Christopher X J. Jensen
Associate Professor, Pratt Institute

# Eco 101: Exponential Growth & Decay

Posted 27 Jan 2017 / 0

In biology, a population is a group of organisms that belong to the same species and occupy a defined land area (or volume of air or water). These populations have the tendency to grow and shrink over time. This is because the rate at which members of a population are producing new offspring is seldom exactly equal to the rate at which members of that population are dying off.

There are a variety of factors that can influence the birth rate and death rate of a given population of organisms. Scarce resources can both depress birth rates and increase death rates. Predators and parasites are also a source of increased death. Often, biologists consider how a population might grow in the absence of these external factors. What growth rate can a population achieve in a given environment if resources are unlimited and there is no threat from parasites or predators? This rate of growth is called the intrinsic growth rate, and can be measured by estimating the maximum birth rate and minimum death rate in the population:

$intrinsic\:growth\:rate = birth\:rate - death\:rate$

It’s important to note that both the birth and death rates are a per capita measure: rather than telling you how fast the entire population is reproducing and dying off, the intrinsic growth rate combines the probability that each individual will produce one or more offspring with the probability that each individual will die over a given time interval. This should make sense: in a larger population, there should be both more births and deaths because there are more individuals reproducing and dying.

Notice that this intrinsic growth rate represents the “ideal scenario” for a population in a particular environment. But even under ideal conditions, there are limitations as to how fast a population can produce offspring and how long individuals can survive. How a given species produces offspring will dictate its maximum birth rate. Some species mature rapidly and produce many offspring, while others take longer to reach reproductive maturity and produce fewer offspring. And all species senesce, which means that even in the absence of external threats a certain proportion of the population will periodically die off.

A higher birth rate makes the intrinsic growth rate of mice much higher than that of humans

### What happens when births exceed deaths?

A species that is well-adapted to its environment should have a positive intrinsic growth rate. To have survived for long periods of evolutionary time, a species will have to have evolved the capacity to reproduce new individuals faster than old individuals die off (at least under ideal conditions). Populations that have a positive intrinsic growth rate display a characteristic pattern (or ‘model’) of change called exponential growth:

Populations with greater intrinsic growth rates grow to disproportionately larger sizes over time

What’s interesting about exponential growth is that it rapidly accelerates over time. In small populations, the population initially expands very slowly. But as a population increases in size over time, the rate of expansion dramatically increases. This is because populations display geometric growth, which is captured in this equation:

$rate\:of\:population\:change =intrinsic\:growth\:rate \times population\:size$

It makes sense that biological populations grow geometrically: both birth and death are per capita processes, so the larger a population is, the larger the difference will be between the actual number of new offspring produced and the actual number of deaths. Even populations with very low intrinsic growth rates expand rapidly once they become very large.

The geometric property of exponential growth also means that very small differences in the intrinsic growth rate can lead to very dramatic differences in population growth. In the graph above, intrinsic growth rate (represented as “r”) varies from 2% (0.02) to 6% (0.06) per time interval. Initially, all populations grow very slowly, and their sizes are nearly indistinguishable. But over time the geometric nature of exponential growth leads to very dramatic differences in the population sizes: although a population growing at 6% has an intrinsic growth rate that’s only 50% greater than a population whose intrinsic growth rate is 4%, after 100 time intervals the population growing at 6% is nearly eight (8) times larger than the population growing at 4%. Should these populations continue to grow exponentially, this difference in size would grow to be even more dramatic.

### What happens when deaths exceed births?

Populations don’t always enjoy ideal conditions. Extinction is a natural process that has taken out most of the species that have ever existed. Why do populations go extinct? The specific causes of extinction are numerous and diverse, but the fundamental phenomena that causes extinction is always the same: if its death rate exceeds its birth rate for a long enough period of time, a population will go extinct. On the way towards extinction, a population with a negative intrinsic growth rate follows a pattern of change (or ‘model’) called exponential decay:

Populations with more negative intrinsic growth rates initially decline more rapidly

Exponential decay looks like the mirror image of exponential growth: the population declines most rapidly early on, and then progressively slows down in its decline. Technically speaking, a population showing exponential decay never gets to zero, but practically speaking — because there is no such thing as a “fraction of an individual” — the population will eventually go extinct.

