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Title: CHAPTER 52 POPULATION ECOLOGY Author: Atkins Last modified by: Atkins Eric Created Date: 8/7/2008 11:41:15 PM Document presentation format – PowerPoint PPT presentation

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The size  and activities of the human population
are now among Earths most significant problems.
With a population of over 6 billion individuals,
our species requires vast amounts of materials
and space, including places to live, land to grow
our food, and places to dump our waste. Endlessly
expanding our presence on Earth, we have
devastated the environment for many other species
and now threaten to make it unfit for ourselves
. To understand human population growth, we must
consider the general principles of population
ecology. It is obvious that no population can
continue to grow indefinitely. Species other than
humans sometimes exhibit population explosions,
but their populations inevitably decline. In
contrast to these radical booms and busts, many
populations are relatively stable over time, with
only minor changes in population size . In our
earlier study of biological populations (see
Chapter 23), we emphasized the relationship
between population genetics--the structure and
dynamics of gene pools--and evolution. Evolution
remains our central theme as we now view
populations in the context of ecology. Population
ecology, the subject of this chapter, is
concerned with measuring changes in population
size and composition, and with identifying the
ecological causes of these fluctuations. Later in
this chapter, we will return to our discussion of
the human population. Lets first examine some of
the structural and dynamic aspects of populations
as they apply to any species, such as the monarch
butterfly population in the photo on this page .
A population is a group of individuals of a
single species that simultaneously occupy the
same general area. They rely on the same
resources, are influenced by similar
environmental factors, and have a high likelihood
of breeding with and interacting with one
another. The characteristics of a population are
shaped by interactions between individuals and
their environments, and natural selection can
modify these characteristics.
Two important characteristics of any population
are density and the spacing of individuals
At any given moment, every population has
geographic boundaries and a population size (the
number of individuals it includes). Ecologists
begin studying a population by defining
boundaries appropriate to the organisms under
study and to the questions being posed. A
populations boundaries may be natural ones, such
as a specific island in Lake Superior where terns
nest, or they may be arbitrarily defined by an
investigator, such as the oak trees within a
specific county in Minnesota. Regardless of such
differences, two important characteristics of any
population are its density and its dispersion.
Population density is the number of individuals
per unit area or volume--the number of oak trees
per square kilometer in the Minnesota county, for
example. Dispersion is the pattern of spacing
among individuals within the geographic
boundaries of the population.
Measuring Density
In rare cases, it is possible to determine
population size and density by actually counting
all individuals within the boundaries of the
population. We could count the number of sea
stars in a tide pool, for example. Herds of large
mammals, such as buffalo or elephants, can
sometimes be counted accurately from airplanes.
In most cases, however, it is impractical or
impossible to count all individuals in a
population. Instead, ecologists use a variety of
sampling techniques to estimate densities and
total population sizes. For example, they might
estimate the number of alligators in the Florida
Everglades by counting individuals in a few
randomly chosen plots. Or they might count the
numbers of oak trees in several randomly placed
circular plots of 10-m diameter. Such estimates
are more accurate when there are many sample
plots and when the habitat is homogeneous.
Aerial census for African buffalo (Syncerus
caffer) in the Serengeti of East Africa.
Biologists can count large mammals and birds in
open habitats from the air, either directly or
from photographs like this one. By repeating
these counts over many years, researchers can
track population trends.
One sampling technique researchers often use to
estimate fish and wildlife populations is the
mark-recapture method. Traps are placed within
the boundaries of the study area, and captured
animals are marked with tags, collars, bands, or
spots of dye and then immediately released. After
a few days or a few weeks--enough time for the
marked animals to mix randomly with unmarked
members of the population--traps are set again.
The proportion of marked (recaptured) animals in
the second trapping is assumed equivalent to the
proportion of marked animals in the total
Thus, if there have been no births, deaths,
immigration, or emigration, the following
proportionality provides an estimate of the
population size N
For example, suppose that 50 snowshoe hares are
captured in box traps, marked with ear tags, and
released. Two weeks later, 100 hares are captured
and checked for ear tags. If 10 hares in this
second catch are already marked and thus are
recaptures, we would estimate that 10 of the
total hare population is marked. Since 50 hares
were originally marked, the entire population
would be about 500 hares. This method assumes
that each marked individual has the same
probability of being trapped as each unmarked
individual. This is not always a safe assumption,
however. An animal that has been caught once may
become wary of the traps later on or may learn to
return to traps to eat the food used as bait.
In some cases, instead of counting individual
organisms, population ecologists estimate density
from some index of population size. This usually
involves counting signs left by organisms, such
as the number of nests, burrows, tracks, or fecal
Patterns of Dispersion
Within a populations geographic range, local
densities may vary substantially because the
environment is patchy (not all areas provide
equally suitable habitat) and because individuals
exhibit patterns of spacing in relation to other
members of the population.
The most common pattern of dispersion is clumped,
with the individuals aggregated in patches.
Plants may be clumped in certain sites where soil
conditions and other environmental factors favor
germination and growth. The eastern red cedar is
often found clumped on limestone outcrops, where
soil is less acidic than in nearby areas.
Mushrooms may be clumped on a rotting log. Some
animals move in herds. Animals often spend much
of their time in a particular micro-environment
that satisfies their requirements. For example,
many forest insects and salamanders are clumped
under logs, where the humidity remains high.
