CHAPTER OVERVIEW 10: Population modeling Learning Objectives Review the basic arithmetic and algebra needed to think quantitatively about populations using mathematical models. Unpack the concept of the demographic rates of a population, including survival, birth rate, immigration, and emigration, and how these rates can be used to determine the growth rate of a population. Introduce two key patterns of population growth: exponential and logistic growth. Show how information on population growth can be used to project change in a population into the future. Detail a classic tool used by human demographers and ecologists to study population change: life tables. 10.1: Prelude - Learning the Math of Population Models 10.2: Demographic rates 10.3: Scientist Spotlight - Erin Satterthwaite 10.4: Overview of Population Growth Models 10.5: Geometric and Exponential Growth 10.5.1: Logistic population growth 10.6: Projecting population growth 10.7: Life Tables 10.8: Population Models Practice Exercises Summary Central to population ecology is the mathematical concept -- and biological reality -- of exponential population growth. The centrality of exponential growth in ecology and evolution was recognized by Darwin and plays a key role today in applied ecological decisions such as the management of invasive species, harvest limits for hunted species, and the management of endangered species. To appreciate the ecology of populations fully we therefore need to do some math. In this section, we will gently ramp up the skills and concepts we need to use math to think quantitatively about how populations change and can be effectively managed and preserved. First, we'll review the basic math we'll use throughout the chapter. Second, we'll build up the ecological concepts of demographic rates such as survival and birthrate and how they are represented mathematically. These demographic rates will then be combined into a mathematical model of population growth that shows how populations change over time due to survival and reproduction. Third, we'll show how once you have determined a population growth rate, you can predict how a population will change over time and demonstrate that it will grow exponentially. Fourth, we'll lay out a classic mathematical tool used both by human demographers and population ecologists: life tables. 10: Population modeling is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by LibreTexts.
10.1: Prelude - Learning the Math of Population Models Math and mathematical models in ecology The processes of population changes in space and time are called population dynamics. Over time, a population or species may expand its geographic range, colonize a new isolated habitat patch like an island, or disappear from areas where it occurred previously (extirpation). Even if a population or species is only found in the same locations as previously, its population size may increase or decrease, become stable, or cycle regularly. Ecologists frequently use mathematical models to describe population dynamics. These models can be used to describe the trajectory of population growth when resources are abundant, its maximum size when resources are limited, or how rapidly in space it expands into new territory. Mathematical models can be intimidating at first, but you can start learning how they work and how to use them yourself - with the basic tools of arithmetic and algebra most people learn in high school. Basic mathematical models can be built using algebra The term mathematical model perhaps sounds fancy, but in many of their forms these are just equations manipulated using standard algebra to allow us to think about how things change. In this book we'll use models for helping us think about things like populations change over time, what impact predators have on a population of its prey, and how two species using the same resource can coexist. Here are some examples of the level of math needed to get started working with ecological models. Elsewhere we'll investigate the applications of math in ecological modeling. Prior to studying ecology, many individuals become familiar with the equation: This equation expresses a relationship between the variables x and y. In biology, we'd call this a mathematical model and it allows us to predict x and y for particular unknowns, such as: x is the number of birds in a population this year and y is the number next year. Or, x could be the size of a plant and y is the number of seeds it produces. A mathematical model contains both variables and parameters. In models like ours above, x and y can be just about anything that varies between organisms or can be measured in nature: height, weight, number of species, size of a habitat, length of a river, the concentration of a toxin. In contrast, M and B are parameters that are generally fixed for a given situation. Solving the variable y given x + B Using the above equation, if I tell you M = 1.2 and B = 10, can you calculate what y is when x = 10? We can plug the values M = 1.2 and B = 10 into the equation: We can then calculate y if x = 10. All the values added to the equation give us this: We then do the multiplication: The last step of addition tells us: Solving for an unknown parameter What if I told you y = 220, x = 100, and B = 10; how would you calculate what M was? We could start like this with our known parameter (B = 10) and our values for x and y: We then do some algebra. First deal with the 10 by subtracting it from both sides:
On the left, 220 - 10 = 210, and on the right 10 - 0 = 0, so this gives us: Now divide both sides by 100. Which gives us: because 100/100 = 1. Then divide 210 by 100: Not too bad? The math behind basic models in ecology is often not much more involved than this. If you can work through the steps above, you can work through the population models discussed in this book. Using mathematical models for prediction A common use of mathematical models in ecology is prediction. For example, you can often predict the number of seeds produced by a plant using an equation like this: Which is the same y = M * x + B equation we just used, where y = number of seeds and x = plant size. If M = 10, B = 15, and plant size = 30 cm tall, how many seeds would be produced? As before we can first add the values for the parameters M and B to to model: We then plug in our particular plant size of 30 cm and do the math. First multiplication: Then addition:
So, if the equation (number of seeds) = 10 * (plant size) + 15 is accurate, a plant that is 30 cm tall would produce 315 seeds. Basic population models Population dynamics can be described with mathematical models Populations change due to processes such as the deaths of current members of the population and the birth of new members (offspring). Discussions of population dynamics typically begin by writing out an equation which describes all of the key processes that impact population change, such as births and deaths. Most organisms live in seasonal environments, and we frequently consider changes in populations over the course of a single year, which we'll call a one-year time step. Let's start building up some basic equations to describe changes in populations (population dynamics), calling the number of organisms right now NNow and the number next year NNext year (The little "Now" and "Next year" are called subscripts). We can write out how a population change like this:
When we write equations like this, we always need to remember that often represent an idealized situation; rarely can we know how large a population currently is (NNow), and it's much harder to determine exactly how many died or were born in a given year (We'll come back to these difficulties and their resolution in the next section on demographic models). In addition to births and deaths, populations can also increase due to the arrival of individuals from different populations (immigration), and decrease due to the exit of current members (emigration). We therefore expand our idealized population equation to be: To make these equations more compact we often write using a more clearly expressed notation, where a subscript of "t" = a certain time, and "t+1" equals the following time period. Often this time step is one year, but it could be any period of time relevant to the biology of an organism or is convenient to the researcher. For example, many insects grow rapidly and some go through multiple generations in a single summer, and a relevant time step could therefore be months or even weeks. Using subscripts we can rewrite our equation as: where and Often in textbooks births and deaths are given their own symbols, B and D, as are immigration (I) and emigration (E). You'll therefore see this equation: or often just with subscripts only on the N's. Notation Alert! Different authors and textbooks unfortunately use different notation, so it's important that everyone is clear what all their symbols mean, and that readers carefully determine what the symbols mean. In this case, it needs to be emphasized that N, D, B, I, and E all represent absolute numbers of individuals - they are meant to represent counts of organisms, not rates. For example, just as Nt is a count of all the individuals in the population and must be a whole number like 1000, B is a count of the number of births and must be a whole number like 5000. In contrast, a rate would be the number of births per individual of that population. In this case the birth rate would be B/N = 5000/1000 = 5. Later we will use rates, like births per year, to build demographic models. It is useful to remember that when you're working with addition and subtraction you can move terms around in the equation and not alter the math. So our previous equation can be changed to this by reordering terms: Similarly, we can add parentheses to help us organize things without changing the meaning. In the equation below, the number of births (B) and deaths (D) are grouped because these are opposing processes; similarly we group E (emigration ) and I
(immigration). Again, this does nothing to change the meaning. I can state the fact that these two equations are identical with an equality like this: \[N_t + B - D + E - I = N_t + {(B - D)} + {(E - I)} Immigration and emigration are hard to study In the prior example, immigration and emigration were ignored. This is because immigration and emigration are very difficult to study in a population. Most populations are demographically open to immigration to some degree, especially animal populations. An open population is one that regularly receives immigrants from a nearby population. Only populations that occur on oceanic islands that are distant from the mainland or occur in other isolated chunks of habitat are likely to be closed and receive few or no immigrants. Many habitats do have fairly rare rates of immigration, such as most islands, lakes, and isolated fragments of habitat that are surrounded by inhospitable conditions such as human structures. For example, Abuko Nature Preserve in The Gambia, West Africa is surrounded on all sides by the suburbs of the capital, Banjul. Except for some birds, all animals found in Abuko were born there, will live their whole lives there, and never leave. It's possible that a brave monkey may run off, but it would have to travel a considerable distance to reach the next nearest fragment of forest. For plants, most seeds will fall onto the forest floor of the preserve except those eaten by birds. These birds may happen to fly to one of the nearest forest fragments and defecate there, but it's unlikely. Definition: Migrant It should be noted that for human populations the general term migrant is often used to describe people moving to a different country. In ecology, the terms migrant and migration are often reserved for species that undergo seasonal movements between habitats.
