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 =
Substituting this in to our original exponential growth model for r gives
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: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:
We can use this to calculate the following year:
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:
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.
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:
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.
