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Biology 310L – Principles of Ecology Lab Manual – Page -61
Chapter 9. Population Growth.
Today's activities:
1. Go to computer lab and work through lab
What you should get out of today's class:
You should understand the basis of population growth (births and deaths, growth rates,
exponential growth). You should be able to recognize the logistic growth equation and say why it
captures the sigmoidal growth pattern of population trajectories. You should understand that many
factors such as resource availability or specific constraints influence population growth and carrying
capacity. You should also develop some skill with entering and manipulating models in a spreadsheet.
Handouts:
1. Ripple, W. J., E. J. Larsen, R. A. Renkin, D. W. Smith. 2001. Trophic cascades among
wolves, elk and aspen on Yellowstone National Park’s northern range. Biological
Conservation 102:227-234.
2. Chapter 10. TBA.
Introduction
A population is a group of organisms that occur in the same area, interbreed, and interact. The
individuals in a population may be dense and gregarious like cliff swallows on the Central Bridge in
Albuquerque or they may be widely dispersed and solitary like lynx. The boundaries of populations
may be fairly concrete, such as on an island, but on large land masses like continents it can be hard to
decide when one population ends and another begins. Nonetheless, we can usually define an area and
examine the population within that area.
The size of a population is usually changing. A population may be increasing, decreasing, or
stable. Most often, populations increase a little one year, decrease a little the next year, and generally
fluctuate through time. The change in a population can be described in a simple mathematical way. A
population grows when individuals are born or immigrate into it, and a population declines when
individuals die or emigrate. This can be expressed as:
N t1=N t  B I − D E 
equation 9.1
Simply, the population size N at a subsequent time period (time plus one unit of time, usually a year) is
equal to the initial population size at time t (Nt) plus the number of births and immigrants minus the
number of deaths and emigrants during the time period. If we do not know what the birth, death,
immigration, and emigration rates are, but just want to model the change from one time step to another,
we can use a simple equation to predict the population growth:
N t =N 0 t
equation 9.2
where t is a discrete time step (such as years or days) and λ (lambda) is the geometric rate of increase.
As written, this model indicates that a population at a particular time is a function of the population
size at an initial time multiplied by λ raised to the power of t. If λ is greater than one, then the
University of New Mexico
Biology 310L – Principles of Ecology Lab Manual – Page -62
Unrestrained populations grow geometrically.
This is easy to understand with dividing cells such as
bacteria. If you start with 1 bacterium, it will divide
into two new cells (λ = 2). In the next generation, the
two cells divide, giving four new cells. In the next
generation, the four cells divide, giving eight new
cells, and so on (Figure 9.1). The number of new
individuals gets larger as the population size
increases, causing the population to grow more and
more quickly.
Number of individuals
population will increase, if λ is one the population does not change, and if λ is less than 1 the
population will decline. Notice that the model predicts open-ended growth if λ is greater than one.
This type of growth is called geometric if the time is discrete or exponential if the time is continuous.
We will continue to use the term geometric here because we will continue to use discrete time-step
models, which are well-suited to population growth in organisms that reproduce once per time step.
For organisms that reproduce year-round or at a frequency greater than the time step, continuous-time
models are more appropriate.
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one of the few real laws in ecology (Turchin 2001).
It may well be that all populations grow Figure 9.1. Exponential growth of a dividing
geometrically ... under the right circumstances. population of cells.
However, no population grows in a vacuum, and
there are always limitations to population growth. In nature there are feedbacks that reduce the rate of
population growth as the size of the population gets larger. Feedbacks include competition for
resources with other individuals of the same species (density dependence), competition for resources
with other species, predators, diseases, or anything that reduces the number of offspring produced or
the survival of individuals. Hence, the underlying process of population growth may be geometric, but
something always limits the population such that the geometric growth is only seen early in a
population's expansion, if at all.
Frequently, a growing population displays a sigmoidal pattern of growth. That is, the
population grows geometrically up to some point, begins to slow, and, finally, settles near a level
known as carrying capacity. Carrying capacity, designated with the letter K, is the number of
individuals that can be supported given the resources available. Organisms need many different
resources, so a particular resource (such as water, nitrogen, or space) usually tends to limit the
population level, even when other resources could support more individuals.
We can express the effects of negative feedback on population growth using a model known as
the logistic growth equation. Here we will subsume the birth, death, emigration, and immigration
parameters into a term known as the intrinsic rate of growth, r, which is like lambda above, but instead
of giving the population size in the next time step, it gives the change in population size (∆N):
 N =r N 
K−N

K
equation 9.3
University of New Mexico
Biology 310L – Principles of Ecology Lab Manual – Page -63
As N gets larger and approaches K, the term on the right gets smaller and smaller as K-N approaches 0.