: Abuko Nature Preserve, The Gambia, West Africa. Surrounding the core forest (dark green) is more open scrub
habitat (dark brown). The preserve is entirely surrounded by residences. Source: Google Earth: https://bit.ly/abuko00
When habitats are isolated like this, dispersal events are rare enough that they can be ignored for many ecological purposes. It should be noted, however, that while rare dispersal between populations doesn't have much impact on population size, it can still have an impact on population genetics and evolution. Population geneticists and evolutionary biologists therefore are often interested in dispersal events that a population ecologists might ignore. A population may therefore be more or less demographically closed, but genetically open.
: Forests nearest to Abuko Nature Preserve, The Gambia, West Africa. Most of the terrain between the forests is either residences or agricultural land. Source: Google Earth: https://bit.ly/abuko1
Ignoring immigration and emigration simplifies population models Research on populations where immigration and emigration are minimal - or can justifiably be ignored - is convenient. This allows us to simplify our model of population dynamics from to one without E and I: We can therefore mathematically define a closed population as one where E = 0 and I = 0, and an open population as one where E > 0 or I > 0. Moving forward, almost all of our population and other ecological models will ignore immigration and emigration. We will therefore be studying closed populations. Again, few populations are truly closed! Leaving E and I out of a model does not mean they never happen, just that they are not key players in the processes we are interested in or we are comfortable ignoring them. While we have used a lot of symbols and equations, what we have done above presents two key creative steps in ecological modeling which are not necessarily reliant on math: 1. Brainstorming and writing out all of the key processes that can impact a population (B, D, E, I) 2. Simplifying the model by applying reasonable assumptions to make it easier to collect relevant data. While math is important to the process, using your ecological imagination and intuition to determine what processes the equations should represent is just as important.
We often frame population dynamics in terms of the amount of change Often our equation of population dynamics gets converted from focusing on the number of individuals
to the change in the number of individuals over time due to births and deaths. We therefore define population change as
Where is the Greek letter, Delta, which is used throughout science to mean "change."
Derivation Dance For the curious: we can arrive at the equation
from both sides of our equation. Our equation was:
Now let's rearrange terms for simplicity:
is usually written as , where is the Greek letter, Delta, which is used throughout science to mean "change".
1. B = D (B is the same as D)? 2. B > D (B is greater than D)? 3. B < D (B is less than D)?
If B = D, the number of births is balanced out by deaths and the population has not experienced a net change in size
When B > D, births exceed deaths, is positive and the population gets bigger. When B < D, is negative and the population
This all may look fairly simple if math comes easily to you, but is actually kind of profound for ecological research: you can ignore how large a population actually is and still know about its population dynamics by keeping track of just births and deaths. Conversely, if you know how much a population has changed in size, you can know whether there were more births than deaths, or if deaths exceed births. As noted before, tracking individual births and deaths is hard, so gaining insights into the net number of births and deaths just from changes in population sizes is very useful. Indeed, a whole branch of population modeling is based on this (Morris and Doak 2002).
10.1: Prelude - Learning the Math of Population Models is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by LibreTexts.
10.2: Demographic rates Population dynamics are often described in terms of demographic rates Population ecologists often collect data on demographic rates: birth rates and death rates (or the converse of death rate, survival rate). Sometimes ecologists call these vital rates. Formally these are called per capita rates because they refer to the frequency of an event per individual of the population, such as births per person. Another common way to calculate rates, though not in ecology, is per annum. This is the frequency of something per year. A model set up in terms of demographic rates is called a demographic model. Study Hint While reading about mathematical models, write out the equations and make sure you understand each step of the arithmetic and algebra. The most direct way to collect this information is if you know all of the individuals in a population (N) and you can count the number of individuals that are born and which subsequently survive until they are old enough to breed. Often we call this the birth rate, but this is not totally correct. It should really be called the "birth and survival until the next census period" rate. We'll stick with convention and call it the birth rate and define it like this:
That is, the per capita birth rate is equal to the number of births in year t ( ) divided by the number of adults ( ) in the
population. is a fraction, and can take on values as low as zero.
indicates that no offspring were born, or that all offspring
died before they could reproduce. indicates that two offspring were born for each adult.
Once we define as the ratio of births to total population size, we can use algebra to set up an equation for the prediction of the actual number of births:
That is, the number of organisms born and entering the population at time t ( ) is equal to the number of organisms in the population ( ) times the number of offspring produced per organism. Stated another way, the total number born ( ) are often expressed as a proportion ( ) of the number alive now ( ). For example, at the beginning of 1996 there were 249 elephants in the Addo Elephant National Park in South Africa (Whitehouse & Hall-Martin 2000). Over the last several decades the average birth rate was 0.0693. We can therefore use our birthrate equation to predict the number of births in 1996:
Indeed, there were 17 elephants born that year.
: Elephants in Addo Elephant National Park. "Elephants at the Hapoor Dam in the park" by NJR ZA is licensed by CC BY-SA 3.0.
The per capita death rate can be similarly defined:
and therefore if we know the death rate and the current population size we can predict the number that will die over the next time period:
The mean death rate in Addo was 0.0175; 248 elephants * 0.01750 = 4.3 predicted deaths. This is slightly fewer than the 5 that actually occurred. So far we've only been doing some minor algebraic rearrangement of these demographic rates, but seeing how this done will help us understand how to formulate demographic models for population dynamics. Now that we're thinking in terms of rates, we can rewrite our main population dynamics equation, which was
in terms of demographic rates, like this:
What this means is that if we can estimate and , we can estimate population change without having to count up the total number of deaths and births. In Addo Elephant National Park extensive data collection allowed researchers in the 1990s to account for all births, deaths, and surviving animals. However, if they had been unable to conduct a full population survey in 1997 they could've predicted the size of the population as
This yields an estimated population size in 1997 of 262; the actual population size was 261.
Math Review If we want, we can factor out the
from the previous equation and get an equation like this:
Ecologists usually calculate survival rates, not death rates
represents the total number of organisms that died in a given year t. The number that survived is therefore
When studying population change ecologists typically work in terms of survival rates, often written as , the Greek letter "Phi." The survival rate can be thought of as either a frequency or a probability. If you have 100 organisms and 50 survive to the next year, the survival rate is 0.50. Similarly, if a single organism has a 50% chance of survival over the next year, the survival rate is 0.50.
The number of deaths in a population plus the number of survivors sums to the total current population size. We therefore can use our death rate and survival rate and write:
Note that in this equation we only have We can do some algebra and move
, and not dealing right now with . using subtraction to the left of the equals sign:
and see that the population size minus those which died
Why all the fuss? The previous equations aren't very profound - we're just accounting for the fate of all of the organisms currently in the population. Functioning population models, though, often rely on many basic mathematical manipulations in order to be set up, and we're taking the time to build up our intuition about how all the various pieces of these models work.
Another useful demonstration is this: we can also start as we just did with
We can distribute the on the right to give us
cancels out on the left and the right. This gives us
mean? Again, mathematically it's a simple statement: the proportion which lived and the proportion which
died over a single time step must total 1. However, keeping this manipulation in mind allows us to write out a demographic
Let's put all of these pieces together. We'll start again with our key demographic equation using rates:
Next, to make this clearer we'll put related terms next to each other:
Now factor out from the stuff in the parentheses:
This means that the number of individuals in the future .
In Addo Elephant National Park mean mortality was 0.0175. Mean survival is therefore 1-0.0175 = 0.9825. We therefore predict population size in the future as
We've now distilled down our demographic equation to predict future population size using the present population size , survival and birth rate . While it's taken a fair bit of working with the equation, we've now reached an important breakthrough: creating a population model without determining population size at all!
