\" alt=\"Two graphs. Left graph, labeled \u201cr=2.46 A 4-cycle.\u201d Vertical measures Population, in increments of 200 from 0 to 1400. Horizontal measures Years, in increments of 1 from 0 to 10. The plotted line moves in sharp up and down cycles, from ~700 in year 0, 1200 in year 1, 600 in year 2, 1200 in year 3, ~600 in year 4, and so forth. Right graph, labeled \u201cr=2.9 Chaos!\u201d Vertical measures Population, in increments of 200 from 0 to 1400. Horizontal measures Years, in increments of 5 from 0 to 30. The plotted line moves in sharp up and down cycles of varying lengths, forming an erratic back and forth between highs and lows.\" width=\"613\" height=\"279\"><\/a><\/center>\n<p>All of the lizard island examples are discussed in this video.<\/p>\n<p>https:\/\/youtu.be\/fuJF_uZGoFc<\/p>\n<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Logistic+growth+of+lizards.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cLogistic growth of lizards\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Use the properties of logarithms to solve exponential models for time<\/li>\n<li>Identify the carrying capacity in a logistic growth model<\/li>\n<li>Use a logistic growth model to predict growth<\/li>\n<\/ul>\n<\/section>\n<h2>Reversing an Exponent<\/h2>\n<div class=\"textbox shaded\"><strong>The Main Idea&nbsp;<\/strong> <\/p>\n<p>Logarithms are mathematical tools that help us work with exponential models, particularly in solving for time or growth rates. They are the inverse of exponentials, meaning they &#8216;undo&#8217; the exponential function.<\/p>\n<p><strong>Key Concepts:<\/strong><\/p>\n<ul>\n<li><strong>Common Logarithm<\/strong>: Denoted as [latex]log(x)[\/latex], it is based on the exponential [latex]10^x[\/latex] and is used to solve equations where the variable is an exponent.\n<ul>\n<li>This means the statement [latex]10^{a} = b[\/latex] is equivalent to the statement [latex]log(b) = a[\/latex]<\/li>\n<li>Exponent property: [latex]\\log\\left({{A}^{r}}\\right)=r\\log\\left(A\\right)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li><strong>Properties of Logarithms<\/strong>: These include product rule, quotient rule, power rule, and more, which are essential for simplifying and solving exponential equations.<\/li>\n<\/ul>\n<p>To solve exponential equations with logarithms:<\/p>\n<ol>\n<li>Isolate the exponential. In other words, get it by itself on one side of the equation. This usually involves dividing by a number multiplying it.<\/li>\n<li>Take the log of both sides of the equation.<\/li>\n<li>Use the exponent property of logs to rewrite the exponential with the variable exponent multiplying the logarithm.<\/li>\n<li>Divide as needed to solve for the variable.<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/Z5myJ8dg_rM\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the&nbsp;<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Logarithms+_+Logarithms+_+Algebra+II+_+Khan+Academy.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cLogarithms | Logarithms | Algebra II | Khan Academy\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=11328604&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=kqVpPSzkTYA&amp;video_target=tpm-plugin-yrfe6wc9-kqVpPSzkTYA\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<p>You can view the&nbsp;<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Logarithms+-+The+Easy+Way!.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cLogarithms &#8211; The Easy Way!\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox example\">Evaluate [latex]log(300)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q517989\">Show Solution<\/button><\/p>\n<div id=\"q517989\" class=\"hidden-answer\" style=\"display: none\">Using a calculator, [latex]log(300)[\/latex] is approximately&nbsp;[latex]2.477121[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">Rewrite&nbsp;[latex]log(25)[\/latex] using the exponent property for logs.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q925419\">Show Solution<\/button><\/p>\n<div id=\"q925419\" class=\"hidden-answer\" style=\"display: none\">[latex]log(25) = log(5^{2}) = 2log(5)[\/latex]<\/div>\n<\/div>\n<\/section>\n<h2>Limits on Exponential Growth<\/h2>\n<div class=\"textbox shaded\"><strong>The Main Idea&nbsp;<\/strong> <\/p>\n<p>In real-world situations, exponential growth often encounters limits, such as resource constraints or environmental factors. The logistic model adjusts the exponential growth formula to account for these limits, introducing the concept of carrying capacity.