Herbivorous animals of a particular species are
likely to be most abundant where their food
plants are concentrated. Clumping of animals may
also be associated with mating behavior. For
example, mayflies often swarm in great numbers, a
behavior that increases mating chances for these
insects, which spend only a day or two as
reproductive adults. There may also be "safety in
numbers" fish swimming in large schools, for
example, are often less likely to be eaten by
predators than fish swimming alone or in small
In contrast to a clumped distribution of
individuals within a population, a uniform, or
evenly spaced, pattern of dispersion may result
from direct interactions between individuals in
the population. For example, a tendency toward
regular spacing of plants may be due to shading
and competition for water and minerals some
plants also secrete chemicals that inhibit the
germination and growth of nearby individuals that
could compete for resources. Animals often
exhibit uniform dispersion as a result of
territorial behavior and aggressive social
interactions. Uniform patterns are not as common
in populations as clumped patterns.
Random spacing (unpredictable dispersion) occurs
in the absence of strong attractions or
repulsions among individuals of a population the
position of each individual is independent of
other individuals. For example, trees in a forest
are sometimes randomly distributed. Random
patterns are not as common in nature as one might
expect most populations show at least a tendency
toward a clumped distribution.
Demography is the study of factors that affect
the growth and decline of populations
Changes in population size reflect the relative
rates of processes that add individuals to the
population and eliminate individuals from it.
Additions occur through births (which we will
define here to include all forms of reproduction)
and immigration, the influx of new individuals
from other areas. Opposing these additions are
mortality (death) and emigration, the movement of
individuals out of a population. Our focus in
this chapter is primarily on factors that
influence birth rates and death rates, but you
should remember that immigration and emigration
may also play a role in population dynamics. The
study of the vital statistics that affect
population size is called demography. Birth rates
vary among individuals (specifically, among
females) within a population, depending, in
particular, on age and death rates depend on
both age and sex. Lets see how these demographic
variables affect population dynamics.
Life Tables and Survivorship Curves
About a century ago, when life insurance first
became available, insurance companies developed
an interest in the mathematics of survival. They
needed to estimate how long, on average, an
individual of a given age could be expected to
live. Some of the greatest demographers of the
past century worked for life insurance companies.
They invented demographic representations called
life tables. A life table is an age-specific
summary of the survival pattern of a population.
Population ecologists adapted this approach for
nonhuman populations and developed quantitative
demography as a branch of biology. The best way
to construct a life table is to follow the fate
of a cohort, a group of individuals of the same
age, from birth until all are dead. The table is
constructed from the number of individuals that
die in each age-group during the defined time
period. Cohort life tables are difficult to
collect on wild animals and plants and are
available only for a limited number of species.
This is a life table for a cohort of Belding
ground squirrels (Spermophilus beldingi ) at
Tioga Pass, in California. Much can be learned
about a population from a life table. The third
column in the table shows the proportion of
individuals in a cohort that are still alive at a
given age. Notice that the death rates are
generally highest among the youngest ground
squirrels and among the oldest individuals and
that males suffer higher rates of loss than
A graphic way of representing the data in a life
table is to draw a survivorship curve, a plot of
the proportion or numbers in a cohort still alive
at each age. Survivorship curves can be
classified into three general types. A Type I
curve is relatively flat at the start, reflecting
low death rates during early and middle life,
then drops steeply as death rates increase among
older age-groups. Humans and many other large
mammals that produce relatively few offspring but
provide them with good care often exhibit this
kind of curve. In contrast, a Type III curve
drops sharply at the left of the graph,
reflecting very high death rates for the young,
but then flattens out as death rates decline for
those few individuals that have survived to a
certain critical age. This type of curve is
usually associated with organisms that produce
very large numbers of offspring but provide
little or no care, such as many fishes and marine
invertebrates. An oyster, for example, may
release millions of eggs, but most offspring die
as larvae from predation or other causes. Those
few that manage to survive long enough to attach
to a suitable substrate and begin growing a hard
shell, however, will probably survive for a
relatively long time. Type II curves are
intermediate, with a constant death rate over the
life span. This kind of survivorship occurs in
some annual plants, various invertebrates such as
Hydra , some lizard species, and some rodents,
such as the gray squirrel.
Many species, of course, fall somewhere between
these basic types of survivorship or show more
complex patterns. In birds, for example,
mortality is often high among the youngest
individuals (as in a Type III curve) but fairly
constant among adults (as in a Type II curve).
Some invertebrates, such as crabs, may show a
"stair-stepped" curve, with brief periods of
increased mortality during molts (caused by
physiological problems or greater vulnerability
to predation), followed by periods of lower
mortality (when the exoskeleton is hard). In
populations without immigration or emigration,
survivorship is one of the two key factors
determining changes in population size. Next we
consider reproductive output, the other key
factor determining population trends.
Idealized survivorship curves. As an example of
a Type I curve, humans in developed countries
experience high survival rates until old age. At
the opposite extreme are Type III curves for
organisms such as oysters, which experience very
high mortality as larvae but decreased mortality
later in life. Type II survivorship curves are
intermediate between the other two types and
result when a constant proportion of individuals
die at each age. Notice that the y axis is
logarithmic and that the x axis is on a relative
scale, so that species with widely varying life
spans can be compared on the same graph.
Reproductive Rates
Demographers who study sexually reproducing
species generally ignore males and concentrate on
females in the population because only females
give birth to offspring. Demographers view
populations in terms of females giving rise to
new females males are important only as
distributors of genes. How can we describe the
reproductive program of a population? The
simplest way is to follow the basic approach of
the life table and ask how reproductive output
varies with age.
A reproductive table, or fertility schedule, is
an age- specific summary of the reproductive
rates in a population. The best way to construct
a fertility schedule is to measure the
reproductive output of a cohort from birth until
death. For sexual species, the reproductive table
tallies the number of female offspring produced
by each age-group. The table below illustrates a
reproductive table for Belding ground squirrels.
