Determining the Carrying Capacity of the United States using Logistic Functions

Determining the Carrying Capacity of the United States using Logistic Functions Eashwar S.C Krishna July 2021

Abstract The carrying capacity of an environment is the number of organisms that it can support without being seriously degraded. Due to the existence of the carrying capacity, population changes in accordance with a logistic curve model, creating a differential equation that can be used to relate carrying capacity, a rate constant, population, and rate of change. Solving this differential equation using calculus, specifically integration, yields a logistic function. In this paper, I demonstrate that logistic functions can be used to determine the carrying capacity of the United States. My work reveals that current trends in population growth and resource management will cause both the population and carrying capacity to approach approximately 420 million U.S residents. This result is in general accordance with the estimates of other academics.

Introduction Although the research on this specific subject is limited, a few academics have proposed figures for the carrying capacity. The late Dr. David Pimentel, professor of agricultural sciences at Cornell University, estimated the American carrying capacity at 200 million, warning that a failure to reduce population numbers now would result in disaster later on.1 Dr. Pimentel and his co-authors took an ecological approach to the problem; in this paper, I take a mathematical approach, utilizing logistic curves and integration to arrive at a reliable model for both population growth and carrying capacity.

Methods According to Strang and Herman’s work with logistic models2, the differential curve of population growth can be represented with the below equation:

Hoffmann, Hillel. “EcoFuture (TM) : Way Too Many for Us, Article by Hillel J. Hoffmann.”

Ecofuture.Org, 4 Dec. 2004, www.ecofuture.org/pk/pkcapcty.html.

2 Strang, G., & Herman, E. “Jed.” (2021, January 2). The Logistic Equation. Retrieved June 2, 2021, from https://math.libretexts.org/@go/page /2559 1

Where capital K represents the carrying capacity, P represents the population, r is the growth constant, and dP/dt is equal to the rate of change in population. Strang and Herman then detail manipulations of this differential equation, eventually yielding the following:

Where P0 is the initial population, P(t) is the population at time “t”, and “t” is the time. It is possible to utilize this equation to determine the carrying capacity, K, given all other variables.

My data set was the population of the United States, as recorded by decennial Census records, between the years 1800 and 2020, with t=0 set at 1800. Initial population was taken as the population in millions in 1800, 5.309. The portion of the data that was used in analysis was between 1820 and 2020.

My value of “r” was calculated using the equation for exponential growth, shown below:

The graph of exponential and logistic growth match each other most closely at “t” approaches zero. Therefore, I used the population in 1800 as P0, and the population in 1810 as P(t), where t=10. I found “r” to be equal to 0.0310217713.

Results

The table above displays the raw results of my calculations. The first column is “t”, the second column is population in millions, the third column is carrying capacity in millions, and the fourth column is the year. Visualizations of certain trends are shown below:

As seen in the above two graphs, population is growing, a predictable trend. In addition, a more interested trend is visible: the change in carrying capacity over time. The carrying capacity

generally increases over time, the major exception being around t=65 (coinciding with the infrastructure damage caused by the American Civil War). There are several key takeaways from the graph and data above: ● The carrying capacity of the United States can be reliable found using the solution to the differential curve ● The difference between the carrying capacity and population is decreasing over time ● Both the population and carrying capacity curves are increasing, yet at a decreasing rate of change.

In order to find the projected carrying capacity, I used U.S population projections until the year 2060, taken from Census Bureau estimations.3 The projections are shown in the table below:

Using these numbers, I created extensions to the previous graph. The data and resulting graph are shown below:

Vespa, J., Medina, L., & Armstrong, D. M. (2020, February). Demographic Turning Points for the United States: Population Projections for 2020 to 2060 [PDF]. https://www.census.gov/content/dam/Census/library/publications/2020/demo/p25-1144.pdf 3

The vertical black line is approximately t=220; the graph beyond that is a projection. We can see that the graph of both lines are beginning to “flatten out” around 420 million.

Discussion We can clearly see that the established logistic growth equation yields a reliable estimate of the carrying capacity of the United States. In fact, the estimate of 420 million people, at current levels of technological advancement, is supported by academics in other fields. Lyman Stone, a population economics research and an agricultural economist at USDA, wrote that “we could feed more than 400 million people, total, merely by consuming locally what we now export.” In other words, the carrying capacity of about 420 million people is a realistic estimate.

In addition, the results of this study reveal that the human carrying capacity can change.

In most ecological models, carrying capacity is a fixed limit that the population continually approaches. In such models, the carrying capacity cannot change. In stark contrast, this study

demonstrates that the carrying capacity of humans changes, and are most likely correlated with changes in infrastructure. As mentioned earlier, the American Civil War caused a drastic decrease in carrying capacity, as large swathes of the country had infrastructure destroyed. On the other hand, we can presume that the carrying capacity is increasing due to technological advancements. For example, the economic expansion following the Second World War led to suburbanization, which directly correlated with increased land usage, as families moved from the cities into the suburbs. Land usage is a critical factor in carrying capacity; the more efficient the land utilization, the greater the carrying capacity. Indeed, we see a steady increase in the carrying capacity in the years following 1945.

If the carrying capacity has changed in the past, then by extension, it can change in the future. If new, society-altering technologies are created in the near future, and these technologies allow us to use our resources more efficiently, then our carrying capacity will certainly increase at a greater rate than the present.

One important trend must be discussed in addition to carrying capacity, and that is the difference between the former and population. As seen in the data, said distance is decreasing, but the population curve shows no indication of drastically increasing above the carrying capacity; in fact, it appears that the population graph is tapering off alongside the carrying capacity. In other words, the United States is not in a population crisis and is not expected to be in one unless there is an abrupt shift in population or societal dynamics.

In conclusion, the carrying capacity of the United States, as estimated using logistic models, is approximately 420 million, and will most likely remain at this level in the absence of revolutionary technological changes.