<\/p>\n<p><strong>Key Concepts:<\/strong><\/p>\n<ul>\n<li><strong>Carrying Capacity ([latex]K[\/latex])<\/strong>: The maximum population or quantity that an environment can sustain.<\/li>\n<li><strong>Adjusted Growth Rate<\/strong>: The growth rate in logistic models decreases as the population approaches the carrying capacity.<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=11328605&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=0BSaMH4hINY&amp;video_target=tpm-plugin-6kfx69zg-0BSaMH4hINY\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<p>You can view the&nbsp;<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Exponential+Growth+a+Commonsense+Explanation.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cExponential Growth: a Commonsense Explanation.\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=11328606&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=ozW7y-y6Ymw&amp;video_target=tpm-plugin-b27f9twt-ozW7y-y6Ymw\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<p>You can view the&nbsp;<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Ecological+Carrying+Capacity-Biology.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEcological Carrying Capacity-Biology\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Logistic Growth<\/h2>\n<div class=\"textbox shaded\"><strong>The Main Idea&nbsp;<\/strong> <\/p>\n<p>The logistic model provides a framework for understanding growth in confined environments, such as population growth in a limited habitat or product adoption in a market.<\/p>\n<p>The logistic growth model can be represented as [latex]P_n=P_{n-1}+r\\left(1-\\dfrac{P_{n-1}}{K}\\right)P_{n-1}[\/latex], where [latex]P_n[\/latex] is the population at time [latex]n[\/latex], [latex]r[\/latex] is the growth rate, and [latex]K[\/latex] is the carrying capacity.<\/p>\n<p>This model can be applied to various scenarios, like predicting population of a specific type of animal in an environment or the spread of a technology in a market.<\/p>\n<\/div>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=11328607&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=zZLRUPAjBaM&amp;video_target=tpm-plugin-3f3mo6ui-zZLRUPAjBaM\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<p>You can view the&nbsp;<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Exponential+vs+Logistic+Growth.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cExponential vs Logistic Growth\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox example\">On an island that can support a population of&nbsp;[latex]1000[\/latex] lizards, there is currently a population of&nbsp;[latex]600[\/latex]. These lizards have a lot of offspring and not a lot of natural predators, so have very high growth rate, around&nbsp;[latex]150\\%[\/latex]. Calculating out the next couple generations<\/p>\n<p style=\"text-align: center;\">[latex]{{P}_{1}}={{P}_{0}}+1.50\\left(1-\\frac{{{P}_{0}}}{1000}\\right){{P}_{0}}=600+1.50\\left(1-\\frac{600}{1000}\\right)600=960[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]{{P}_{2}}={{P}_{1}}+1.50\\left(1-\\frac{{{P}_{1}}}{1000}\\right){{P}_{1}}=960+1.50\\left(1-\\frac{960}{1000}\\right)960=1018[\/latex]<\/p>\n<p>Interestingly, even though the factor that limits the growth rate slowed the growth a lot, the population still overshot the carrying capacity. We would expect the population to decline the next year.<\/p>\n<p style=\"text-align: center;\">[latex]{{P}_{3}}={{P}_{2}}+1.50\\left(1-\\frac{{{P}_{3}}}{1000}\\right){{P}_{3}}=1018+1.50\\left(1-\\frac{1018}{1000}\\right)1018=991[\/latex]<\/p>\n<p>Calculating out a few more years and plotting the results, we see the population wavers above and below the carrying capacity, but eventually settles down, leaving a steady population near the carrying capacity.<\/p>\n<div style=\"text-align: center;\"><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/12\/21175945\/Screen-Shot-2016-12-21-at-12.59.24-PM.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-910\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/12\/21175945\/Screen-Shot-2016-12-21-at-12.59.24-PM.png\" alt=\"Graph. Vertical measures Population, in increments of 200 from 0 to 1200. Horizontal measures Years, in increments of 1 from 0 to 10. Year 0 shows population of 600, jumping to ~1000 in year 1, a little over 1000 in year 2, and staying close to 1000 in every subsequent year.