Chapter 14: Population Growth and Regulation
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Transcript Chapter 14: Population Growth and Regulation
Robert E. Ricklefs
The Economy of Nature,
Fifth Edition
Chapter 14:
Population Growth
and Regulation
2
Human Population Growth 1
Growth of the human population is one of the
most significant ecological developments in the
earth’s history.
Early population growth was very slow:
1 million individuals lived a million years ago
3-5 million individuals lived at the start of the
agricultural revolution (10,000 years ago)
3
Human Population Growth 2
More recent population changes have been
quite rapid:
population increased 100-fold from 10,000 years ago
to start of eighteenth century
in the past 300 years, population has increased from
300 million to 6 billion, a 20-fold increase
the most recent doubling (3 billion to 6 billion) has
taken place in the last 40 years
4
How many humans?
Has the human population exceeded
the ability of the earth to support it?
there is no consensus on this point
clearly, continued growth will further stress
the biosphere
When, and at what level, will the
human population cease to grow?
there are many unknowns
the United Nations estimates a plateau at 9
billion
5
Demography
Demography is the study of populations:
involves the use of mathematical techniques to
predict growth of populations
involves intensive study of both laboratory and
natural populations, with emphasis on:
causes of population fluctuations
effects of crowding on birth and death rates
6
Populations grow by
multiplication.
A population increases in proportion to its
size, in a manner analogous to a savings
account earning interest on principal:
at a 10% annual rate of increase:
a population of 100 adds 10 individuals in 1 year
a population of 1000 adds 100 individuals in 1 year
allowed to grow unchecked, a population growing at a
constant rate would rapidly climb toward infinity
7
Two Models of Population
Growth
Because of differences in life histories
among different kinds of organisms,
there is a need for two different
models (mathematical expressions)
for population growth:
exponential growth: appropriate when
young individuals are added to the
population continuously
geometric growth: appropriate when
young individuals are added to the
population at one particular time of the
year or some other discrete interval
8
Exponential Population Growth
1
A population exhibiting exponential growth has a smooth
curve of population increase as a function of time.
The equation describing such growth is:
N(t) = N(0)ert
where:N(t) = number of individuals after t time units
N(0) = initial population size
r = exponential growth rate
e = base of the natural logarithms (about 2.72)
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Exponential Population Growth 2
Exponential growth results in a continuously
accelerating curve of increase (or continuously
decelerating curve of decrease).
The rate at which individuals are added to the
population is:
dN/dt = rN
This equation encompasses two principles:
the exponential growth rate (r) expresses population
increase on a “per individual basis”
the rate of increase (dN/dt) varies in direct proportion to N
10
Geometric Population Growth 1
Geometric growth results in seasonal patterns of
population increase and decrease.
The equation describing such growth is:
N(t + 1) = N(t)
where:N(t + 1) = number of individuals after 1 time unit
N(t) = initial population size
= ratio of population at any time to that 1 time
unit earlier, such that λ = N(t + 1)/N(t)
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Geometric Population
Growth 2
To calculate the growth of a population
over many time intervals, we multiply the
original population size by the geometric
growth rate for the appropriate number
of intervals t:
N(t) = N(0) t
For a population growing at a geometric
rate of 50% per year ( = 1.50), an initial
population of N(0) = 100 would grow to
N(10) = N(0) 10 = 5,767 in 10 years.
12
Exponential and geometric
growth are related.
Exponential and geometric growth equations
describe the same data equally well.
These models are related by:
= er
and
loge = r
13
Varied Patterns of
Population Change
A population is:
growing when > 1 or r > 0
constant when = 1 or r = 0
declining when < 1 (but > 0) or r < 0
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Per Individual Population
Growth
Rates or per capita growth rates
The per individual
of a population are functions of component
birth (b or B) and death (d or D) rates:
r=b-d
and
=B-D
While these per individual or per capita rates
are not meaningful on an individual basis, they
take on meaning at the population level.
15
Age structure determines
population growth rate.
When birth and death rates vary with the age of
individuals in the population, contributions of
younger and older individuals must be
calculated separately.
Age specific schedules of survival and fecundity
enable us to project the population’s size and
age structure into the future.
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Stable Age Distribution
When a population grows with
constant schedules of survival and
fecundity, the population
eventually reaches a stable age
distribution (each age class
represents a constant percentage
of the total population):
Under a stable age distribution:
all age classes grow or decline at the
same rate,
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Life Tables
Life tables summarize
demographic information
(typically for females) in a
convenient format, including:
age (x)
number alive
survivorship (lx): lx = s0s1s2s3 ... sx-1
mortality rate (mx)
probability of survival between x and
x+1 (sx)
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Cohort and Static Life Tables
Cohort life tables are based on data
collected from a group of individuals
born at the same time and followed
throughout their lives:
difficult to apply to mobile and/or long-lived
animals
used by Grants to construct life tables for
Darwin’s finches on Galápagos Islands
Static life tables consider survival of
individuals of known age during a single
time interval:
require some means of determining ages of
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The Intrinsic Rate of Increase
1
The Malthusian parameter (rm) or
intrinsic rate of increase is the
exponential rate of increase (r)
assumed by a population with a
stable age distribution.
rm is approximated (ra) by
performing several computations
on a life table, starting with
computation of R0, the net
reproductive rate, (Σlxbx) across all
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The Intrinsic Rate of Increase
2
The net reproductive rate, R0, is the expected
total number of offspring of an individual over the
course of her life span.