Reproductive output for sexual species like birds
and mammals is a product of the fraction of
females of a given age that are breeding and the
number of female offspring of those breeding
females. By multiplying these together, we can
obtain the average output of daughters for each
individual in a given age class (the last
column). For these ground squirrels, which begin
to reproduce at age 1 year, reproductive output
rises to a peak at 4 years of age and then falls
off in older females.
Reproductive tables vary greatly, depending on
the species. Squirrels have a litter of two to
six young once a year, whereas oak trees drop
thousands of acorns each year for tens or
hundreds of years. Salmon lay thousands of eggs
when they spawn, and mussels and other
invertebrates may release hundreds of thousands
of eggs in a spawning cycle. Why does one type of
life cycle rather than another evolve in a
particular population? This is one of the many
questions at the interface of population ecology
and evolutionary biology.
Natural selection will favor traits in organisms
that improve their chances of survival and
reproductive success. Organisms that survive a
long time but do not reproduce are not at all
"fit" in the Darwinian sense. In every species,
there are trade-offs between survival and traits
such as frequency of reproduction, investment in
parental care, and the number of offspring
produced (seed crops for seed plants and litter
size or clutch size for animals). The traits that
affect an organisms schedule of reproduction and
survival (from birth through reproduction to
death) make up its life history. Of course, a
particular life history, like most
characteristics of an organism, is the result of
natural selection operating over evolutionary
time. Life history traits help determine how
populations grow.
Life histories are highly diverse, but they
exhibit patterns in their variability
Because of varying environmental contexts for
natural selection, life histories are very
diverse. Pacific salmon, for example, hatch in
the headwaters of a stream, then migrate to open
ocean, where they require one to four years to
mature. They eventually return to freshwater
streams to spawn, producing thousands of small
eggs in a single reproductive opportunity, and
then they die. Ecologists call this big-bang
reproduction.  This figure illustrates big-bang
reproduction in agaves. The agave, or century
plant, grows in arid climates with sparse and
unpredictable rainfall. Agaves grow vegetatively
for several years, then send up a large flowering
stalk, produce seeds, and die. (We introduced the
big-bang reproduction of century plants on the
opening page of Chapter 38.) The shallow roots of
agaves catch water after rain showers but are dry
during droughts. This unpredictable water supply
may prevent seed production or seedling
establishment for several years at a time. By
growing and storing nutrients until an unusually
wet year and then putting all its resources into
reproduction, the agaves big-bang strategy is a
life history adaptation to erratic climate. In
another example of big-bang reproduction, annual
desert wildflowers generally germinate, grow,
produce many small seeds, and then die, all in
the span of a month after spring rains. Big-bang
(one-time) reproduction is also called
semelparity (from the Latin semel , once, and
parito , to beget).
An example of big-bang reproduction. Agaves, or
century plants, grow without reproducing for
several years and then produce a gigantic
flowering stalk and many seeds. After this
onetime reproductive effort, the plant dies.
In contrast to big-bang reproduction, some
lizards produce only a few large eggs during
their second year of life, then repeat the
reproductive act annually for several years. And
some species of oaks do not reproduce until the
tree is 20 years old, but then produce vast
numbers of large seeds each year for a century or
more. Ecologists call this repeated reproduction
or iteroparity (from the Latin itero , to repeat).
What factors contribute to the evolution of
semelparity versus iteroparity? That is, how much
will an individual gain in reproductive success
through one strategy versus the other? The key
demographic effect of big-bang reproduction is
higher reproductive rates. Plants like agaves
that reproduce only once typically produce two to
five times as many seeds as closely related
species that reproduce repeatedly. The critical
factor in the evolutionary dilemma of big-bang
versus repeated reproduction is the survival rate
of the offspring. If their chance of survival is
poor or inconsistent, repeated reproduction will
be favored.
Limited resources mandate trade-offs between
investments in reproduction and survival
Darwinian fitness is measured not by how many
offspring are produced but by how many survive to
produce their own offspring Heritable
characteristics of life history that result in
the most reproductively successful descendants
will become more common within the population. If
we were to construct a hypothetical life history
that would yield the greatest lifetime
reproductive output, we might imagine a
population of individuals that begin reproducing
at an early age, produce many offspring each time
they reproduce, and reproduce many times in a
lifetime. However, natural selection cannot
maximize all these variables simultaneously,
because organisms have finite resources, and
limited resources mean trade-offs. Ecologists who
study the evolution of life histories focus on
how these trade-offs operate in specific
populations. For example, the production of many
offspring with little chance of survival may
result in fewer descendants than the production
of a few well-cared-for offspring that can
compete vigorously for limited resources in an
already dense population.
The life histories we observe in organisms
represent an evolutionary resolution of several
conflicting demands. Time, energy, and nutrients
that are used for one thing cannot be used for
something else. In the broadest sense, there is a
trade-off between reproduction and survival, and
this has been demonstrated by several studies.
For example, in red deer on the Scottish island
of Rhum, females that reproduce in one summer
suffer higher mortality over the following winter
than do females that did not reproduce. This cost
of reproduction was found even in red deer in the
prime of life, but was particularly severe in the
older females. And in many insect species,
females that lay fewer eggs live longer,
suggesting a similar trade-off between investing
in current reproduction and survival.
Cost of reproduction in female red deer on the
island of Rhum, in Scotland. Mortality in winter
is higher for females that reproduced during the
previous summer, no matter what the age of the
There can also be trade-offs between current and
future reproduction. When perennial plants
produce more seeds in one year, they grow less
and have reduced seed production the next year.
