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# Exponential Population Growth

Unlike the expansion of the universe, no population of living organisms can go on and on, forever increasing. Living creatures require too many resources and encounter too many confounding factors to expand indefinitely at a constant rate. However, for short periods of time, some organisms can experience very fast and constant growth rates. When this occurs, it is known as exponential growth!

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• In the following article, we will:
• discuss how and why some populations may experience exponential growth,
• provide some examples,
• detail the significance of population growth to ecology, and
• provide the formulas and models used to illustrate exponential growth.

## What is Population Growth?

In order to understand population growth, we must first understand what a population is and how it relates to ecology.

A population is a group of individuals of a particular species living in a specific area.

Population ecology is a field of science (a subfield of synecology, which deals with species groups relative to their ecosystems) interested in how and why certain factors (e.g., birth rates, death rates, immigration, and emigration) influence populations over periods of time.

Birth rates and immigration rates are collectively known as a population's recruitment rates. A population's size refers to the total number of individuals of a certain species in a certain area and a population's density is its size relative to its habitat.

Finally, population growth involves population dynamics, which deal with the variability in a given population's size over time.

Population growth involves population dynamics, which deal with the variability in a given population's size over time.

• A population's size refers to the total individuals of a certain species in a certain area and a population's density is its size relative to its habitat.

## What is Exponential Population Growth?

There are two kinds of population growth recognized: exponential and logistic. Logistic population growth is, by far, the most common kind observed in nature.

A population experiences exponential growth when the per capita rate of its growth remains constant independent of the size of the population. This results in the population growing larger at a very fast rate.

This is in contrast to logistic population growth, where the per capita growth rate for a population decreases as it approaches carrying capacity.

• Carry capacity, referred to as “K”, is a population’s maximum size dependent on limiting factors.

Logistic population growth occurs when the per capita growth rate decreases as its size increases and gradually approaches its carrying capacity, which is primarily influenced by resource limitations.

For a more in-depth explanation of logistic growth, take a look at the article on "Logistic Population Growth"!

In the natural world, exponential population growth is rare and always temporary, as it is not sustainable and all populations (even humans) are limited by density-dependent factors, mainly the depletion of natural resources, and all populations have a carrying capacity.

Density-dependent factors are limiting factors that will affect a population depending upon its density (e.g., individuals per km2). Examples include resource depletion and the increased spread of disease as populations increase in density.

In unnatural settings, exponential population growth can occur when a population has limitless resources, no natural predators, no competitors, and no other factors limiting its growth!

## The Relevance of Exponential Population Growth to Population Ecology

Understanding exponential growth is important because it helps us to predict future population sizes, estimate resource consumption, and evaluate the impact of population growth on the environment. Furthermore, exponential population growth can have significant consequences for population dynamics, such as competition for resources, changes in habitat availability, and the potential for population crashes.

Overall, understanding the relevance of exponential population growth to population ecology is crucial to develop a comprehensive understanding of how ecological systems work and how human activities can affect them.

## Exponential Population Growth Example

In living organisms, exponential population growth is most frequently observed in bacteria. However, there is another example that you are likely to be much more familiar with.

In recent centuries, the human population has experienced exponential population growth (Fig. 1). In fact, over the past 50 years, the human population has more than doubled, from 3.85 billion people in 1972 to 7.95 billion in 2022, and has more than quadrupled over the past century. This is a rare example of exponential growth in a mammalian species!

Thanks to modern medical and technological advances, much of the human population has been temporarily and unnaturally able to mitigate the negative impact that some population-depleting density-dependent factors (e.g., food availability and predation) would have on population growth.

Despite this, these factors still have a major impact on many human populations, particularly in parts of the developing world, where overcrowding, poverty, starvation, and increased pollution are largely fueled by this unsustainable increase in population on a global scale.

Eventually and inevitably, the human population will level off and produce a logistic growth curve, due to the increasing intensity of these limiting factors as the population increases. The problem is, how much damage will be done before we reach that point?

Figure 1: Exponential human population growth. Source: Population Connection

Bacteria more commonly experience exponential population growth than any other kind of organism, particularly when placed into an ideal medium. Bacteria have very fast generation times, allowing them to breed and evolve at a very high rate (this is how some bacteria quickly evolve antibiotic resistance).

Take, for example, the bacteria species Vibrio natriegens, which is the fastest multiplying bacteria known to man. V. natriegens is a gram-negative species discovered in salt marshes, such as those in the Bay of Bengal, and can double its population in under 10 minutes under optimal conditions in a lab!

Due to its extremely fast growth (twice as fast as Escherichia coli), V. natriegens has been suggested as a replacement for E. coli as a model prokaryotic organism.

Nonliving organisms, such as viruses, can also experience exponential population growth. The coronavirus, COVID-19, for example, experienced exponential growth following the beginning of the pandemic in late 2019/early 2020. This exponential growth of the virus population occurred alongside the exponential increase in the number of people infected.

A virus is a small infectious agent that can replicate only inside the living cells of an organism. Because of this, viruses are not considered living beings. Viruses consist of genetic material, either DNA or RNA, surrounded by a protein coat called a capsid. Some viruses also have a lipid envelope surrounding the capsid.

Mitigation techniques, such as social distancing and the wearing of masks, can substantially reduce the exponential population growth of the virus and the number of people infected with it (Fig. 2).

Figure 2: Exponential growth of COVID-19 cases and the potential effect of mitigation techniques. Source: Robert Signer and Gary Warshaw

## Exponential Population Growth Function

Finally, let's talk about the formula for the population growth rate.

The formula for a population's growth rate is concerned with the change in the population's size over a period of time.

This formula can be displayed as dN (difference in population size) divided by dT (difference in time), resulting in rN (per capita population growth rate).

$rN = \frac{dN}{dt}$

Sometimes, in exponential population growth, "r" is referred to as "rmax", but they both signify the same thing - the growth rate.

The equation for rN is different for exponential and logistic population growth.

• In exponential population growth, no matter how large population growth is, per capita growth rate remains constant. Therefore, the equation is simply rN.

• In logistic population growth, the population size decreases as it grows larger and approaches its carrying capacity. Therefore, in logistic population growth, we must subtract the carrying capacity (K) from the population size (N), and then divide by the carrying capacity (K) and multiply by the population size (N). So, the formula in this case is $$\frac{dN}{dt} = r_{max}(\frac{K-N}{K})N$$.

In addition, when plotting a graph for exponential population growth, a J-shaped curve is produced, while logistic population growth produces an S-shaped curve (Fig. 3).

• Exponential population growth produces a J-shaped curve because the population's growth rate remains the same as the population grows in size.

• Logistic population growth results in an S-shaped curve because the population's growth rate tapers off gradually as the population approaches its carrying capacity.

Over a long enough period of time, virtually all populations will have an S-shaped curve, even populations which may have experienced exponential growth for a short period of time previously. Thus, no populations ever experienced permanent exponential growth, as it simply is not possible on a planet with finite resources.

Figure 3: Exponential (J-shaped) and logistic (S-shaped) population growth curves. Source: Encyclopedia Britannica, Inc.

## Exponential Population Growth - Key takeaways

• A population experiences exponential growth when the per capita rate of its growth remains constant independent of the size of the population.
• In the natural world, exponential population growth is rare and always temporary, as all populations (even human) are limited by density-dependent factors.
• Over the past 50 years, the human population has more than doubled, from 3.85 billion people in 1972 to 7.95 billion in 2022. This is a rare example of exponential growth in a large organism.
• The formula for a population's growth rate is displayed as dN (difference in population size) divided by dT (difference in time), resulting in rN (per capita population growth rate).
• When plotting a graph for exponential population growth, a J-shaped curve is produced.

#### Flashcards in Exponential Population Growth 16

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When can exponential growth occur in a population?

Exponential growth can occur in a population when resources are unlimited.

Which population is most likely to have exponential growth?

Usually, bacteria and viruses exhibit exponential growth.

What is exponential population growth?

A population experiences exponential growth when the per capita rate of its growth remains constant independent of the size of the population. This results in the population growing larger at a very fast rate.

When does the exponential growth of a population stop?

The exponential growth of a population usually stops when the number of individuals is big enough to make a dent in the resources. As resources get used up, population growth slows down.

Is the human population growth exponential or logistic?

In recent centuries, the human population has experienced exponential population growth. In fact, over the past 50 years, the human population has more than doubled, from 3.85 billion people in 1972 to 7.95 billion in 2022, and has more than quadrupled over the past century. This is a rare example of exponential growth in a mammalian species!

## Test your knowledge with multiple choice flashcards

What shaped curve does logistic population growth produce?

What is the formula for a population's growth rate?

What shaped curve does exponential population growth produce?

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