Deer die continuously! The same textbook uses aphids as the paradigmatic example of an exponentially growing population because their births are continuous. But, exponential growth assumes deaths and births occur at the same rate, and aphid birth and death rates vary wildly with age. If I chose a time step of an hour, then the births would be pulsed again. Moreover, I can start with continuous rates, and calculate the equivalent discrete time model that makes identical predictions.
Neither model is perfectly correct. We are mistaking a difference in the map discrete vs. Ecologists seem particularly prone to map-territory errors maybe more on that later.
Maybe this is because of our physics envy. So if the emissions are growing exponentially, they are growing by about 2. As a note on rounding, notice that if we had rounded the growth rate to 2. A very small difference in the growth rates gets magnified greatly in exponential growth.
For this reason, it is recommended to round the growth rate as little as possible. If you need to round, keep at least three significant digits — numbers after any leading zeros. A growth rate of 0. In the previous example, we had to calculate the 10th root of a number.
Many scientific calculators have a button for general roots. It is typically labeled like:. Try it on yours to see which to use — you should get an answer of 2. If your calculator does not have a general root button, all is not lost.
You can instead use the property of exponents which states that:. To do this, type:. They are necessary to force the desired order of operations when using your calculator. The number of users on a social networking site was 45 thousand in February when they officially went public, and grew to 60 thousand by October.
If the site is growing exponentially, and growth continues at the same rate, how many users should they expect two years after they went public? Looking back at the last example, for the sake of comparison, what would the carbon emissions be in if emissions grow linearly at the same rate? To find d , we could take the same approach as earlier, noting that the emissions increased by million metric tons in 10 years, giving a common difference of 22 million metric tons each year.
Alternatively, we could use an approach similar to that which we used to find the exponential equation. This tells us that if the emissions are changing linearly, they are growing by 22 million metric tons each year. The yearly increase in the northern YNP bison population between and can be described as exponential growth. A population that grows exponentially adds increasingly more individuals as the population size increases.
The original adult bison mate and have calves, those calves grow into adults who have calves, and so on. This generates much faster growth than, say, adding a constant number of individuals to the population each year.
Exponential growth works by leveraging increases in population size, and does not require increases in population growth rates. This meant that the herd only added between 4 and 9 individuals in the first couple of years, but added closer to 50 individuals by when the population was larger and more individuals were reproducing. Speaking of reproduction, how often a species reproduces can affect how scientists describe population growth see Figure 2 to learn more.
Figure 2: Bison young are born once a year — how does periodic reproduction affect how we describe population growth?
The female bison in the YNP herd all have calves around the same time each year — in spring from April through the beginning of June Jones et al. This type of periodic reproduction is common in nature, and very different from animals like humans, who have babies throughout the year. When scientists want to describe the growth of populations that reproduce periodically, they use geometric growth. Geometric growth is similar to exponential growth because increases in the size of the population depend on the population size more individuals having more offspring means faster growth!
Exponential growth and geometric growth are similar enough that over longer periods of time, exponential growth can accurately describe changes in populations that reproduce periodically like bison as well as those that reproduce more constantly like humans.
Photo courtesy of Guimir via Wikimedia Commons. The power of exponential growth is worth a closer look. If you started with a single bacterium that could double every hour, exponential growth would give you ,,,, bacteria in just 48 hours! The YNP bison population reached a maximum of animals in Plumb et al.
That's more than thirteen times larger than the largest population ever thought to have roamed the entire plains region! The potential results may seem fantastic, but exponential growth appears regularly in nature.
When organisms enter novel habitats and have abundant resources, as is the case for invading agricultural pests, introduced species , or during carefully managed recoveries like the American bison, their populations often experience periods of exponential growth. In the case of introduced specie s or agricultural pests, exponential population growth can lead to dramatic environmental degradation and significant expenditures to control pest species Figure 3.
Figure 3: If this much money is being spent on something, it must be important! Understanding population growth is important for predicting, managing, monitoring, and eradicating pest and disease outbreaks.
Many introduced species, including agricultural pests and infectious diseases, grow exponentially as they invade new areas, and billions of dollars are spent predicting and managing the population growth and dispersal of species that have the potential to destroy crops, harm the health of humans, wildlife, and livestock, and affect native species and natural ecosystem functioning.