Moreover, experimental transfers of eggs or
nestlings in bird populations have measured the
trade-off between reproductive effort and
survival. When nestlings of European kestrels
were transferred among nests to produce broods of
three or four (reduced), five or six (normal),
and seven or eight (enlarged), adult kestrels
that raised the enlarged broods survived poorly
over the following winter.
Probability of survival over the following year
for European kestrels after raising a modified
brood. A total of 200 birds were studied from
1985 to 1990 in the Netherlands. Adults with
experimentally enlarged broods die more often
over the following winter. (Both males and
females provide parental care for the nestlings.)
As in our red deer and kestrel examples, many
life history issues involve balancing the profit
of immediate investment in offspring against the
cost to future prospects of survival and
reproduction. These issues can be summarized by
three basic "decisions" when to begin
reproducing, how often to breed, and how many
offspring to produce during each reproductive
episode. The various "choices" are integrated
into the life history patterns we see in nature.
It is important to clarify our use of the word
choice . Organisms do not choose consciously when
to breed and how many offspring to have. (Humans
are an important exception we will consider later
in the chapter.) Life history traits are
evolutionary outcomes reflected in the
development, physiology, and behavior of an
organism. Age at maturity and the number of
offspring produced during a given reproductive
episode are usually maintained within narrow
ranges by stabilizing selection. Natural
selection molds reproductive patterns in
populations such patterns are not consciously
chosen by the organism.
As with all life history adaptations, the number
and size of offspring depend on the selective
pressures under which the organism evolved.
Plants and animals whose young are subject to
high mortality rates often produce large numbers
of relatively small offspring. Thus, plants that
colonize disturbed environments usually produce
many small seeds, most of which will not reach a
suitable environment. Small size might actually
benefit such seeds if it enables them to be
carried long distances. Birds and mammals that
suffer high predation rates also produce large
numbers of offspring examples include quail,
rabbits, and mice.
Variation in seed crop size in plants. Most weedy
plants, such as this dandelion, grow quickly and
produce a large number of seeds. Although most of
the seeds will not produce mature plants, their
large number and ability to disperse to new
habitats ensure that at least some will grow and
eventually produce seeds themselves.
In other organisms, extra investment on the part
of the parent greatly increases the offsprings
chances of survival. Oak, walnut, and coconut
trees all have large seeds with a large store of
energy and nutrients that the seedlings can use
to become established. In animals, parental
investment in offspring does not always end with
incubation or gestation. Primates generally have
only one or two offspring at a time. Parental
care and an extended period of learning in the
first several years of life are very important to
offspring fitness in these mammals.
Now that we have analyzed some patterns that
underlie diverse life histories, lets examine
the effects of these life history traits on the
growth of populations.
Variation in seed crop size in plants. Some
plants, such as this coconut palm, produce a
moderate number of very large seeds. The large
endosperm provides nutrients for the embryo (a
plants version of parental care), an adaptation
that helps ensure the success of a relatively
large fraction of offspring. Animal species
exhibit similar trade-offs between number of
offspring and the amount of nutrients provided to
each offspring.
To begin to understand the potential for
population increase, consider a single bacterium
that can reproduce by fission every 20 minutes
under ideal laboratory conditions. At the end of
this time, there would be two bacteria, four
after 40 minutes, and so on. If this continued
for only a day and a half--a mere 36 hours--there
would be enough bacteria to form a layer a foot
deep over the entire Earth. At the other life
history extreme, elephants may produce only six
young in a 100-year life span. Still, Darwin
calculated that it would take only 750 years for
a single pair of elephants to produce a
population of 19 million. Obviously, indefinite
population increase does not occur for any
species, either in the laboratory or in nature. A
population that begins at a low level in a
favorable environment may increase rapidly for a
while, but eventually the numbers must, as a
result of limited resources and other factors,
stop growing.
As we discussed in Chapter 50, finding the
answers to ecological questions depends on a
combination of observation and experimentation.
The two major forces affecting population
growth--birth rates and death rates--can be
measured in many populations and used to predict
how the populations will change in size over
time. Small organisms can be studied in the
laboratory to determine how various factors
affect their population growth rates, and natural
populations can be experimentally manipulated to
answer the same questions. Mathematical models
can be used for testing hypotheses about the
effects of different factors on population growth
once we understand how birth and death rates
change over time. We can begin to understand
population growth by looking at a few simple
models of how a population can grow.
The exponential model of population growth
describes an idealized population in an
unlimited environment
Imagine a hypothetical population consisting of a
few individuals living in an ideal, unlimited
environment. Under these conditions, there are no
restrictions on the abilities of individuals to
harvest energy, grow, and reproduce, aside from
the inherent physiological limitations that are
the result of their life history. The population
will increase in size with every birth and with
the immigration of individuals from other
populations and decrease in size with every death
and with the emigration of individuals out of the
population. For simplicity, lets ignore the
effects of immigration and emigration (a more
complex formulation would certainly include these
factors). We can define a change in population
size during a fixed time interval with the
following verbal equation
Using mathematical notation, we can express this
relationship more concisely. If N represents
population size and t represents time, then ?N is
the change in population size and ?t is the time
interval (appropriate to the life span or
generation time of the species) over which we are
evaluating population growth. (The Greek letter
delta, ?, indicates change, such as change in
time.) We can now rewrite the verbal equation as
where B is the number of births in the population
during the time interval and D is the number of
Similarly, the per capita death rate, symbolized
as d , allows us to calculate the expected number
of deaths per unit time in a population of any
size. If d 0.016 per year, we would expect 16
deaths per year in a population of 1,000
individuals. (Using the formula D dN , how many
deaths would you expect per year if d 0.010
annually in populations of 500, 700, and 1,700?)