It is clear that being subject to excessive predation or parasitism, or the complete loss of resources, can lead to death rates that exceed the birth rate. But do populations ever experience exponential decay in the absence of these factors? They can if their abiotic environment shifts radically enough to render their evolved adaptations inadequate. A great example of this phenomenon is coral bleaching. Corals can bleach for a variety of reasons, but warming ocean temperatures and increasing acidification are major causes. These abiotic changes can dramatically drive up death rates and lead to an exponential decay of coral populations, even as the corals are not subject to increased predation/parasitism or increased competition for scarce resources.

Increased ocean temperatures and ocean acidification both can cause corals to bleach, dramatically increasing their death rates

### Real-world limitations of the exponential growth model

The ideal conditions that we assume when we depict a population as growing exponentially are rarely present for very long. A population introduced to a recently-disturbed, open habitat may be able to grow exponentially for a time, but eventually its rate of growth will slow.

Imagine a flask full of bacterial media (an optimal mix of necessary resources) into which we introduce a few individual bacterial cells. Initially the supply of resources in that flask will be so great compared to the number of individual bacteria that the population will grow as though resources are infinite. But that drives exponential growth, and in rapidly-reproducing bacteria that means a very rapid spike in population density. It won’t take long for the population to consume all of the resources available. We can design a lab set-up that partially deals with this problem if we continuously drip in a supply of resources. But even this set-up can’t keep up with an exponentially growing bacterial population, because as the population grows it will require an exponentially-increasing supply of resources. It’s inevitable that the growth of the bacterial population will slow down. What’s true of bacteria in the controlled conditions of the lab is equally true for organisms in the wild.

Bacteria in nutrient media initially grow exponentially, but eventually their growth is limited by resource availability

Why do all populations eventually stop growing exponentially? The general answer is that their population growth rates are density-dependent. That is to say that birth and death rates are not unchanging (as the density-independent model of exponential growth assumes): as the population increases in density, its birth rate is likely to decline and its death rate is likely to increase. There are a variety of reasons why this is so, but chief among them is is that there is more intense competition for resources at higher densities, and if you have a lower chance of obtaining the resources you need you are less likely to make offspring and more likely to die.

Has any population been able to maintain a pattern of exponential growth for extended periods of time? One candidate may be our own species, Homo sapiens. Since the advent of agriculture our population has expanded for many thousands of years, displaying a growth pattern that appears to be exponential. How have we avoided the density-dependent limitations that seem to hold back every other biological population on the globe? Well, we have a remarkable propensity for inventing cultural practices that allow us to expand our resource base. Chief among these are dramatically-improved farming techniques that allow us to extract more and more food out of our farms (“intensifying” to increase our resource supply). We are also the species most adept at engineering our own environment, a property of our species that allows us to capitalize on our expanding population as an expanding workforce that converts wild lands to domesticated farms (“extensifying” to increase our resource supply). But that doesn’t mean that the human population has the potential to grow exponentially forever. If we continue to have birth rates that exceed our death rates, eventually we will run into some biophysical limitation on our ability to continually increase our resource supply.

### Real-world limitations of the exponential decay model

Does the model of exponential decay also unrealistically ignore density effects? At first glance, the importance of density dependence in a shrinking population is less apparent. After all, don’t smaller populations experience reduced competition for resources than larger ones? They do, but there are other density-dependent factors that can shift the pattern of growth in exponentially decaying populations. Once a population’s density becomes quite low, individuals may have a very hard time finding a mate, especially if the population remains spread throughout its original range. And small biological populations also suffer from lowered genetic diversity, which can lead to lowered survival rates of offspring. This means that as a population decays exponentially, at lower densities it may experience even lower birth rates and/or even higher death rates. This property of low-density populations — called the Allee effect — has important implications for conserving populations that are at risk of extinction.

This post was written for students in my Ecology course. As such, a lot of nuance is excluded from this post. If you are interested in a more in-depth discussion the Wikipedia page on exponential growth includes a lot more, including the formal mathematrical representations of this kind of growth.

This post is part of my Eco 101 series

Image credits: