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)
9
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)
11
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
14
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.
16
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, 
17
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)
18
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
19
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
20
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
21
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
22
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:
23
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
24
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.
26
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
28
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:
31
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
32
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
34
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
35
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
36
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