For natural populations or those in the
laboratory, the per capita birth rates and death
rates can be calculated from estimates of
population size and data given in life tables and
reproductive tables.
We can revise the population growth equation
again, this time using per capita birth rates and
death rates rather than the numbers of births
and deaths                 
One final simplification is in order. Population
ecologists are concerned with overall changes in
population size, using r to identify the
difference in the per capita birth rates and
death rates           
This value, the per capita growth rate, tells
whether a population is actually growing
(positive value of r ) or declining (negative
value of r ). Zero population growth (ZPG) occurs
when the per capita birth rates and death rates
are equal (r 0). Note that births and deaths
still occur in the population, but they balance
each other exactly. (Later in this chapter, we
will discuss the relevance of ZPG for the human
population and the factors preventing the human
population from leveling off.)
Using the per capita growth rate, we rewrite the
equation for change in population size as
Finally, most ecologists use the notation of
differential calculus to express population
growth in terms of instantaneous growth rates
If you have not yet studied calculus, dont be
intimidated by the form of the last equation it
is essentially the same as the previous one,
except that the time intervals ?t are very short
and are expressed in the equation as dt . (Do not
confuse this use of d to symbolize very small
change with our earlier use of d to represent per
capita death rate.)
We started this section by describing a
population living under ideal conditions, where
organisms are constrained only by their life
history. In such a situation, the population
grows rapidly, because all members have access to
abundant food and are free to reproduce at their
physiological capacity. Population increase under
these ideal conditions is called exponential
population growth, or geometric population
growth. Under these conditions the per capita
growth rate may assume the maximum growth rate
for the species, called the intrinsic rate of
increase, denoted as rmax . And the equation for
exponential population growth is then
The size of a population that is growing
exponentially increases rapidly, resulting in a
J-shaped growth curve when population size is
plotted over time. Although the intrinsic rate of
increase is constant as the population grows, the
population actually accumulates more new
individuals per unit of time when it is large
than when it is small thus, the curves in FIGURE
52.8 get progressively steeper over time. This
occurs because population growth depends on N as
well as r , and larger populations experience
more births (and deaths) than small ones growing
at the same per capita rate. It is also clear
from FIGURE 52.8 that a population with a higher
intrinsic rate of increase (dN/dt 1.0N ) will
grow faster than one with a lower rate of
increase (dN/dt 5 0.5N ).
Population growth predicted by the exponential
model. The exponential growth model predicts
unlimited population increase under conditions of
unlimited resources. This graph compares growth
in populations with two different values of r
1.0 and 0.5.
The J-shaped curve of exponential growth is
characteristic of some populations that are
introduced into a new or unfilled environment or
whose numbers have been drastically reduced by a
catastrophic event and are rebounding. For
example, this figure illustrates exponential
population growth in the whooping crane, an
endangered species now recovering from the impact
of habitat loss due to agriculture.
Example of exponential population growth in
nature. The whooping crane is an endangered
species that has been recovering from near
extinction since 1940. Counts of adults are made
annually on the wintering grounds at Aransas,
Texas. In the year 2000-2001, there were 179
birds in the wintering population in Texas, the
population having declined slightly from the
preceding year. The overall average rate of
increase has been 4 per year since the 1950s.
The logistic model of population growth
incorporates the concept of carrying capacity
The exponential growth model assumes unlimited
resources, which is never the case in the real
world. No population--neither bacteria nor
elephants nor any other organisms--can grow
exponentially indefinitely. As any population
grows larger in size, its increased density may
influence the ability of individuals to harvest
sufficient resources for maintenance, growth, and
reproduction. Populations subsist on a finite
amount of available resources, and as the
population becomes more crowded, each individual
has access to an increasingly smaller share.
Ultimately, there is a limit to the number of
individuals that can occupy a habitat. Ecologists
define carrying capacity as the maximum
population size that a particular environment can
support at a particular time with no degradation
of the habitat. Carrying capacity, symbolized as
K , is not fixed, but varies over space and time
with the abundance of limiting resources. For
example, the carrying capacity for bats may be
high in a habitat where flying insects are
abundant and there are caves for roosting but
lower in a habitat where food is abundant but
suitable shelters are less common. Energy
limitation is one of the most significant
determinants of carrying capacity, although other
factors, such as shelters, refuges from
predators, soil nutrients, water, and suitable
nesting and roosting sites, can be limiting.
Crowding and resource limitation can have a
profound effect on the population growth rate. If
individuals cannot obtain sufficient resources to
reproduce, per capita birth rate will decline. If
they cannot find and consume enough energy to
maintain themselves, per capita death rates may
also increase. A decrease in b or an increase in
d results in a smaller r and a lower overall rate
of population growth.
Yellow bacterial colonies with EPS of
Xanthomonas campestris pv. vesicatoria grown on
sucrose-peptone-agar medium
The Logistic Growth Equation
We can modify our mathematical model of
population growth to incorporate changes in
growth rate as the population size nears the
carrying capacity (as N grows toward K ). The
logistic population growth model incorporates the
effect of population density on the per capita
rate of increase, allowing this rate to vary from
a maximum at low population size to zero as
carrying capacity is reached. When a populations
size is below the carrying capacity, population
growth is rapid, but as N approaches K ,
population growth slows down.