Math Review If we want to be fancy we can do a bit of algebra and factor out the
Let's take stock of what we've done. So far we've gone from our initial population models that show up in most biology textbooks but isn't really used by working ecologists:
and then simplified it by ignoring immigration comfortably be ignored. This gives us:
and emigration , or found a population where they don't apply or can
Tracking total B and D is really hard -- harder than even some algebra to get:
-- so we've done some simple thinking about population processes and
where and are usually estimated from a subset of the population. If we are confident that our survival and birth rate estimates are good, we can predict population size in the future. However, we're still relying on estimates of population size which are still costly. For example, estimating the population sizes of large animals that live in open habitats such as polar bears and elephants often requires using aircraft. For animals that live in forests it can be very difficult to determine population sizes except over small areas. In the case of plants, populations are often so large that population size can only be determined for small, isolated populations. Luckily, there are some mathematical tools we can use to build meaningful population models that don't require population sizes to be estimated. To set this type of model up we'll introduce a core concept in ecology: the population growth rate. Population dynamics are frequently described using the population growth rate Population ecologists are frequently interested in both the absolute number of organisms (e.g. , ) and also the rate of population change over time, usually referred to as the population growth rate. The Greek letter "L" called "lambda" ( ) is used to represent this rate. If you have been following a population closely over time and have complete censuses you can calculate ( ) directly using the size of the population at one time point and at a previous time point :
is therefore the ratio between two population sizes. In situations where population sizes have been estimated, can be calculated directly from these data and used in subsequent models. In other cases, demographic rates (e.g. and ) are used to calculate it.
Case study: Calculating lambda ( ) for Kirtland's Warbler:
The Kirtland's warbler has a small geographic range and is a habitat specialist. It therefore occurs in very specific habitats, so the
approximate total number of individuals could be monitored relatively easily. In 2011 there were 1828 Kirkland's warbler males,
and in 2012 there were 2090. We can therefore set up our equation with
While the warbler population grew from 2011 to 2012, the next year in 2013 only 2020 singing males were counted. Therefore
Once we have estimates for , we can make predictions about future population sizes and project population dynamics into the
future. For as many years as possible, we can calculate
; this isn't always possible because of gaps in the data, e.g. we can't
A small bird with a pointed beak, yellow belly, and gray back is sitting on the ground littered with twigs. : Male Kirtland's Wabler. "Kirtland" by Bjamoros is licensed under CC BY-SA 3.0.
The value of lambda summarizes population dynamics
means that both adult survival and reproduction are 0. If
Below is a plot of the population time series next to a histogram of all of the values calculated from the time series. The time series starts in the 1970s on the left when researchers began conducting counts of all singing male Kirthland's Warblers each year. For each pair of years, was calculated to make the histogram.
Figure growth rates
: A time series (left) of the number of male Kirtland's warblers recorded and a histogram (right) of all population calculated from the time series. The histogram shows us the distribution of observed lambda values over the ~50 year time series when KIWA males were counted every year.
Exercise What do you notice in the KIWA Lambda histogram? Why is the red line plotted and what is happening to the population at the red line?
Answer The red line 1.0 is plotted because when by births.
, a population is staying the same size because deaths are being balanced out
Population growth rates can be calculated from demographic rates As noted previously, it is often very difficult - if not impossible - to determine actual population sizes. For KIWA, researchers worked very hard to determine the number of males that were singing each year, but they were never completely sure if they found all of them. KIWA population is currently growing, and researchers are no longer collecting population size data, which allows allocation of research money to answer other questions, such as the expansion of the species' range into Wisconsin and Canada. While still very challenging, it's often easier to determine demographic rates for a subset of the population, such as survival and reproduction, rather than population size. Several studies have captured KIWA males, marked them with bird bands and attempted to re-capture them each year to estimate survival rates . This approach for calculating survival using mark-recapture data is similar to the one discussed in the previous chapter for calculating population size. Researchers have also found nests and
determined how many baby KIWA are born per nest, and what the survival rate is for those birds until they are one year old and can breed. This gives an estimate of .
: Adult Male Kirtland's Warbler being banded. Source: Source: Hanna (2015): Kirtland's Warbler Banded as Nestling in Wisconsin Confirmed in Bahamas A Field Update. Photo by J. Trick. https://www.fws.gov/midwest/GreenBay...April2015.html
With these estimates we can calculate population growth rate without knowing population size as:
Derivation Dance To show that can be defined with demographic rates as
We can distribute the division of like this
This cancels out entirely from the right-hand side
we start with our previous demographic equation:
represents a combination of both survival of adults from one year to the next plus how many offspring are produced per adult and survive to reproduce themselves (b). We now have a quantity we are very interested in, the population growth rate , in terms of parameters that aren't too hard to calculate: the survival rate and birth rate ( and b). This means we can understand population dynamics without needing to conduct a complete census and count every single organism -- just as long as we can track survival and reproduction on a representative subset of the population.
Alternative Derivation We could also do our math this way. We can start with and factor out on the right to be: We then divide both sides by This again gives us which we write as Case study: Calculating Kirthland's Warbler with demographic data We can estimate for any species if we have estimates of its adult survival rate and its birth rate. Survival for Kirtland's Warbler (KIWA) is around 67%, or 0.67. This means on average that if we have 100 adult KIWA nesting in a forest, we'd expect to see 67 of them again next year. Equivalently, we can say that an adult bird has a probability of surviving of 0.67. The birth rate (b) is tricky to estimate for a number of reasons. Recall that b should really be called the "born and survives to reproduce" rate. Incorporating both the number of baby KIWA that hatch from eggs and their probability of surviving for one year until they can reproduce, the birth rate (b) is about 0.74. Population growth rate is therefore Since >1 we'd predict that the population of KIWA will be growing. We aren't actually counting all the birds, however. 10.2: Demographic rates is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by LibreTexts.
10.3: Scientist Spotlight - Erin Satterthwaite Relative to the offspring of other species, humans enter the world utterly defenseless. Emerging from a 9-month gestation period, we arrive completely dependent upon our caregivers. In contrast, newly hatched (or birthed) marine larvae are amazingly independent. Swimming, something humans may never learn in their lifetime, is an elementary skill required for the survival of mobile and sessile marine organisms. Sessile means, "permanently attached or established : not free to move about" (Merriam-Webster n.d.). Something does not add up here. Sessile organisms move? Yes, when they are in the midst of the larval life stage. Both sessile and mobile marine animals, including species of fish, sponge, crab, sea star, worm, and snail experience this microscopic phase. According to Dr. Erin Satterthwaite, a marine ecologist with California Sea Grant at Scripps Institution of Oceanography , dispersal of marine young "is one of the primary determinants of recruitment of new individuals into populations and can be an important driver of population dynamics," (Drake et al. 2018). In other words, larval movement affects how larvae settle into new populations. In turn, population size and structure influence how those populations interact with the environment and the rest of their community (Khan Academy n.d.). Dr. Satterthwaite and her colleagues found that when larvae swam toward shore (as opposed to not swimming toward shore), there was "a substantial increase in nearshore larval supply," which meant that there would be more young available to settle into suitable habitat (Drake et al. 2018). This demonstrated that larval behavior may be an important factor shaping marine population dynamics. Overall, in order to understand and conserve marine animals, we need to better understand where they are born from, where they end up, and how they got there. Dr. Satterthwaite's favorite part of being a marine ecologist has been exploring nature and sharing its wonders with others. As a first-generation college and graduate student, her "path has been a winding journey," and she "has had to rely on others for support and guidance." For Dr. Satterthwaite, it has been this sense of community and relationships she has built along the way that has been the best part of her career.
: A photo of Dr. Erin Satterthwaite contributed to Project Biodiversify by Dr. Satterthwaite.
Scientist Spotlight inspiration from Project Biodiversify
Drake, P.T., Edwards, C.A., Morgan, S.G., & Satterthwaite, E.V. (2018). Shoreward swimming boosts modeled nearshore larval
<https://www.sciencedirect.com/science/article/pii/S0924796317304955>. Accessed October 5, 2021.
Khan Academy. n.d. <https://www.khanacademy.org/science/biology/ecology/population-ecology/a/population-size-density-anddispersal>. Accessed October 5, 2021.