As the population approaches its carrying capacity, the forces driving the population to grow are
reduced. We can use this equation to make an analogy to equation 9.2 as follows:
N t1=r N t 
K −N t
 N t
K
equation 9.4
This equation tells us what the population size is at a given time by estimating the change in the
population and adding it to the previous population size.
Today we are going to look at population growth in recovering populations of Osprey (Pandion
haliaetus). Osprey are a globally-occurring predatory bird that eats only fish (Poole et al. 2002). In
the 1960's and 1970's, Osprey populations in North America declined precipitously as a result of the
toxic effects of DDT (a pesticide), loss of nesting substrates (trees), and persecution from humans
(Poole et al. 2002). These three major impacts reduced reproduction, constrained the number of
territories, and increased mortality, respectively. The populations began to rebound after DDT was
taken off the market in the United States, and after people stopped shooting, trapping, and poisoning
the birds. People also began constructing artificial nesting platforms near lakes and rivers, and with
the nest-site constraint reduced, the species rebounded (Poole et al. 2002). We will model Osprey
population growth using geometric and logistic models, attempting to predict as accurately as possible
the growth of a recovering Osprey population in Wisconsin (data from Eckstein et al. 2003).
Methods
1. Open up your spreadsheet entitled “Osprey population data.xls”. You will see three columns of
data – year, number of pairs, and number of individuals – and several titled but empty columns
to the right. The data set is a time series of the population size of Osprey in Wisconsin from
1973 through 2004. The surveyors who collected this data tallied the number of pairs rather
than the number of individuals because in the field it is often impossible to see both adults. In
raptors, females often will stay close to the nest to defend the young against predators or feed
or shade the young – they are therefore easy to spot. Males, on the other hand, are often off
hunting, so they are not likely to get counted. For our purposes, we will use the number of
birds column, which is just the number of pairs multiplied by two, meaning that we will assume
that for all pairs a male and female exist.
2. Make a time series plot with year on the x-axis and number of birds on the y-axis. Examine the
growth curve. Does there appear to be geometric growth? Is there a leveling off of the
population size somewhere? Where? What is the approximate carrying capacity of the
population?
3. Now let's try to fit our model of geometric growth to the time series.
a) Calculate λ. Notice in equation 9.2 that if you set t = 1 (one time step) and solved for λ you
would get Nt/N0. Therefore, we could calculate λ for each successive time step over the
University of New Mexico
Biology 310L – Principles of Ecology Lab Manual – Page -64
period of geometric growth. The average λ would be an excellent choice for use in equation
9.2. Of course, we already know that after year ______ the population is not growing
geometrically, so only calculate λ for the period of time up until that year. In your
spreadsheet, under the heading “lambda”, enter a formula that will calculate λ. Note that
you must skip 1973, because we do not have data for 1972. What is the average λ?
________. Given the value of λ, should the population grow, stay the same, or decline?
_________________.
b) Predict the geometric population growth. Use equation 9.2. Multiply the population size in
1973 by λ raised to t. In 1974 the time step is 1, so you should raise λ to the power of 1. In
1975, two time steps have passed, so multiply the population size of 1973 by λ raised to the
power of 2. Continue this process until the population is no longer growing geometrically.
Enter λ into your equation with the cell address for λ on the right, and be sure to put a $ in
front of the cell designators, e.g. $J$3. Do the same for the initial population size year (e.g.,
$C$3). You can also make use of the time step column in your formula. Add the predicted
values to the graph (right click graph, click source data, add a series, and select the
appropriate range of x and y values, then hit ok). Does this simple model capture the basic
shape of the recovering Osprey population? _______ Does it appear to miss any important
features?
c) Now extend the predicted series through 2004. If you right-click on the predicted growth
series on the graph, the data columns should light up, and you can extend those series to
include 2004 by dragging the boxes down. Although we saw geometric growth in the
Osprey population for a little while, what is happening to the Osprey population after, say,
1994?
d) Now calculate the λ for the period of time where the population oscillates around carrying
capacity. Enter the value here ________. Given that we can see the population is more or
less stable, is λ what we expect?