Mathematically, we construct the logistic model
by starting with the model of exponential
population growth and creating an expression that
reduces the rate of population increase as N
increases. If the maximum sustainable population
size is K , then K -N tells us how many
additional individuals the environment can
accommodate, and (K -N )/K tells us what fraction
of K is still available for population growth. By
multiplying the exponential rate of increase
rmaxN by (K -N )/K , we reduce the actual growth
rate of the population as N increases
Reduction of population growth rate with
increasing population size (N ). The logistic
model of population growth assumes that the
population growth rate dN/dt decreases as N
increases. When N is close to 0, the population
grows rapidly. However, as N approaches K (the
carrying capacity of the environment), the
population growth rate approaches 0, and
population growth slows. If N is greater than K ,
then the population growth rate is negative, and
population size decreases. An equilibrium is
reached at the white line when N K .
The table below shows hypothetical calculations
for the rate of population increase and changes
in N at various population sizes for a population
growing according to the logistic model. Notice
that when N is small compared to K , the term (K
-N )/K is large, and the actual rate of
population increase (dN/dt ) is close to the
intrinsic (maximum) rate of increase. But when N
is large and resources are limiting, then (K -N
)/K is small, and so is the rate of population
increase. Zero population growth occurs when the
numbers of births and deaths are equal--when N
equals K .
The logistic model of population growth produces
a sigmoid (S-shaped) growth curve when N is
plotted over time. New individuals are added to
the population most rapidly at intermediate
population sizes, when there is not only a
breeding population of substantial size, but also
lots of available space and other resources in
the environment. The population growth rate slows
dramatically as N approaches K .
Notice that we havent said anything about what
makes the population growth rate change as N
approaches K . Either the birth rate b must
decrease, the death rate d must increase, or
both. Later in the chapter, we will go into some
detail about some of the factors affecting b and
d .
Population growth predicted by the logistic
model. The logistic growth model assumes that
there is a maximum population size that the
environment can support--the carrying capacity K
. The rate of population growth slows as the
population approaches the carrying capacity of
the environment. The red line shows logistic
growth in a population where rmax 1.0 and K
1,500 individuals. For comparison, the blue line
illustrates a population continuing to grow
exponentially with the same rmax .
How Well Does the Logistic Model Fit the Growth
of Real Populations?
The growth of laboratory populations of some
small animals, such as beetles or crustaceans,
and of microorganisms, such as paramecia, yeasts,
and bacteria, fit S-shaped curves fairly well
(FIGURE a). These experimental populations are
grown in a constant environment lacking predators
and other species that may compete for resources,
conditions that rarely occur in nature. Even
under these laboratory conditions, not all
populations show logistic growth patterns.
Laboratory populations of water fleas (Daphnia )
, for example, show exponential growth and
overshoot their carrying capacity before settling
down to a relatively stable density (FIGURE b).
Most populations show some deviations from a
smooth sigmoid curve. And while many natural
populations increase in approximately logistic
fashion, the stable carrying capacity is rarely
observed. FIGURE c shows population changes in
the song sparrow on a small island in southern
British Columbia. The population increases
rapidly but suffers periodic catastrophes in
winter, so that there is no stable population
How well do these populations fit the logistic
population growth model? The dots on these
graphs are the actual data points.
Some of the basic assumptions built into the
logistic model clearly do not apply to all
populations. For example, the model incorporates
the idea that even at low population levels, each
individual added to the population has the same
negative effect on population growth rate. Some
populations, however, show an Allee effect (named
after W. C. Allee, of the University of Chicago,
who first described it), in which individuals may
have a more difficult time surviving or
reproducing if the population size is too small.
For example, a single plant standing alone may
suffer from excessive wind but would be protected
in a clump of individuals. And some seabirds that
breed in colonies require large numbers at their
breeding grounds to provide the necessary social
stimulation for reproduction. Moreover,
conservation biologists fear that populations of
solitary animals, such as rhinoceroses, may
become so small that individuals will not be able
to locate mates in the breeding season. In all
these cases, a greater number of individuals in
the population has a positive effect, up to a
point, on population growth, rather than a
negative effect as assumed by the logistic model.
The logistic model also makes the assumption that
populations adjust instantaneously and approach
carrying capacity smoothly. In most natural
populations, however, there is a lag time before
the negative effects of an increasing population
are realized. For example, as some important
resource, such as food, becomes limiting for a
population, reproduction will be reduced, but the
birth rate may not be affected immediately
because the organisms may use their energy
reserves to continue producing eggs for a short
time. This may cause the population to overshoot
the carrying capacity. Eventually, deaths will
exceed births, and the population may then drop
below carrying capacity even though reproduction
begins again as numbers fall, there is a delay
until new individuals actually appear. Many
populations fluctuate strongly, which makes it
difficult to define what is meant by carrying
capacity (see FIGURE c). Others overshoot it at
least once before attaining a relatively stable
size (see FIGURE b). We will examine some
possible reasons for these fluctuations later in
the chapter.
As you will see in the next section, some
populations do not necessarily remain at, or even
reach, levels where population density is an
important factor. In many insects and other
small, quickly reproducing organisms that are
sensitive to environmental fluctuations, physical
variables such as temperature or moisture usually
reduce the population well before resources
become limiting. Overall, the logistic model is
a useful starting point for thinking about how
populations grow and for constructing more
complex models. Although it fits few, if any,
real populations closely, the logistic model is
useful in conservation biology and in pest
control to estimate how rapidly a particular
population might increase in numbers after it has
been reduced to a small size. And like any good
starting hypothesis, the logistic model has
stimulated much research and many discussions
that, whether they support the model or not, lead
to a greater understanding of the factors
affecting population growth.