Merriam-Webster. n.d. <https://www.merriam-webster.com/dictionary/sessile>. Accessed October 5, 2021.
10.3: Scientist Spotlight - Erin Satterthwaite is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by LibreTexts.
10.4: Overview of Population Growth Models Skills to Develop: By the end of this section, you will be able to do the following: Explain the characteristics of and differences between exponential and logistic growth patterns Give examples of exponential and logistic growth in natural populations Describe how natural selection and environmental adaptation led to the evolution of particular life history patterns Although life histories describe the way many characteristics of a population (such as their age structure) change over time in a general way, population ecologists make use of a variety of methods to model population dynamics mathematically. These more precise models can then be used to accurately describe changes occurring in a population and better predict future changes. Certain long-accepted models are now being modified or even abandoned due to their lack of predictive ability, and scholars strive to create effective new models.
Exponential Growth Charles Darwin, in his theory of natural selection, was greatly influenced by the English clergyman Thomas Malthus. Malthus published a book in 1798 stating that populations with unlimited natural resources grow very rapidly, which represents an exponential growth, and then population growth decreases as resources become depleted, indicating a logistic growth.
The best example of exponential growth is seen in bacteria. Bacteria reproduce by prokaryotic fission. This division takes about an
hour for many bacterial species. If 1000 bacteria are placed in a large flask with an unlimited supply of nutrients (so the nutrients
will not become depleted), after an hour, there is one round of division and each organism divides, resulting in 2000 organisms--an
increase of 1000. In another hour, each of the 2000 organisms will double, producing 4000, an increase of 2000 organisms. After
the third hour, there should be 8000 bacteria in the flask, an increase of 4000 organisms. The important concept of exponential
growth is the accelerating population growth rate--the number of organisms added in each reproductive generation--that is, it is
increasing at a greater and greater rate. After 1 day and 24 of these cycles, the population would have increased from 1000 to more
than 16 billion. When the population size, N, is plotted over time, a J-shaped growth curve is produced (Figure
The bacteria example is not representative of the real world where resources are limited. Furthermore, some bacteria will die during the experiment and thus not reproduce, lowering the growth rate. Therefore, when calculating the growth rate of a population, the death rate (D) (number organisms that die during a particular time interval) is subtracted from the birth rate (B) (number organisms that are born during that interval). This is shown in the following formula:
= Change in time, = birth rate and = death rate
The birth rate is usually expressed on a per capita (for each individual) basis. Thus, B (birth rate) = bN (the per capita birth rate "b" multiplied by the number of individuals "N") and D (death rate) = dN (the per capita death rate "d" multiplied by the number of individuals "N"). Additionally, ecologists are interested in the population at a particular point in time, an infinitely small time interval. For this reason, the terminology of differential calculus is used to obtain the "instantaneous" growth rate, replacing the change in number and time with an instant-specific measurement of number and time.
Notice that the "d" associated with the first term refers to the derivative (as the term is used in calculus) and is different from the death rate, also called "d." The difference between birth and death rates is further simplified by substituting the term "r" (intrinsic rate of increase) for the relationship between birth and death rates:
The value "r" can be positive, meaning the population is increasing in size; or negative, meaning the population is decreasing in size; or zero, where the population's size is unchanging, a condition known as zero population growth. A further refinement of the
formula recognizes that different species have inherent differences in their intrinsic rate of increase (often thought of as the potential for reproduction), even under ideal conditions. Obviously, a bacterium can reproduce more rapidly and have a higher intrinsic rate of growth than a human. The maximal growth rate for a species is its biotic potential, or , thus changing the equation to:
: When resources are unlimited, populations exhibit exponential growth, resulting in a J-shaped curve. When
resources are limited, populations exhibit logistic growth. In logistic growth, population expansion decreases as resources become
scarce, and it levels off when the carrying capacity of the environment is reached, resulting in an S-shaped curve.
Logistic Growth Exponential growth is possible only when infinite natural resources are available; this is not the case in the real world. Charles Darwin recognized this fact in his description of the "struggle for existence," which states that individuals will compete (with members of their own or other species) for limited resources. The successful ones will survive to pass on their own characteristics and traits (which we know now are transferred by genes) to the next generation at a greater rate (natural selection). To model the reality of limited resources, population ecologists developed the logistic growth model.
Carrying Capacity and the Logistic Model In the real world, with its limited resources, exponential growth cannot continue indefinitely. Exponential growth may occur in environments where there are few individuals and plentiful resources, but when the number of individuals gets large enough, resources will be depleted, slowing the growth rate. Eventually, the growth rate will plateau or level off (Figure 10.3.1). This population size, which represents the maximum population size that a particular environment can support, is called the carrying capacity, or K. The formula we use to calculate logistic growth adds the carrying capacity as a moderating force in the growth rate. The expression "K - N" indicates how many individuals may be added to a population at a given stage, and "K - N" divided by "K" is the fraction of the carrying capacity available for further growth. Thus, the exponential growth model is restricted by this factor to generate the logistic growth equation:
or 1, and the right side of the equation reduces to
which means the population is growing exponentially and is not influenced by carrying capacity. On the other hand, when N is
comes close to zero, which means that population growth will be slowed greatly or even stopped. Thus,
population growth is greatly slowed in large populations by the carrying capacity K. This model also allows for the population of a
negative population growth, or a population decline. This occurs when the number of individuals in the population exceeds the
A graph of this equation yields an S-shaped curve (Figure
), and it is a more realistic model of population growth than
exponential growth. There are three different sections to an S-shaped curve. Initially, growth is exponential because there are few
individuals and ample resources available. Then, as resources begin to become limited, the growth rate decreases. Finally, growth
levels off at the carrying capacity of the environment, with little change in population size over time.
Role of Intraspecific Competition The logistic model assumes that every individual within a population will have equal access to resources and, thus, an equal chance for survival. For plants, the amount of water, sunlight, nutrients, and the space to grow are the important resources, whereas in animals, important resources include food, water, shelter, nesting space, and mates. In the real world, phenotypic variation among individuals within a population means that some individuals will be better adapted to their environment than others. The resulting competition between population members of the same species for resources is termed intraspecific competition (intra- = "within"; -specific = "species"). Intraspecific competition for resources may not affect populations that are well below their carrying capacity--resources are plentiful and all individuals can obtain what they need. However, as population size increases, this competition intensifies. In addition, the accumulation of waste products can reduce an environment's carrying capacity.
Yeast, a microscopic fungus used to make bread and alcoholic beverages, exhibits the classical S-shaped curve when grown in a
). Its growth levels off as the population depletes the nutrients. In the real world, however, there are
variations to this idealized curve. Examples in wild populations include sheep and harbor seals (Figure
the population size exceeds the carrying capacity for short periods of time and then falls below the carrying capacity afterwards.
This fluctuation in population size continues to occur as the population oscillates around its carrying capacity. Still, even with this
oscillation, the logistic model is confirmed.
: (a) Yeast grown in ideal conditions in a test tube show a classical S-shaped logistic growth curve, whereas (b) a natural population of seals shows real-world fluctuation.
If the major food source of the seals (Figure likely occur?
) declines due to pollution or overfishing, which of the following would
a. The carrying capacity of seals would decrease, as would the seal population. b. The carrying capacity of seals would decrease, but the seal population would remain the same.
c. The number of seal deaths would increase but the number of births would also increase, so the population size would remain the same. d. The carrying capacity of seals would remain the same, but the population of seals would decrease. Answer a. The carrying capacity of seals would decrease, as would the seal population. Contributors and Attribution: This material is edited from the following sources OpenStax Biology 2e section 45.3 Environmental Limits to Population Growth 10.4: Overview of Population Growth Models is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by LibreTexts.
