{"id":2026,"date":"2023-04-18T18:44:27","date_gmt":"2023-04-18T18:44:27","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&p=2026"},"modified":"2024-09-12T05:27:25","modified_gmt":"2024-09-12T05:27:25","slug":"logarithms-and-logistic-growth-learn-it-4","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/logarithms-and-logistic-growth-learn-it-4\/","title":{"raw":"Logarithms and Logistic Growth: Learn It 4","rendered":"Logarithms and Logistic Growth: Learn It 4"},"content":{"raw":"

Logistic Growth<\/h2>\r\n
\r\n
\r\n

logistic growth<\/h3>\r\n

If a population is growing in a constrained environment with carrying capacity\u00a0[latex]K[\/latex], and absent constraint would grow exponentially with growth rate\u00a0[latex]r[\/latex], then the population behavior can be described by the logistic growth model:<\/p>\r\n

[latex]{{P}_{n}}={{P}_{n-1}}+r\\left(1-\\frac{{{P}_{n-1}}}{K}\\right){{P}_{n-1}}[\/latex]<\/p>\r\n<\/div>\r\n<\/section>\r\n

There is another form of this model that you will be introduced to later in the module. It is the continuous logistic model in the form:\r\n\r\n\r\n

[latex]P_{t}=\\dfrac{c}{1+\\left(\\dfrac{c}{P_{0}}-1\\right)e^{-rt}}[\/latex]<\/p>\r\n

where [latex]t[\/latex] stands for time in years, [latex]c[\/latex] is the carrying capacity (the maximal population), [latex]P_0[\/latex] represents the starting quantity, and [latex]r[\/latex] is the rate of growth.<\/p>\r\n\r\n\r\nFor now we will use the model in the form [latex]{{P}_{n}}={{P}_{n-1}}+r\\left(1-\\frac{{{P}_{n-1}}}{K}\\right){{P}_{n-1}}[\/latex] but it is important to know both forms of the model.<\/section>\r\n

Unlike linear and exponential growth, logistic growth behaves differently if the populations grow steadily throughout the year or if they have one breeding time per year. The recursive formula provided above models generational growth, where there is one breeding time per year (or, at least a finite number); there is no explicit formula for this type of logistic growth.<\/p>\r\n

A forest is currently home to a population of\u00a0[latex]200[\/latex] rabbits. The forest is estimated to be able to sustain a population of\u00a0[latex]2000[\/latex] rabbits. Absent any restrictions, the rabbits would grow by\u00a0[latex]50\\%[\/latex] per year. Predict the future population using the logistic growth model.
\r\n[reveal-answer q=\"543594\"]Show Solution[\/reveal-answer]
\r\n[hidden-answer a=\"543594\"]Modeling this with a logistic growth model,\u00a0[latex]r = 0.50[\/latex],\u00a0[latex]K = 2000[\/latex], and [latex]P\u00ad_0 = 200[\/latex]. Calculating the next year:\r\n\r\n\r\n

[latex]{{P}_{1}}={{P}_{0}}+0.50\\left(1-\\frac{{{P}_{0}}}{2000}\\right){{P}_{0}}=200+0.50\\left(1-\\frac{200}{2000}\\right)200=290[\/latex]<\/p>\r\n

We can use this to calculate the following year:<\/p>\r\n

[latex]{{P}_{2}}={{P}_{1}}+0.50\\left(1-\\frac{{{P}_{1}}}{2000}\\right){{P}_{1}}=290+0.50\\left(1-\\frac{290}{2000}\\right)290\\approx414[\/latex]<\/p>\r\n