Let's think about the conditions that allowed the bison population to grow between and The total number of bison in the YNP herd could have changed because of births, deaths, immigration and emigration immigration is individuals coming in from outside the population, emigration is individuals leaving to go elsewhere.
The population was isolated, so no immigration or emigration occurred, meaning only births and deaths changed the size of the population.
Because the population grew, there must have been more births than deaths, right? Right, but that is a simple way of telling a more complicated story. Births exceeded deaths in the northern YNP bison herd between and , allowing the population to grow, but other factors such as the age structure of the population, characteristics of the species such as lifespan and fecundity , and favorable environmental conditions, determined how much and how fast.
Changes in the factors that once allowed a population to grow can explain why growth slows or even stops. Figure 4 shows periods of growth, as well as periods of decline, in the number of YNP bison between and Growth of the northern YNP bison herd has been limited by disease and predation, habitat loss and fragmentation, human intervention, and harsh winters Gates et al. Figure 4: The YNP bison population has increased and decreased in size over the past century in response to factors such as disease, predation, habitat loss, human intervention, and environmental conditions.
Scientists with the National Park Service and Colorado State University recently published these data showing both the number of bison counted in YNP on an annual basis blue dots and the number of bison removed from the population grey columns for the purposes of herd management.
Management of the bison population in YNP has been fairly controversial — to learn more about this controversy check out Plumb et al.
Factors that enhance or limit population growth can be divided into two categories based on how each factor is affected by the number of individuals occupying a given area — or the population's density. As population size approaches the carrying capacity of the environment, the intensity of density-dependent factors increases. For example, competition for resources, predation, and rates of infection increase with population density and can eventually limit population size.
Other factors, like pollution, seasonal weather extremes, and natural disasters — hurricanes, fires, droughts, floods, and volcanic eruptions — affect populations irrespective of their density, and can limit population growth simply by severely reducing the number of individuals in the population. The idea that uninhibited exponential growth would eventually be limited was formalized in by mathematician Pierre-Francois Verhulst.
While studying how resource availability might affect human population growth, Verhulst published an equation that limits exponential growth as the size of the population increases. Verhulst's equation is commonly referred to as the logistic equation , and was rediscovered and popularized in when Pearl and Reed used it to predict population growth in the United States. Figure 5 illustrates logistic growth: the population grows exponentially under certain conditions, as the northern YNP bison herd did between and , but is limited as the population increases toward the carrying capacity of its environment.
Check out the article by J. Vandermeer for a more detailed explanation of the equations that describe exponential and logistic growth. Figure 5: This curve describes logistic growth. The population size grows exponentially for a while like the bison in Figure 1 , but then it slows down and levels off when as it approaches the carrying capacity K. Logistic growth is commonly observed in nature as well as in the laboratory Figure 6 , but ecologists have observed that the size of many populations fluctuates over time rather than remaining constant as logistic growth predicts.
Fluctuating populations generally exhibit a period of population growth followed a period of population decline, followed by another period of population growth, followed by Figure 6: Logistic growth curves as seen in real populations. Populations growing according to logistic growth are observed in laboratory populations Paramecium and Daphnia as well as in nature fur seals.
In the Daphnia example, it appears that the population size grew to more than individuals and then declined, leveling off at around — individuals.
What factors might have caused this pattern? Populations can fluctuate because of seasonal or other regular environmental cycles e. For example, Elton observed that snowshoe hare and lynx populations in Canadian boreal forests fluctuated over time in a fairly regular cycle Figure 7. More importantly, they fluctuated, one after the other, in a predictable way: when the snowshoe hare population increased, the lynx population tended to rise plentiful food for the lynx! Many populations, over time, exhibit periods of growth and decline.
Cyclic changes in population growth can be caused by seasonal, or other environmental changes, or can be driven by density-dependent processes, such as predation, like the snowshoe hare and lynx example depicted here.
It is also possible for populations to decline to extinction if changing conditions cause death rates to exceed birth rates by a large enough margin or for a long enough period of time.
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