R0 = 1 represents the replacement rate
R0 < 1 represents a declining population
R0 > 1 represents an increasing population
The generation time for the population is
calculated as T = Σxlxbx/Σlxbx
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The Intrinsic Rate of Increase
3
Computation of ra is based on R0 and T as follows:
ra = logeR0/T
Clearly, the intrinsic rate of natural increase depends on both
the net reproductive rate and the generation time:
large values of R0 and small values of T lead to the most rapid
population growth
Most populations have a
great biological growth
potential.
Consider the population growth of the
ring-necked pheasant:
8 individuals introduced to Protection Island,
Washington, in 1937, increased to 1,325 adults in
5 years:
166-fold increase
r = 1.02, = 2.78
another way to quantify population growth is
through doubling time:
t2 = loge2/loge = 0.69/loge = 0.675 yr or 246 days for
the ring-necked pheasant
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Environmental conditions
and intrinsic rates of
increase.
The intrinsic rate of increase
depends on how individuals
perform in that population’s
environment.
Individuals from the same
population subjected to different
conditions can establish the
reaction norm for intrinsic rate of
increase across a range of
conditions:
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Intrinsic rate of increase is
balanced by extrinsic
factors.
Despite potential for exponential increase, most
populations remain at relatively stable levels why?
this paradox was noted by both Malthus and Darwin
for population growth to be checked requires a
decrease in the birth rate, an increase in the death
rate, or both
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25
Consequences of Crowding
for Population Growth
Crowding:
results in less food for individuals and their offspring
aggravates social strife
promotes the spread of disease
attracts the attention of predators
These factors act to slow and eventually halt
population growth.
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The Logistic Equation
In 1910, Raymond Pearl and L.J. Reed analyzed
data on the population of the United States since
1790, and attempted to project the population’s
future growth.
Census data showing a decline in the
exponential rate of population growth suggested
that r should decrease as a function of increasing
N.
27
Behavior of the Logistic
Equation
The logistic equation describes a
population that stabilizes at its carrying
capacity, K:
populations below K grow
populations above K decrease
a population at K remains constant
A small population growing according to
the logistic equation exhibits sigmoid
growth.
An inflection point at K/2 separates the
accelerating and decelerating phases
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The Proposal of Pearl and
Reed
Pearl and Reed proposed that the
relationship of r to N should take
the form:
r = r0(1 - N/K)
in which K is the carrying capacity
of the environment for the
population.
The modified differential equation
for population growth is then the
29
Pearl
and
Pearl and
ReedReed’s
projected aProjections
U.S. population stabilized at
197,273,000.
The U.S. population reached this level between 1960 and
1970 and has continued to grow vigorously.
Pearl and Reed could not have foreseen improvements in
public health and medical treatment that raised survival
rates.
30
Population size is regulated by
density-dependent factors.
Only density-dependent factors,
whose effects vary with crowding,
can bring a population under
control; such factors include:
food supply and places to live
effects of predators, parasites, and
diseases
Density-independent factors may
influence population size but
cannot limit it; such factors include:
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Density Dependence in
Animals
Evidence for density-dependent
regulation of populations comes from
laboratory experiments on animals such
as fruit flies:
fecundity and life span decline with increasing
density in laboratory populations
Populations in nature show variation
caused by density-independent factors,
but also show the potential for regulation
by density-dependent factors:
song sparrows exhibit density dependence of
territory acquisition, fledging of young, and
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Density Dependence in Plants 1
Plants experience increased mortality and
reduced fecundity at high densities, like
animals.
Plants can also respond to crowding with
slowed growth:
as planting density of flax seeds is increased, the
average size achieved by individual plants declines
and the distribution of sizes is altered
33
Density Dependence in Plants 2
When plants are grown at very high
densities, mortality results in declining
density:
growth rates of survivors exceed the rate of decline
of the population, so total weight of the planting
increases:
in horseweed, a thousand-fold increase in average
plant weight offsets a hundred-fold decrease in density
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Self-Thinning Curve
A graph of log (average weight)
versus log (density) for plants
undergoing density-induced
mortality has points falling on a line
with slope of approximately -3/2:
this kind of graphical representation is
known as a self-thinning curve
similar patterns are seen for a wide
variety of plants:
this relationship is known as the -3/2
power law
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Summary 1
Population growth can be
described by both exponential
and geometric growth equations.
When birth and death rates vary
by age, predicting future
population growth requires
knowledge of age-specific survival
and fecundity.
Life tables summarize
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Summary
2 have potential for explosive
Populations
growth, but all are eventually regulated
by scarcity of resources and other densitydependent factors. Such factors restrict
growth by decreasing birth and survival
rates.
Density-dependent population growth is
described by the logistic equation.
Both laboratory and field studies have
shown how population regulation may be
brought about by density-dependent