Limits on Exponential Growth
In our basic exponential growth scenario, we had a recursive equation of the form
[latex]P_{n}= P_{n-1} + r P_{n-1}[/latex]
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.
For our fish, the carrying capacity is the largest population that the resources in the lake can sustain. If the population in the lake is far below the carrying capacity, then we would expect the population to grow essentially exponentially. However, as the population approaches the carrying capacity, there will be a scarcity of food and space available, and the growth rate will decrease. If the population exceeds the carrying capacity, there won’t be enough resources to sustain all the fish and there will be a negative growth rate, causing the population to decrease back to the carrying capacity.
If the carrying capacity was [latex]5000[/latex], the growth rate might vary something like that in the graph shown.
Slope, the measure of the steepness of a line, is given be the difference in the vertical distance over the distance in the horizontal distance between [latex]2[/latex] points. The line graphed above falls [latex]0.1[/latex] in growth rate for a corresponding increase in population of [latex]5000[/latex]. This gives the stated slope of
[latex]\dfrac{-0.1}{5000}[/latex], which we write conveniently as [latex]-\dfrac{0.1}{5000}[/latex].
This slope remains constant across the population change since, as the population increases another [latex]5000[/latex] to [latex]10,000[/latex], the growth rate falls again by [latex]0.1[/latex].
Note that this is a linear equation with intercept at [latex]0.1[/latex] and slope [latex]-\frac{0.1}{5000}[/latex], so we could write an equation for this adjusted growth rate as:
[latex]r_{adjusted} =[/latex] [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 [latex]r[/latex] 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.
The video below provides a demonstration, but it may be helpful to recap the process in writing as well.
The recursive formula for exponential growth (listed again at the top of this page), [latex]P_n=P_{n-1}+rP_{n-1}[/latex], may be rewritten by factoring out the [latex]P_{n-1}[/latex] from both terms on the right hand side of the equation. This gives
[latex]P_n=P_{n-1}\left(1+r\right)[/latex]
[latex]P_n=\left(1+r\right)P_{n-1}[/latex], equivalently.
We can write an equation of the line formed in the graph above. It’s vertical intercept is [latex]0.1[/latex] and slope is [latex]\dfrac{-0.1}{5000}[/latex]. In the form of a linear equation, [latex]y=mx+b[/latex] with [latex]y=r[/latex] for growth rate and [latex]x=P[/latex] for population, this gives
[latex]r=-\dfrac{0.1}{5000}P+0.1[/latex]
[latex]r=0.1-\dfrac{0.1}{5000}P[/latex]
[latex]r=0.1\left(1-\dfrac{P}{5000}\right)[/latex] by factoring [latex]0.1[/latex] from both terms.
Now we can build the adjusted exponential growth model for this situation.
[latex]P_n=P_{n-1}+\left(r\right)P_{n-1}[/latex]
[latex]P_n=P_{n-1}+0.1\left(1-\dfrac{P_{n-1}}{5000}\right)P_{n-1}[/latex], substituting the equivalent value for r in our situation, and the initial population [latex]P_{n-1}[/latex] for [latex]P[/latex].
Putting this in general terms for any such situation, we can replace the particular growth rate with the variable [latex]r[/latex] and the maximal population of [latex]5000[/latex] in this case with a variable [latex]K[/latex] that represents carrying capacity. This gives us a general model for constrained growth called the logistic model.
[latex]P_n=P_{n-1}+r\left(1-\dfrac{P_{n-1}}{K}\right)P_{n-1}[/latex]
You can view the transcript for “Logistic model” here (opens in new window).