10.5: Geometric and Exponential Growth INTRODUCTION We will begin by developing a population model in discrete time. That is, we will treat time as if it moved in steps, rather than continuously. This allows us to use difference equations rather than differential equations, and thereby avoid the calculus. This assumption is realistic for many populations that have seasonal, synchronous reproduction. Strictly speaking, the discrete-time model represents geometric population growth. Later in the chapter, we will develop a continuous-time model, properly called an exponential model. Model Development To begin, we can write a very simple equation expressing the relationship between population size and the four demographic processes. Let: represent the size or density of the population at some arbitrary time (we will ignore the distinction between population size and population density) represent population size one arbitrary time-unit later represent the total number of births in the interval from time to time represent the total number of deaths in the same time interval represent the total number of immigrants in the same time interval represent the total number of emigrants in the same time interval Then we can write For simplicity, this exercise ignores immigration and emigration. Our equation becomes This equation is easy to understand but inconvenient for modeling. The problem lies in the use of "raw" birth and death rates ( and ). We have no obvious, biologically reasonable starting assumptions about these numbers. However, if we switch from raw birth and death rates to per capita birth and death rates, we can do some fruitful modeling. Geometric (Discrete-Time) Model of Population Growth A per capita rate is a rate per individual; that is, the per capita birth rate is the number of births per individual in the population per unit time, and the per capita death rate is the number of deaths per individual in the population per unit time. Per capita birth rate is easy to understand, and seems a reasonable thing to model because reproduction (giving birth) is something individuals rather than whole populations do. Per capita death rate may seem strange at first; after all, an individual can die only once. But remember, this rate is calculated per unit time. You can think of per capita birth and death rates as each individual's probability of giving birth or dying in a given unit of time. Keeping in mind that per capita rates are per individual rates, we can translate the raw rates Bt and Dt into per capita rates, which we will represent with lower-case letters (bt and dt) to distinguish them from the raw numbers. To calculate per capita rates, we divide the raw numbers by the population size. Thus,
Now we can rewrite our model in terms of per capita rates:
Perhaps this seems to have gotten us nowhere, but it turns out to be a very informative model if we make one further assumption. Let us assume, just to see what happens, that per capita rates of birth and death remain constant over time. In other words, let us assume that average number of births per unit time per individual in the population and the average risk of dying per unit time remain unchanged over some period of time. What will happen to population size? Because we assume constant per capita birth and death rates, we can make one further minor modification to our equation by leaving off the time subscripts on b and d:
At this point, you're probably thinking that this assumption is unrealistic--that per capita rates of birth and death are likely to change over time for a variety of reasons. You are quite correct, but the model is still useful for three reasons: It provides a starting point for a more complex and realistic model in which per capita rates of birth and death do change over time. It is a good heuristic model--that is, it can lead to insights and learning despite its lack of realism. Many populations do in fact grow as predicted by this model, under certain conditions and for limited periods of time.
You may also wonder why we use this complex model (Equation 1) rather than the simpler forms of the geometric and exponential models presented in most textbooks (and developed here beginning with Equation 2). We prefer Equation 1 for three reasons:
It emphasizes the roles of per capita birth and death rates rather than the more abstract quantities or (explained later).
It allows you to manipulate per capita birth and death rates directly and separately, and discover that neither alone, but
rather the difference between them, determines population growth rate.
It allows you to discover that the per capita rate of population growth
is a constant, which you can then relate to
Because per capita birth and death rates do not change in response to the size (or density) of the population, this model is said to be density-independent. We can further simplify Equation 1 by factoring out of the birth and death terms:
The term (b - d) is so important in population biology that it is given its own symbol, the geometric rate of increase. Substituting for (b - d) gives us
To further define , we can calculate the amount of change in population size, 2:
, by subtracting from both sides of Equation
In words, the amount of change in population size is proportional to the population size, and the constant of proportionality is . We can convert this to per capita rate of change in population size if we divide both sides by :
In other words, the parameter represents the (discrete-time) per capita rate of change in the size of the population.
Notation note A common source of confusion when learning about ecological is the fact that different books and other resources will use slightly different notation and alternative -- but mathematically equivalent -- ways to set up equations (parametrizations). For example, the source material for this chapter used instead of .
Moving on, we can simplify Equation 2 ( get
) even further by factoring out of the terms on the right-hand side, to
is often given its own symbol, (lambda), and its own name: the finite rate of increase. Substituting , we
The quantity can be very useful in analyzing real population data. Some additional algebra will show us how.
If we divide both sides of Equation 5 by , we get
In words, is the ratio of the population size at one time to its size one time-unit earlier. We can calculate counts at successive times, even if we do not know per capita rates of birth and death.
In Equations 2 and 5, we showed how to calculate the size of the population one time unit into the future. What if you wanted to know how big the population will be at some distant future time? You could carry out the one-time-step calculations many times, until you arrived at the desired answer. But there is also a shortcut. Population size at time 1 is 1N0, at time 2 it is 2N0, and at time 3 it is 3N0. In general, we can write
Notation note is called "N-naught." In population ecology it is often used to denote an initial population size.
The previous expression may strike you as rather abstract. One way to understand its impact is to use Equation 7 to calculate
--that is, the time required for the population to double in size. If we plug the doubling time into Equation 7,
We can derive doubling time by exploiting the fact that the population at time tdouble is, by definition, twice the population at time 0:
Substituting 2N0 for Nt double gives us If we divide both sides by N0, we get Taking the logarithm of both sides gives us Dividing both sides by , we get
What does this mean? Suppose = 0.1 individuals/individual/year. Therefore, increases by 10% per year, which doesn't sound like much. But, if you plug this value of population doubles in about 7.27 years, which seems more impressive.
. This implies that the population into Equation 8, you'll find that the
You may be wondering how a population that grows in discrete intervals of a year can double in a non-integer number of years. It can't, of course. This calculation really means that the population will not quite double in 7 years, and will more than double in 8 years.
Exponential (Continuous-Time) Model of Population Growth Population growth can also be modeled in continuous time, which is more realistic for populations that reproduce continuously, rather than seasonally. Continuous-time models also allow use of the calculus, which provides many powerful analytical tools. Here, we will eschew the calculus, and simply present some results. Most textbooks begin with the continuous-time analog of Equation 3:
The left-hand side of Equation 9 represents the instantaneous rate of change in population size, which is different from the rate of
, that we looked at in Equation 7. Therefore, we use a lowercase instead of to
distinguish the continuous-time exponential model from the discrete-time geometric model. The symbol is called
the instantaneous rate of increase or the intrinsic rate of increase. The parameters and are not equal, although they are
As we did with the discrete-time model, we can calculate the per capita rate of population growth by dividing both sides of Equation 9 by :
You can use the calculus to operate on Equation 10 and calculate the size of the population at any time. We will spare you the derivation, but the resulting equation is
where is the root of the natural logarithms
You can derive the relationship between and as follows. Suppose we start two populations with the same initial number of individuals, N0, and both grow at the same rate. However, one grows in continuous time and the other grows in discrete time.
Because they grow at the same rate, at some later time, t, they will have reached the same size, Nt. If we write the discrete-time population on the left and the continuous-time population on the right we can derive as follows:
So, we can convert back and forth between continuous-and-discrete time models. Remember that
Suppose we have a population growing in continuous time with some value of r, and a population growing in discrete time with the
. Which will grow faster? As we did with the geometric model, we can derive the doubling time for the
exponential model (Gotelli 2001). We begin with Equation 11, and plug in
and taking the natural logarithm of both sides yields
Finally, we divide both sides by , and rearrange, to get
Parallel to our earlier example, let us suppose
individuals/individual/year. As before, this implies a 10% annual increase in
the population, but now this increase occurs continuously rather than in discrete time intervals. How long does it take for this
population to double? Plugging in the value 0.1 for yields a doubling time of 6.93 years, somewhat faster than indicated by the
: Exponential growth models have a faster growth rate than geometric models, so the population size of exponentially growing populations outpaces geometrically growing populations over time.
EXPLORE THIS MODEL Before moving on to the next section, explore this Exponential Growth Shiny App developed by Dr. Aaron Howard to better understand how changes to the initial population size (N) and the population growth rate (r) impact population size over time.
References Donovan, T.M., & Welden, C. (2002). Spreadsheet exercises in ecology and evolution. Sinauer Associates, Inc. Sunderland, MA, USA. Gotelli, N.J. (2001). A Primer of Ecology, 3rd Edition. Sinauer Associates, Sunderland, MA. 10.5: Geometric and Exponential Growth is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by LibreTexts.
