Logistic Growth
Learning Outcomes
- Evaluate and rewrite logarithms using the properties of logarithms
- Use the properties of logarithms to solve exponential models for time
- Identify the carrying capacity in a logistic growth model
- Use a logistic growth model to predict growth
Limits on Exponential Growth
In our basic exponential growth scenario, we had a recursive equation of the formPn = Pn-1 + r Pn-1
In a confined environment, however, the growth rate may not remain constant. In a lake, for example, there is some maximum sustainable population of fish, also called a carrying capacity.Carrying Capacity
The carrying capacity, or maximum sustainable population, is the largest population that an environment can support.radjusted = [latex]0.1-\frac{0.1}{5000}P=0.1\left(1-\frac{P}{5000}\right)[/latex]
Substituting this in to our original exponential growth model for r gives[latex]{{P}_{n}}={{P}_{n-1}}+0.1\left(1-\frac{{{P}_{n-1}}}{5000}\right){{P}_{n-1}}[/latex]
View the following for a detailed explanation of the concept.
https://youtu.be/-6VLXCTkP_cLogistic Growth
If a population is growing in a constrained environment with carrying capacity K, and absent constraint would grow exponentially with growth rate r, then the population behavior can be described by the logistic growth model: [latex-display]{{P}_{n}}={{P}_{n-1}}+r\left(1-\frac{{{P}_{n-1}}}{K}\right){{P}_{n-1}}[/latex-display]Examples
A forest is currently home to a population of 200 rabbits. The forest is estimated to be able to sustain a population of 2000 rabbits. Absent any restrictions, the rabbits would grow by 50% per year. Predict the future population using the logistic growth model.Answer: Modeling this with a logistic growth model, r = 0.50, K = 2000, and P0 = 200. Calculating the next year:
[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]
We can use this to calculate the following year:[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]
A calculator was used to compute several more values:n | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
Pn | 200 | 290 | 414 | 578 | 784 | 1022 | 1272 | 1503 | 1690 | 1821 | 1902 |
On an island that can support a population of 1000 lizards, there is currently a population of 600. These lizards have a lot of offspring and not a lot of natural predators, so have very high growth rate, around 150%. Calculating out the next couple generations:
[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]
[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]
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.[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]
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.Try It
A field currently contains 20 mint plants. Absent constraints, the number of plants would increase by 70% each year, but the field can only support a maximum population of 300 plants. Use the logistic model to predict the population in the next three years.Answer: [latex]P_1=P_0+0.70(1-\frac{P_0}{300})P_0=20+0.70(1-\frac{20}{300})20=33[/latex] [latex-display]P_2=54[/latex-display] [latex-display]P_3=85[/latex-display]
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 – about 205%. 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. 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]http://users.rcn.com/jkimball.ma.ultranet/BiologyPages/P/Populations2.html[/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 Jurassic Park?). All of the lizard island examples are discussed in this video. https://youtu.be/fuJF_uZGoFcLicenses & Attributions
CC licensed content, Original
- Revision and Adaptation. Provided by: Lumen Learning License: CC BY: Attribution.
CC licensed content, Shared previously
- Logistic Growth. Authored by: David Lippman. Located at: http://www.opentextbookstore.com/mathinsociety/. License: CC BY-SA: Attribution-ShareAlike.
- fishes-colourful-beautiful-koi. Authored by: sharonang. License: CC0: No Rights Reserved.
- Logistic model. Authored by: OCLPhase2's channel. License: CC BY: Attribution.
- Logistic growth of rabbits. Authored by: OCLPhase2's channel. License: CC BY: Attribution.
- Logistic growth of lizards. Authored by: OCLPhase2's channel. License: CC BY: Attribution.
- Question ID 54627. Authored by: Hartley,Josiah. License: CC BY: Attribution. License terms: IMathAS Community License.
- Question ID 6589. Authored by: Lippman,David. License: CC BY: Attribution. License terms: IMathAS Community License CC-BY + GPL.