A calculator was used to compute several more values:<\/p>\r\n\r\n\r\n\r\n\r\n
[latex]n[\/latex]<\/td>\r\n[latex]0[\/latex]<\/td>\r\n[latex]1[\/latex]<\/td>\r\n[latex]2[\/latex]<\/td>\r\n[latex]3[\/latex]<\/td>\r\n[latex]4[\/latex]<\/td>\r\n[latex]5[\/latex]<\/td>\r\n[latex]6[\/latex]<\/td>\r\n[latex]7[\/latex]<\/td>\r\n[latex]8[\/latex]<\/td>\r\n[latex]9[\/latex]<\/td>\r\n[latex]10[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]P_n[\/latex]<\/td>\r\n[latex]200[\/latex]<\/td>\r\n[latex]290[\/latex]<\/td>\r\n[latex]414[\/latex]<\/td>\r\n[latex]578[\/latex]<\/td>\r\n[latex]784[\/latex]<\/td>\r\n[latex]1022[\/latex]<\/td>\r\n[latex]1272[\/latex]<\/td>\r\n[latex]1503[\/latex]<\/td>\r\n[latex]1690[\/latex]<\/td>\r\n[latex]1821[\/latex]<\/td>\r\n[latex]1902[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

 <\/p>\r\n

Plotting these values, we can see that the population starts to increase faster and the graph curves upwards during the first few years, like exponential growth, but then the growth slows down as the population approaches the carrying capacity.<\/p>\r\n

\"Graph.<\/center>\r\n

View more about this example below.<\/p>\r\n

[embed]https:\/\/youtu.be\/dPOlEgJ2QX0[\/embed]<\/p>\r\n

You can view the transcript for \u201cLogistic growth of rabbits\u201d here (opens in new window).<\/a><\/p>\r\n\r\n\r\n[\/hidden-answer]<\/section>\r\n

A field currently contains\u00a0[latex]20[\/latex] mint plants. Absent constraints, the number of plants would increase by\u00a0[latex]70\\%[\/latex] each year, but the field can only support a maximum population of\u00a0[latex]300[\/latex] plants. Use the logistic model to predict the population in the next three years.
\r\n[reveal-answer q=\"726473\"]Show Solution[\/reveal-answer]
\r\n[hidden-answer a=\"726473\"]
[latex]P_1=P_0+0.70(1-\\frac{P_0}{300}) \\quad P_1=20+0.70(1-\\frac{20}{300})20=33[\/latex]<\/center>
[latex]P_2=54[\/latex]<\/center>
[latex]P_3=85[\/latex]<\/center>[\/hidden-answer]<\/section>\r\n
[ohm2_question hide_question_numbers=1]6929[\/ohm2_question]<\/section>\r\n
[ohm2_question hide_question_numbers=1]6931[\/ohm2_question]<\/section>","rendered":"

Logistic Growth<\/h2>\n
\n
\n

logistic growth<\/h3>\n

If a population is growing in a constrained environment with carrying capacity\u00a0[latex]K[\/latex], and absent constraint would grow exponentially with growth rate\u00a0[latex]r[\/latex], then the population behavior can be described by the logistic growth model:<\/p>\n

[latex]{{P}_{n}}={{P}_{n-1}}+r\\left(1-\\frac{{{P}_{n-1}}}{K}\\right){{P}_{n-1}}[\/latex]<\/p>\n<\/div>\n<\/section>\n

There is another form of this model that you will be introduced to later in the module. It is the continuous logistic model in the form:<\/p>\n

[latex]P_{t}=\\dfrac{c}{1+\\left(\\dfrac{c}{P_{0}}-1\\right)e^{-rt}}[\/latex]<\/p>\n

where [latex]t[\/latex] stands for time in years, [latex]c[\/latex] is the carrying capacity (the maximal population), [latex]P_0[\/latex] represents the starting quantity, and [latex]r[\/latex] is the rate of growth.<\/p>\n

For now we will use the model in the form [latex]{{P}_{n}}={{P}_{n-1}}+r\\left(1-\\frac{{{P}_{n-1}}}{K}\\right){{P}_{n-1}}[\/latex] but it is important to know both forms of the model.<\/section>\n

Unlike linear and exponential growth, logistic growth behaves differently if the populations grow steadily throughout the year or if they have one breeding time per year. The recursive formula provided above models generational growth, where there is one breeding time per year (or, at least a finite number); there is no explicit formula for this type of logistic growth.<\/p>\n

A forest is currently home to a population of\u00a0[latex]200[\/latex] rabbits. The forest is estimated to be able to sustain a population of\u00a0[latex]2000[\/latex] rabbits. Absent any restrictions, the rabbits would grow by\u00a0[latex]50\\%[\/latex] per year. Predict the future population using the logistic growth model.<\/p>\n