10.5.1: Logistic population growth INTRODUCTION
This material in this chapter has been adapted from Donovan and Welden (2002). Donovan, T. M. and C. Welden. 2002. Spreadsheet exercises in ecology and evolution. Sinauer Associates, Inc. Sunderland, MA, USA. As in the previous section on Geometric and Exponential Growth, we begin with a model of population dynamics in discrete time, with explicit parameters for per capita rates of birth and death. In the previous section, we developed the following geometric model of population dynamics:
where =population at time = population at one time unit later = per capita birth rate = per capita birth rate
As you discovered in the earlier exercise, this model produces geometric population growth (the discrete-time analog of
exponential growth) if and are held constant and
However, the assumption that per capita rates of birth and death remain
constant is unrealistic, so we will now develop a model in which these rates change. Specifically, we will consider only one cause
of changes in per capita birth and death rates: the size of the population itself. In other words, we will assume that environmental
conditions, food supply, and so on remain constant; only the size of the population itself changes. These additions result in
Because per capita rates of birth and death do change in response to population size or density, logistic models are densitydependent, in contrast to geometric and exponential models, which are density-independent. As the population grows, less food and water, fewer nesting and hiding sites, and fewer resources in general are available to each individual, affecting both an individual's rate of reproduction and its risk of death. Our model will thus include intraspecific competition (competition among members of the same species) for resources. These models are used to inform practical decisions in the management of fisheries and game animal populations and are used to predict the growth of the human population. Later exercises will develop models of interspecific (between two species) competition and predator-prey dynamics.
Logistic growth models include an equilibrium population size in this model. In other words, populations grow until they reach a stable size. The population is at equilibrium when total deaths equal total births and when per capita rates of birth and death are equal. This equilibrium populations size is so important in population biology, it is given its own name--the carrying capacity. The carrying capacity is defined as the largest population that can be supported indefinitely, given the resources available in the environment. This carrying capacity is represented by the parameter .
If we begin with a very small population, the term
is very nearly equal to or 1. The model will then behave like a
geometric model, and the population will grow, provided . The population will grow slowly at first, because the parameter is
also being multiplied by a number that is nearly equal to zero, but it will grow faster and faster, at least for a while. At some
point, however, population growth will begin to slow because the term closer to .
is getting smaller and smaller as gets larger and
At the other extreme, imagine a population that starts out at a size very close to its carrying capacity, K. The term
nearly equal to zero, and population growth is extremely slow. When
, the population stops growing altogether.
EXPLORE THIS MODEL Before moving on to the next section, explore the Logistic growth Shiny App developed by Dr. Aaron Howard to better understand how changes to the initial population size , carrying capacity , and the population growth rate impact population size over time.
References Donovan, T. M. and C. Welden. 2002. Spreadsheet exercises in ecology and evolution. Sinauer Associates, Inc. Sunderland, MA, USA. 10.5.1: Logistic population growth is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by LibreTexts.
10.6: Projecting population growth Predicting changes in population size using lambda Once we have a value for and the current size of the population from a complete census, we can estimate the future size of the population. Recall that we arrived at lambda previously by setting up this equation Where is the survival rate of adults, and is the number of offspring produced per adult that live to reproduce themselves. We can rearrange this equation for making projections like this:
Derivation Dance indicates that our concept of lambda is based on the ratio of two population sizes. Let's start with this:
cancels out from the left side and we get
This tells us that the population size next year
Case Study: Projecting the number of Kirtland's Warblers In 1990 it was estimated that there were 265 KIWA males with territories. If \[\lambda] = 1.3 as we calculated earlier, we'd predict that in 1991 the population size would be
So we'd predict there to be 344 or 345 birds in 1991 (you can't have 0.5 of an organism. Indeed, researchers observed 347 birds. We can repeat this using our estimate for 1991 to estimate 1992 and so on. When you plug the output of an equation (eg N1991=344) back into itself repeatedly this is called recursion. The equation is therefore sometimes referred to as a recursion equation. (Similar terms are iterate and iteratively.) Exponential population growth If you take the population size of 344 KIWA from our last calculation and plug it back into the population growth equation like this to estimate the number of warblers in the next year, 1992:
The estimated population the following year will be about 447 KIWA males. In reality, there were 497 males counted, so our projection is a bit low If you repeat this process of plugging the results of one calculation into the next calculation 8 more times (for 10 total years of population change) we can project the population size for a decade. Note the convex, upward curvature of the line. This upward, accelerating curve is an exponential growth curve. Exponential growth occurs when something increases by multiplication. This is in contrast to arithmetic growth, which occurs when something increases by addition.
: Exponential growth increases by multiplication and accelerates the population size over time without limit.
Many things in the world change arithmetically. For example, if you are being paid by the hour to do a job and don't have to work an entire day, the amount you earn increases arithmetically. Some populations can grow arithmetically if individuals from other populations are being introduced intentionally to the population. For example, the Peregrine falcon (Falco peregrinus) has a geographic range that encompases the entire world. However, due to the impacts of pesticides such as DDT the falcon was almost extirpated from many places in the United States. In addition to banning DDT, falcon populations were restored by taking young falcons from large populations and introducing them to smaller populations. If 10 falcons were introduced each year to a population, the fixed number of new individuals results in arithmetic contribution to growth.
In contrast to the fixed increases of arithmetic growth, exponential growth occurs multiplicatively - that is, by multiplying the current population size by a growth rate. If a population of falcons has 100 birds and a growth rate of 1.10, then it's population size the next year is 100*1.10 = 110. The next year if the growth rate is still 1.10 the population would be 110*1.10 = 121. Therefore the first time step the change in population size (delta N) was 10 individuals (110-100 = 10), while over the second time step delta N was 11 individuals (121-110 = 11). If we projected a third year the population would be 121*1.10 = 133.1. Delta N is therefore increasing each year from 10, to 11, to about 12. This is why we use the term "accelerating" to describe this type of growth: each year the amount of change increases.
: Male peregrine falcon (Falco peregrinus) in Humber Bay Park West in Toronto. "Falco peregrinus" by is licensed under CC BY-SA 3.0.
: The peregrine falcon's range covers most of the world, besides some areas of Africa, the Arabian Peninsula, South
America, Greenland, and Asia. The birds' large home range allows for alternative options from limiting factors. "Range map
for Falco peregrinus (including F. (p.) pelegrinoides, the Barbary falcon)" by MPF is licensed under CC BY-SA 3.0.
Case study: The Montserrat Oriole The Montserrat Oriole (Icterus oberi) is a relative of the Baltimore Oriole (Icterus galbula). While the Baltimore Oriole occurs throughout eastern and central North America, the Montserrat Oriole lives only on the Caribbean island of Montserrat. Montserrat is a small volcanic island that is only about 10 miles long and 7 miles wide and lies east of Puerto Rico and north of Barbados. Since the Montserrat Oriole only occurs on the island of Montserrat. It therefore is an entirely closed population. The Montserrat Oriole is therefore called a single-island endemic species, like the Cozumel Thrasher.
: The red box identifies Montserrat. "Location of Montserrat" by TUBS is licensed under CC BY-SA 3.0 / adapted from original.
: The Montserrat Oriole is only present on the island of Montserrat in a closed population. "Male Montserrat Oriole (Icterus oberi), London Zoo" by Neil Phillips is licensed under CC BY 2.0.
Volcanic activity in the 1990s on Montserrat has destroyed its forest habitat and rained ash down across the island, destroying nests and killing the birds' insect prey. N was therefore negative due to a low number of births of new birds and a high death rate of members of the existing population due to starvation. The species is therefore threatened with extinction - death of all members of the population. Biologists have become very interested in determining how many orioles are left on the island and what is happening with the population.
The Montserrat Oriole population size is small, but researchers have never been able to determine how many there are. What they have done, however, is marked a subset of adult orioles with bird bands and re-captured them each year to estimate survival rates. They have also found some nests and determined how many baby orioles are born per nest, and what the survival rate is for those birds until they are one year old and can breed. This gives an estimate of b.
Survival for the Montserrat Oriole is currently around 70%, or 0.70. This means that if there were 100 birds banded, 70 of them survived to the following year. Alternatively, we can think of this in terms of probability: a single bird banded this year has a 70% chance of surviving to the next year to be re-captured.