Female with ootheca (egg case)

The Logistic Population Growth Model and Life
The logistic model predicts different growth
rates for low-density populations and
high-density populations, relative to the
carrying capacity of the environment. At high
densities, each individual has few resources
available, and the population can grow slowly, if
at all. At low densities, the opposite is true
Resources per capita are relatively abundant, and
the population can grow rapidly. Different life
history features will be favored under these
different conditions. At high population density,
selection favors adaptations that enable
organisms to survive and reproduce with few
resources. Thus, competitive ability and maximum
efficiency of resource utilization should be
favored in populations that are at or near their
carrying capacity. At low population density, on
the other hand, even in the same species, the
"empty" environment should favor adaptations that
promote rapid reproduction. Increased fecundity
and earlier maturity, for example, would be
selected for.
Thus, the life history traits that natural
selection favors may vary with population density
and environmental conditions. Selection for life
history traits that are sensitive to population
density can be called K -selection, or
density-dependent selection. In contrast,
selection for life history traits that maximize
reproductive success in uncrowded environments
(low densities) can be called r -selection, or
density-independent selection. These names follow
from the variables of the logistic equation. K
-selection tends to maximize population size and
operates in populations living at a density near
the limit imposed by their resources (the
carrying capacity K ). By contrast, r -selection
tends to maximize r , the rate of increase, and
occurs in variable environments in which
population densities fluctuate well below
carrying capacity or in open habitats where
individuals are likely to face little competition.
Characteristics of r- and K-selected organisms Characteristics of r- and K-selected organisms
r-organisms K-organisms
short-lived long-lived
small large
weak strong or well-protected
waste a lot of enrgey energy efficient
less intelligent more intelligent
have large litters have small litters
reproduce at an early age reproduce at a late age
fast maturation slow maturation
little care for offspring much care for offspring
strong sex drive weak sex drive
small size at birth large size at birth
In laboratory experiments, researchers have shown
that different populations of the same species
may show a different balance of K -selected and r
-selected traits, depending on conditions. For
example, cultures of the fruit fly Drosophila
melanogaster raised under crowded conditions with
minimal food for 200 generations are more
productive at high density than populations
raised in uncrowded conditions with maximal food.
Larvae from cultures selected for living in
crowded conditions feed faster than larvae
selected for living in uncrowded cultures. The
fruit fly genotypes that are most fit at low
density do not have high fitness at high density,
as predicted by r- and K - selection theory.
There are two general questions that we can ask
about population growth. First, why do all
populations eventually stop increasing?
Exponential population growth is rare in nature
and always of short duration. What environmental
factors stop a population from growing? If we
have an introduced weed that is spreading
rapidly, what should we do to stop its population
growth? Second, why is the population density of
a particular species greater in some habitats
than in others? Every bird-watcher can tell you
what the favorable and unfavorable habitats are
for any particular bird species. What determines
a favorable habitat, and how do we turn an
unfavorable habitat into a good one?
These questions have many practical applications.
A conservation biologist might want to turn a
declining species population into an increasing
one. And in agriculture, the objective may be to
get a pest population to decrease. Moreover,
agricultural pests may have severe effects in
some areas and negligible effects in others. Why?
Endangered species, meanwhile, such as humpback
whales, require good habitats for survival. What
environmental factors create a favorable feeding
habitat for humpbacks? All these practical issues
involve population-limiting factors. Regulation
is one of this books ten themes (see Chapter 1).
In this section, we apply that theme to
The first step in understanding why a population
stops growing is to find out how the rates of
birth, death, immigration, and emigration change
as population density rises. If immigration and
emigration offset each other, then a population
grows when the birth rate exceeds the death rate
and declines when the death rate exceeds the
birth rate. This graph shows a simple graphical
model of how a population may stop increasing and
reach equilibrium. A death rate that rises as
population density rises is said to be density
dependent, as is a birth rate that falls with
rising density. Density-dependent rates are an
example of negative feedback, a type of
regulation you learned about in Chapter 1. In
contrast, a birth rate or death rate that does
not change with population density is said to be
density independent. With density-independent
rates, there is no feedback to slow down
population growth.
Graphic model showing how equilibrium may be
determined for population density. Population
density reaches equilibrium only when the per
capita birth rate equals the per capita death
rate, and this is possible only if the birth or
death rate (or both) changes with density (is a
density-dependent rate). In this simple model,
immigration and emigration are assumed to be
either zero or equal.
Negative feedback prevents unlimited population
No population stops growing without some type of
negative feedback between population density and
the vital rates of birth and death. Once we know
how birth and death rates change with population
density, we need to determine the mechanisms
causing these changes. Because populations are
affected by a variety of factors that cause
negative feedback, it can be a challenge to
pinpoint the exact factors at work in a
particular population.
Although field studies may eventually shed light
on the most important factors producing negative
feedback in specific cases, they have not yet
provided many generalizations. First, much of the
research on populations has been conducted in the
temperate zone, and we need many more studies of
tropical and polar organisms to complete the
picture. Second, birds and mammals have been the
subjects of much more research than have other
organisms. In particular, insects, which form the
dominant group of species on Earth, have not been
studied in proportion to their species richness.
Finally, long time periods are required for
experimental work on population dynamics, with
definitive studies routinely taking 10 to 20
years for completion. With these reservations in
mind, let us look at several examples of how
birth and death rates change with population
density--in some cases where the mechanisms
behind these changes are well understood.
Resource limitation in crowded populations can
stop population growth by reducing reproduction.
For example, crowding can reduce seed production
by plants (FIGURE a). And available food supplies
often limit the reproductive output of songbirds
as bird population density increases in a
particular habitat, each female lays fewer eggs,
a density-dependent response (FIGURE b). In both
of these examples, increasing population density
intensifies intraspecific competition for
declining nutrients, resulting in a lower birth
Decreased fecundity at high population densities.