\" width=\"350\" height=\"277\" \/><\/a><\/div>\n<\/section>\n<section class=\"textbox example\">On a neighboring island to the one from the previous example, there is another population of lizards, but the growth rate is even higher \u2013 about&nbsp;[latex]205\\%[\/latex].Calculating out several generations and plotting the results, we get a surprise: the population seems to be oscillating between two values, a pattern called a 2-cycle.<\/p>\n<div style=\"text-align: center;\"><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/12\/21184924\/Screen-Shot-2016-12-21-at-1.47.24-PM.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-911\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/12\/21184924\/Screen-Shot-2016-12-21-at-1.47.24-PM.png\" alt=\"Graph. Vertical measures Population, in increments of 200 from 0 to 1200. Horizontal measures Years, in increments of 1 from 0 to 10. Year 0 shows population of 600, jumping to ~1100 in year 1, down to ~900 in year 2, and vacillating from 1100 to 800 in alternating years through the rest of the graph.\" width=\"350\" height=\"276\" \/><\/a><\/div>\n<p>While it would be tempting to treat this only as a strange side effect of mathematics, this has actually been observed in nature. Researchers from the University of California observed a stable 2-cycle in a lizard population in California.<a class=\"footnote\" title=\"http:\/\/users.rcn.com\/jkimball.ma.ultranet\/BiologyPages\/P\/Populations2.html\" id=\"return-footnote-276-1\" href=\"#footnote-276-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a>Taking this even further, we get more and more extreme behaviors as the growth rate increases higher. It is possible to get stable 4-cycles, 8-cycles, and higher. Quickly, though, the behavior approaches chaos (remember the movie <em>Jurassic Park<\/em>?).<\/p>\n<div style=\"text-align: center;\"><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/12\/21185438\/Screen-Shot-2016-12-21-at-1.50.01-PM.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-913 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/12\/21185438\/Screen-Shot-2016-12-21-at-1.50.01-PM.png\" alt=\"Two graphs. Left graph, labeled \u201cr=2.46 A 4-cycle.\u201d Vertical measures Population, in increments of 200 from 0 to 1400. Horizontal measures Years, in increments of 1 from 0 to 10. The plotted line moves in sharp up and down cycles, from ~700 in year 0, 1200 in year 1, 600 in year 2, 1200 in year 3, ~600 in year 4, and so forth. Right graph, labeled \u201cr=2.9 Chaos!\u201d Vertical measures Population, in increments of 200 from 0 to 1400. Horizontal measures Years, in increments of 5 from 0 to 30. The plotted line moves in sharp up and down cycles of varying lengths, forming an erratic back and forth between highs and lows.\" width=\"613\" height=\"279\" \/><\/a><\/div>\n<p>All of the lizard island examples are discussed in this video.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Logistic growth of lizards\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/fuJF_uZGoFc?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Logistic+growth+of+lizards.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cLogistic growth of lizards\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-276-1\"><a>http:\/\/users.rcn.com\/jkimball.ma.ultranet\/BiologyPages\/P\/Populations2.html<\/a> <a href=\"#return-footnote-276-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":6,"menu_order":10,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":266,"module-header":"","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/pressbooks\/v2\/chapters\/276"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":1,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/pressbooks\/v2\/chapters\/276\/revisions"}],"predecessor-version":[{"id":300,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/pressbooks\/v2\/chapters\/276\/revisions\/300"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/pressbooks\/v2\/parts\/266"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/pressbooks\/v2\/chapters\/276\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/wp\/v2\/media?parent=276"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/pressbooks\/v2\/chapter-type?post=276"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/wp\/v2\/contributor?post=276"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/wp\/v2\/license?post=276"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}