The birth rate is tricky to estimate for a number of reasons; incorporating both the number of Orioles that hatch from eggs and their probability of surviving for one year until they can reproduce, the birth rate (b) is about 0.42. The population growth rate is therefore
Since >1 we'd predict that the population of Orioles will be growing. As before, we aren't actually counting all the birds, but instead using demographic rates to estimate lambda.
In 2012 it was estimated that there might be as few as 300 Orioles on the island. If that in 2013 the population size would be
In 2001 a second, very small population of Montserrat Orioles was found very near to the crater of the volcano on the island. Researchers haven't studied this population much, but they estimate that there were about 100 birds in 2020.
: Green spots on the island of Montserrat identify habitats. "Range map of Montserrat Oriole (Icterus oberi)" by Cephas is licensed under CC BY-SA 4.0.
If you start with 100 Orioles and you estimate that for this population is 1.10 you can start with this equation
We can make this more specific with subscripts:
and get the project population the following year
So, if you have 100 orioles in 2020 you'd expect to have 2021 next year. Exponential population growth of Orioles If you take the population size of 110 orioles from our last calculation and plug it back into the population growth equation you can estimate the number of orioles in next year, 2022:
the estimated population the following year will be 121 orioles.
If you repeat this 8 more times (for 10 total years of population change since 2020) you will see a graph like the one below. Note the slight, convex, upward curvature of the line.
: Exponential growth models can be used to graph the predicted population sizes of species.
Contributors and Attributions This chapter was written by NL Brouwer (University of Pittsburgh)
10.6: Projecting population growth is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by LibreTexts.
10.7: Life Tables This material in this chapter has been adapted from Donovan and Welden (2002). Donovan, T. M. and C. Welden. 2002. Spreadsheet exercises in ecology and evolution. Sinauer Associates, Inc. Sunderland, MA, USA. Objectives Use age or stage-specific abundance data to calculate a variety of life table parameters. Discover how patterns of survivorship relate to the classic three types of survivorship curves. Learn how patterns of survivorship relate to life expectancy. Explore how patterns of survivorship and fecundity affect the rate of population growth. 10.7.1: Introduction A life table is a record of survival and reproductive rates in a population, broken out by age, size, or developmental stage (e.g., egg, hatchling, juvenile, adult). Ecologists and demographers (scientists who study human population dynamics) have found life tables useful in understanding patterns and causes of mortality, predicting the future growth or decline of populations, and managing populations of endangered species. Predicting the growth and decline of human populations is one very important application of life tables. As you might expect, whether the population of a country or region increases or decreases depends in part on how many children each person has and the age at which people die. But it may surprise you to learn that population growth or decline also depends on the age at which they have their children. Another use of life tables is in species conservation efforts, such as in the case of the loggerhead sea turtle of the southeastern United States (Crouse et al., 1987). Generally speaking, the loggerhead population is declining and mortality among loggerhead eggs and hatchlings is very high. These facts led conservation biologists to advocate for the protection of nesting beaches. When these measures proved ineffective in halting the population decline, compiling and analyzing a life table for loggerheads indicated that reducing mortality of older turtles would have a greater probability of reversing the population decline. Therefore, management efforts shifted to persuading fishermen to install turtle exclusion devices on their nets to prevent older turtles from drowning. 10.7.2: Life Table Varieties Life tables come in two varieties: cohort and static. A cohort life table follows the survival and reproduction of all members of a cohort from birth to death. A cohort is the set of all individuals born, hatched, or recruited into a population during a defined time interval. Cohorts are frequently defined on an annual basis (e.g., all individuals born in 1978), but other time intervals can be used as well. A static life table records the number of living individuals of each age in a population and their reproductive output. The two varieties have distinct advantages and disadvantages, some of which we discuss below. Life tables (whether cohort or static) that classify individuals by age are called age-based life tables. Such life tables treat age the same way we normally do: that is, individuals that have lived less than one full year are assigned age zero; those that have lived one year or more but less than two years are assigned age one; and so on. Life tables represent age by the letter x, and use x as a subscript to refer to survivorship, fecundity, and so on, for each age. Size-based and stage-based life tables classify individuals by size or developmental stage, rather than by age. Size-based and stage-based tables are often more useful or more practical for studying organisms that are difficult to classify by age, or whose
ecological roles depend more on size or stage than on age. 10.7.2.1: Cohort Life Tables To build a cohort life table for, let's say, humans born in the United States during the year 1900, we would record how many individuals were born during the year 1900, and how many survived to the beginning of 1901, 1902, etc., until there were no more survivors. This record is called the survivorship schedule. We must also record the fecundity schedule--the number of offspring born to individuals of each age. The total number of offspring is usually divided by the number of individuals in the age, giving the average number of offspring per individual, or per capita fecundity. Many life tables count only females and their female offspring; for animals with two sexes and equal numbers of males and females of each age, the resulting numbers are the same as if males and females were both counted. For most plants, hermaphroditic animals, and many other organisms, distinctions between the sexes are nonexistent or more complex, and life table calculations may have to be adjusted. 10.7.2.2: Static Life Tables A static life table is similar to a cohort life table but introduces a few complications. For many organisms, especially mobile animals with long life spans, it can be difficult or impossible to follow all the members of a cohort throughout their lives. In such cases, population biologists often count how many individuals of each age are alive at a given time. That is, they count how many members of the population are currently in the 0-1- year-old class, the 1-2-year-old class, etc. These counts can be used as if they were counts of survivors in a cohort, and all the calculations described below for a cohort life table can be performed using them. In doing this, however, the researcher must bear in mind that she or he is assuming that agespecific survivorship and fertility rates have remained constant since the oldest members of the population were born. This is usually not the case and can lead to some strange results, such as negative mortality rates. These are often resolved by averaging across several ages, or by making additional assumptions. 10.7.3: Life Table Parameters Survivorship and fecundity schedules are the raw data of any life table. From them we can calculate a variety of other quantities, including age-specific rates of survival, mortality, fecundity, survivorship curves, life expectancy, generation time, net reproductive rate, and intrinsic rate of increase. Which of these quantities you calculate will depend on your goals in constructing the life table. Key Parameters and When conducting a study, these are the data generally collected on populations. We then calculate the rest of the life table from these data. The first column, represents the age classes. This column could represent days, minutes, or life stages (eggs, juveniles, adults, etc.). The number of individuals from the original cohort that are alive at the specified age, age class, or life stage ( ). From information on the number of individuals at each age, we can calculate a variety of survival and mortality rates. The difference between the number of individuals alive for any age class ( ) and the next older age class ( ) is the number of individuals that have died during that time intervals. is a measure of age-specific mortality. The number of individuals that died during any given time interval ( ) divided by the number alive at the beginning of that interval ( ) provides an age-specific mortality rate.
The age-specific survival rate for age interval x is the proportion of individuals that survive during any given time interval. The number of individuals surviving to any given life stage as a proportion of the original cohort size. represents the probability at birth of surviving to any given life stage. Calculating life expectancy requires calculating two additional parameters, and . The number of individuals that are alive in the middle of the first age class - 0.5 years old, or 1.5 years old. The total years lived into the future by individuals in age class x. This value is calculated by summing the values of Lx cumulatively from age to the end of the life table. The number of time units left for all individuals to live from age x onward - obtained by summing all values of . The life expectancy for an individual of age x is age-specific life expectancy divided by number of individuals at age . Life expectancy represents the average additional length of times than an individual will live once it has reached age x. A typical life table is shown in Figure 1. If we were to build a cohort life table for a population born during the year 1900, we would record how many individuals were born during the year 1900, and how many survived to the beginning of 1901, 1902, etc., until there were no more survivors. This record is called the survivorship schedule. We would also record the fecundity schedule: the number of offspring born to members of each age class. The total number of offspring is usually divided by the number of individuals in the age class, giving the average number of offspring per individual, which is represented by .
: A cohort of 3751 individuals tracked over time. The number alive at the beginning of each year is given in Column
B, and the average number of offspring per female is given in Column C. Columns D through G are calculated from information in
: "Mt. Olivet Cemetery" was taken by Daniel Lobo and is licensed under Creative Commons Attribution 2.0 Generic.