Factors other than intraspecific competition for
food or nutrients can also cause
density-dependent behavior of populations. In
many vertebrates and some invertebrates,
territoriality, the defense of a well-bounded
physical space, may set a limit on density, so
that the space that constitutes the territory
becomes the resource for which individuals
compete. For oceanic birds such as gannets, which
nest on rocky islands where they are relatively
safe from predators, the limited number of
suitable nesting sites allows only a certain
number of pairs to nest and reproduce. Up to a
certain population size, most birds can find a
suitable nest site, but few birds beyond that
threshold breed successfully. Thus, the limiting
resource that determines breeding population size
for gannets is safe nesting space. As this space
fills, birds that cannot obtain a nesting spot do
not reproduce. Surplus, or nonbreeding,
individuals are a good indication that
territoriality is restricting population growth,
as it does in many bird populations.
Population density also influences the health and
thus the survival of organisms. Plants grown
under crowded conditions tend to be smaller and
less robust than those grown at lower densities.
Small plants are less likely to survive, and
those that do survive produce fewer flowers,
fruit, and seeds. Gardeners who recognize this
phenomenon thin their seed lings to produce the
best possible yield. Animals, too, experience
increased mortality at high population densities.
In laboratory studies of flour beetles, for
example, the percentage of eggs that hatch and
survive to adulthood decreases steadily as
density increases from moderate to high levels.
The main cause of this density-dependent effect
is cannibalism of eggs by adult beetles and large
Decreased survivorship at high population
densities. The percentage of flour beetles
(Tribolium confusum ) surviving from egg stage to
adult in a laboratory culture decreases as
density increases from moderate to high
population densities, reducing the numbers of
adults in the next generation.
Predation may also be an important cause of
density- dependent mortality for some prey
populations if a predator encounters and captures
more food as the population density of the prey
increases. Many predators, for example, exhibit
switching behavior They begin to concentrate on
a particularly common species of prey when it
becomes energetically efficient to do so. For
example, trout may concentrate for a few days on
a particular species of insect that is emerging
from its aquatic larval stage, then switch as
another insect species becomes more abundant. As
a prey population builds up, predators may feed
preferentially on that species, consuming a
higher percentage of individuals this can cause
density- dependent regulation of the prey
The accumulation of toxic wastes is another
component that can contribute to
density-dependent regulation of population size.
In laboratory cultures of small microorganisms,
for example, metabolic by-products accumulate as
the population grows, poisoning the population
within this limited, artificial environment.
Indeed, ethanol accumulates as a waste product
when yeast ferments sugar. The alcohol content of
wine is usually less than 13, the maximum
ethanol concentration that most wine-producing
yeast cells can tolerate.
The impact of a disease on a population can be
density dependent if the transmission rate of the
disease depends on a certain level of crowding in
the population. For example, tuberculosis strikes
a greater percentage of people living in cities
than in rural areas.
For some animal species, intrinsic factors,
rather than the extrinsic factors just discussed,
appear to regulate population size. White-footed
mice in a small field enclosure will multiply
from a few to a colony of 30 to 40 individuals,
but eventually, reproduction will decline until
the population ceases to grow. This drop in
reproduction is associated with aggressive
interactions that increase with population
density, and it occurs even when food and shelter
are provided in abundance. Although the exact
mechanisms by which aggressive behavior affects
reproductive rate are not yet understood, we do
know that high population densities in mice
induce a stress syndrome in which hormonal
changes can delay sexual maturation, cause
reproductive organs to shrink, and depress the
immune system. In this case, high densities cause
both an increase in mortality and a decrease in
birth rates. Similar effects of crowding occur in
wild populations of woodchucks and other rodents.
Population dynamics reflect a complex interaction
of biotic and abiotic influences
In these various examples of population
regulation by negative feedback, we have seen how
increased densities cause population growth rates
to decline by affecting reproduction, growth, and
survivorship in the individuals that make up the
populations. This helps us answer our first
question about populations why all populations
eventually stop increasing. Now lets turn to our
second question why certain habitats favor
greater population densities.
There are good and bad habitats for every
species. Carrying capacities can vary in space
some parts of a lake provide better fishing than
other parts, for example. Carrying capacities can
also vary in time grasshoppers may be serious
pests for farmers in some years and nearly absent
in other years. Over the long term, most
populations exhibit change. Some remain fairly
stable in size, but most populations for which we
have long-term data show fluctuations in numbers.
Although we can determine an average population
size for many species, the average is often of
less interest than the year-to-year or
place-to-place trend in numbers. For example,
this graph illustrates the fluctuation in the
number of northern pintail ducks from 1955 to
1998. Pintails nest in the prairie regions of the
United States and Canada, and their present
populations have declined considerably below the
levels of the 1950s. Wildlife managers need to
find out why these changes occur. Researchers
have identified the loss of prairie ponds (either
by drying up in droughts or by draining for
agriculture) as one key abiotic factor
contributing to the pintail duck decline shown in
the graph. However, when heavy rains kept prairie
ponds full in the 1990s, the pintails did not
recover. Pintails nest in the stubble left after
grain is harvested. More intensive agriculture in
recent years has resulted in early cultivation of
stubble fields and the destruction of many
pintail nests. As more and more land is taken
over by agricultural fields, duck nests also
become concentrated in the remaining natural
vegetation, enabling predators like foxes and
skunks to steal eggs from the nests more
Decline in the breeding population of the
northern pintail (Anas actua ) from 1955 to 1998.
\Wildlife managers conduct extensive aerial
surveys and ground counts each June throughout
the breeding range in Canada and the United
States to set hunting regulations for the autumn
of each ye