4. Now let's try to capture the leveling off of the population using the logistic model (equation
9.4).
a) Estimate K. We will do this by taking the average of the population sizes for the stable part
of the time series, say 1993-2003. Write in your estimate of K here ________ and enter it
in the spreadsheet in the designated cell.
b) Calculate r. It turns out that λ = er, so we should take the natural log of λ to estimate r.
Enter it in the spreadsheet in the designated cell and write in down here ________.
c) Predict population growth. Enter equation 9.4 in the column headed “Predicted birds for
University of New Mexico
Biology 310L – Principles of Ecology Lab Manual – Page -65
logistic growth”. Be sure to use $ signs where necessary, and be sure to use the observed
1973 population size for the original, and then the predicted population size for all
subsequent operations. Drag the equation down through 2003. Add this series of data to
your graph (see directions in 3b).
•
Does this model capture the basic shape of the recovering Osprey population? _______
•
How does the predicted growth differ from the observed population growth?
•
Why do you think this model is failing to accurately predict the population growth?
Hint: Why was there such a quick change from the geometric growth to the steady-state
at carrying capacity?
b) Ecologists say that a model is sensitive to a parameter if changing the parameter a little
causes a large change in the outcome. Is the logistic growth equation sensitive to the value
of r? Experiment with different values of r by simply entering them into the r cell in the
spreadsheet. What happens to the dynamics of the population when r is increased to, say, 2,
or up to 3? What happens if r goes above 3? We will say there are three general types of
patterns – what are they?
d) The intrinsic rate of growth, r, is defined as the birth rate, b, minus the death rate, d. In the
Wisconsin Osprey population, the annual birth rate is 1.22 young per territory, and the
mortality rate of young birds is ~0.6 per year. What is r? _________. This value is
_______ than our empirically estimated r. What could be happening to these surplus
young?
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Biology 310L – Principles of Ecology Lab Manual – Page -66
e) Ecologists and mathematicians have devised a slew of competing models of population
growth that add realism and special circumstances. Frequently a basic model like the
logistic growth model captures the basic pattern but fails to capture some of the subtleties.
To make our models more accurate, we usually need to add more parameters, and in some
cases, change approach entirely. Are there any variables that you think would be important
to add to make our model more realistic?
Literature Cited
Eckstein, R., G. Dahl, B. Glenzinski, B. Ishmael, J. Nelson, P. Manthey, M. Meyer, L. Tesky. 2004.
Wisconsin Bald Eagle and Osprey surveys: 2004.
Unpublished report available at
http://www.dnr.state.wi.us/org/land/wildlife/harvest/Reports/eagleospreysurv04.pdf.
Gotelli, N. J. 2001. A Primer of Ecology. Sinauer Associates, Inc., Sunderland, Massachusetts.
Poole, A. F., R. O. Bierregaard, and M. S. Martell. 2002. Osprey (Pandion haliaetus). In The Birds
of North America, No. 683 (A. Poole and F. Gill, eds.). The Birds of North America, Inc.,
Philadelphia, PA.
Turchin, P. 2001. Does population ecology have general laws? Oikos 94:17-26.
University of New Mexico
Biology 310L – Principles of Ecology Lab Manual – Page -67
Homework # 8 – Population assignment (10 points)
In this assignment you will extend our work on predicting Osprey population trajectories with a new
model.
Model concept
Because it is clear that the logistic growth model fits the Osprey population data very poorly, we need
to try another model. Let's assume that there is no slow, gradual decline in resource availability at this
range of density, and rather, the population is still growing geometrically. Let's also hypothesize that
whenever the population exceeds the number of spaces available on the landscape, excess, young
individuals emigrate to another place, more or less permanently (this is actually true with Osprey).
Thus, we could write a simple model that has the population grow geometrically when the population
size is lower than the maximum, and to lose all excess individuals above the maximum:
If N t  K , then N t1=rN t N t
equation 9.5
If N t  K , then N t1=rN t N t −E t equation 9.6
Assignment
Use the spreadsheet with which we have been working. Enter formulas to make the model predict
population size according to equations 9.5 and 9.6. Hint: you will need to use Excel's if function. The
function allows you to specify if something is true (i.e., is the population size less than the maximum?
- K in our spreadsheet) and to specify what happens under that circumstance, as well as what happens
if the population is greater than the maximum. Just be sure that what happens if the statement is true
or not is an equation that gives you a predicted number of individuals. Finally, insert another predicted
population trajectory in the graph of trajectories. Print out thte graph along with the data (hint:
highlight area with graph and data and set print area), and then answer the following question. Why
does this model still not capture all of the dynamics of the population?