Calculating Key Parameters Standardized Survival Schedule ( ). Because we want to compare cohorts of different initial sizes, we standardize all cohorts to their initial size at time zero, . We do this by dividing each by . This proportion of original numbers surviving to the beginning of each interval is denoted , and calculated as
We can also think of as the probability that an individual survives from birth to the beginning of age with all the individuals born during the year (or other interval), always begins at a value of one (i.e., decrease with time. At the last age, , is zero.
Age-Specific Survivorship ( ). Standardized survivorship, , gives us the probability of an individual surviving from birth to the
beginning of age . But what if we want to know the probability that an individual who has already survived to age will survive
to age ? We calculate this age-specific survivorship as
Life Expectancy ( ). You may have heard another demographic statistic, life expectancy, mentioned in discussions of human populations. Life expectancy is how much longer an individual of a given age can be expected to live beyond its present age. Life expectancy is calculated in three steps. First, we compute the proportion of survivors at the mid-point of each time interval ( --note the capital L here); that is, Second, we sum all the Lx values from the age of interest ( ) up to the oldest age, : Finally, we calculate life expectancy as Life expectancy is age-specific--it is the expected number of time-intervals remaining to members of a given age. The statistic most often quoted (usually without qualification) is the life expectancy at birth ( ). 10.7.4: Survivorship Curves There are three classic survivorship curves, called Type I, Type II, and Type III (Figure 2). To understand survivorship curves you can use survivorship schedules (Sx) to calculate and graph standardized survivorship ( ), age-specific survivorship ( ), and life expectancy ( ).
: Hypothetical survivorship curves. Note that the y-axis has a logarithmic scale. Type 1 organisms have high
survivorship throughout life until old age sets in, and then survivorship declines dramatically to 0. Humans are type 1 organisms.
Type III organisms, in contrast, have very low survivorship early in life, and few individuals live to old age.
Population Growth or Decline We frequently want to know whether a population can be expected to grow, shrink, or remain stable, given its current age-specific rates of survival and fecundity. We can determine this by computing the net reproductive rate ( ). To predict long-term changes in population size, we must use this net reproductive rate to estimate the intrinsic rate of increase ( ).
Net Reproductive Rate (R0) We calculate net reproductive rate ( ) by multiplying the standardized survivorship of each age ( ) by its fecundity (( )), and summing these products:
The net reproductive rate is the lifetime reproductive potential of the average female, adjusted for survival. Assuming survival and
fertility schedules remain constant over time, if R0 > 1, then the population will grow exponentially. If
, the population size will not change over time. You may be tempted to conclude the
intrinsic rate of increase of the exponential model. However, this is not quite correct, because r measures population change in
absolute units of time (e.g., years) whereas R0 measures population change in terms of generation time. To convert into , we
must first calculate generation time G, and then adjust .
Generation Time. Generation time is calculated as
For organisms that live only one year, the numerator and denominator will be equal, and generation time will equal to one year. For all longer-lived organisms, generation time will be greater than one year, but exactly how much greater will depend on the survival and fertility schedules. A long-lived species that reproduces at an early age may have a shorter generation time than a shorter-lived one that delays reproduction.
Intrinsic Rate of Increase. We can use our knowledge of exponential population growth and our value of to estimate the
intrinsic rate of increase ( ) (Gotelli 2001). The size of an exponentially growing population at some arbitrary time is
where e is the base of the natural logarithms and is the intrinsic rate of increase. If we consider the growth of such a population
from time zero through one generation time, , it is
as roughly equivalent to ; both are estimates of the rate of population growth over the period of one
Taking the natural logarithm of both sides gives us
and dividing through by gives us an estimate of :
Finally, we can use our estimate of r (uncorrected or corrected) to predict the size of the population in the future. This kind of analysis is done for human populations to predict the effects of changes in medical care and birth control programs. If we assume that all age groups are roughly equivalent in size, a similar analysis can be done for endangered species to determine what intervention may be most effective in promoting population growth. The same analysis can be applied to pest species to determine what intervention may be most effective in reducing population size.
Reproductive Value The idea that different individuals have different "value" in terms of their contribution to future generations is called their reproductive value (Fisher 1930). As Caswell (2001) states, "The amount of future reproduction, the probability of surviving to realize it, and the time required for the offspring to be produced all enter into the reproductive value of an age-class." The reproductive value of an individual of age is designated at , and is the number of offspring that an individual is expected to produce over its remaining life span (after adjusting for the growth rate of the population). Biologists are often interested in knowing the "value" of the different individuals from a practical standpoint because knowing something about the reproductive value can suggest which individuals should be harvested, killed, transplanted, etc. from a conservation or management perspective. The reproductive value of different ages is strongly tied to an organism's life history. Typically, reproductive value is low at birth, increases to a peak near the age of first reproduction, and then declines (Caswell 2001).
Life History Table Practice x age individuals alive at start of age x
individuals alive at start of age x --- 125 58 32 16 4 0
Life History Table Practice x age individuals alive at start
10.7.5: References Caswell, H. (2001). Matrix Population Models, 2nd Ed. Sinauer Associates, Sunderland, MA. Connell, J.H. (1970). A predator-prey system in the marine intertidal region. I. Balanus glandula. Ecological Monographs, 40: 49- 78. (Reprinted in Ecology: Individuals, Populations and Communities, 2nd edition, M. Begon, J. L. Harper and C. R. Townsend.. (1990) Blackwell Scientific Publications, Oxford.) Crouse, D.T., Crowder, L.B., & Caswell, H. (1987). A stage-based population model for loggerhead sea turtles and implications for conservation. Ecology, 68, pp. 1412-1423. Deevey, E.S., Jr. (1947). Life tables for natural populations of animals. The Quarterly Review of Biology, 22, pp. 283-314. (Reprinted in Readings in Population and Community Ecology, W.E. Hazen (ed.). 1970, W.B. Saunders, Philadelphia.) Fisher, R.A. (1930). The Genetical Theory of Natural Selection. Clarendon Press, Oxford. Gotelli, N.J. (2001). A Primer of Ecology, 3rd Edition. Sinauer Associates, Sunderland, MA. 10.7: Life Tables is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by LibreTexts.
10.8: Population Models Practice Exercises Put your knowledge and comprehension to the test with these practice problems! Some of these questions may require you to use a calculator and draw out models while you practice. Be sure to complete all questions included in each section before you click "Answer" for that section because the answers for all the questions in that section will be revealed together. General Population Growth Equation Your ecology class starts the semester out with a student population of 42. By the end of the semester, two students dropped out of the college, one student dropped the class, one student died on a fieldtrip, and five students joined the course late. What is the class population at the end of the semester? Answer = 42 + (0-1) + (5-3) = 43 students Choosing the Right Model Which type of model (exponential, logistic, geometric, Leslie Matrix) belongs at each point in this figure?
Answer a) Leslie Matrix Model b) Logistic Model c) Geometric Model d) Exponential Model
Reading Results How does the population change under the following conditions (increase, decrease, or remain stable)? 1) = 1.02 2) = 1 3) N > K
4) r = -0.10 Answer 1) increase 2) remains stable 3) decrease (population overshoots carrying capacity) 4) decrease Geometric Model Given (population size at initial time) = 500, B = 300, and D = 350, calculate . 1) Is this population increasing, decreasing, or stable? 2) What will the population be in 10 years? What about in 30 years? Answer 1) = + B -D = 500 + 300 - 350 = 450 = / = 450/500 = 9/10 = 0.9 Since < 1, the population is decreasing. 2) = t = 500*(0.9)10 = 174 = 500*(0.9)30 = 21 Exponential Model A new species you are studying has continuous reproduction in a newly invaded habitat. The species' population is still growing without limit. 1) If the initial population size is 3000, and the growth rate is 0.02, what is the expected population in 25 years? How about in 50 years? 2) If at , the population size is 500, and r = 0.035, how many years until the population reaches 4,000 individuals? Answer 1) = e t r = 3000, r = 0.02 = 3000*e(25)(0.02) = 4,946 = 3000*e(50)(0.02) = 8,154 2) Population doubling time = 70/3.5 = 20: 20 years to reach 1000 individuals, 40 years to reach 2000 individuals, 60 years to reach 4000 individuals. 10.8: Population Models Practice Exercises is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.