Life Table Analysis
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:
Biology 208
Andrew Derocher
– all age classes grow or decline at the same rate, λ
– the population also grows or declines at this
constant rate, λ
Chapter 14, p. 277-292
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)
fecundity (bx)
Life Tables
• Provide insight into important life process
(schedules of births and deaths)
• Note: many different notations are used for
life tables
Dall Sheep (Ovis dalli dalli)
Age interval
(years)
# surviving at start
of age interval, nx
# surviving as
fraction of
newborn, lx
0-1
608
1.000
1-2
487
0.801
2-3
480
0.789
3-4
472
0.776
4-5
465
0.764
5-6
447
0.734
6-7
419
0.688
7-8
390
0.640
8-9
348
0.571
9-10
268
0.441
10-11
154
0.252
11-12
59
0.096
12-13
4
0.060
13-14
2
0.003
14-15
0
0.000
In this example,
l1 = 608/608 = 1
l2 = 487/608 = 0.801
l3 = 480/608 = 0.789
1
Dall Sheep (Ovis dalli dalli)
Age interval
(years)
# surviving at start
of age interval, nx
# surviving as
fraction of
newborn, lx
0-1
608
1.000
1-2
487
0.801
2-3
480
0.789
3-4
472
0.776
4-5
465
0.764
5-6
447
0.734
6-7
419
0.688
7-8
390
0.640
8-9
348
0.571
9-10
268
0.441
10-11
154
0.252
11-12
59
0.096
12-13
4
0.006
13-14
2
0.003
14-15
0
0.000
Female Himalayan thar
Age
Frequency*
lx
0
1
2
3
4
5
6
7
8
9
10
11
12
12+
205
95.83
94.43
88.69
79.41
67.81
55.2
42.85
31.71
22.37
15.04
9.64
5.9
11
1.000
0.467
0.461
0.433
0.387
0.331
0.269
0.209
0.155
0.109
0.073
0.047
0.029
0.054
Himalayan thar
- Introduced into New Zealand
Survivorship curves
• Common for of life table visualization
– Easier to see patterns
• Typically, a plot of lx
* Correct data – “smoothed”
Juvenile mortality
Juvenile mortality (v. low)
Males in U.S.A. (1929-31)
Dall sheep
2
Juvenile mortality
Prawns
Log # surviving
Black-tailed deer
Ty
pe
II
Birds
Ty
pe
I
Mammals
Ty
pe
I
I
I
Invertebrates
AGE
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
• Static life tables consider survival of individuals
of known age during a single time interval:
– require some means of determining ages of
individuals
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, λ = 2.78
– another way to quantify population growth is through
doubling time:
• t2 = loge2/loge λ = 0.69/loge λ = 0.675 yr or 246 days for the
ring-necked pheasant
3
Doubling time
Intrinsic rate of increase
r (per day)
tdouble
300
3.3 minutes
rat
0.0148
46.8 days
cow
0.001
1.9 years
polar bear
0.000134
14.2 years
beech tree
0.000075
25.3 years
Organism
virus
Environmental conditions and
intrinsic rates of increase.
• The intrinsic rate of increase depends on how
individuals perform in that population’s
environment.
• rm – intrinsic rate of increase or Malthusian
parameter (after Thomas Malthus the 18th
century English economist)
• Defined as the exponential rate of increase for a
population with a stable age distribution
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
Consequences of Crowding for
Population Growth
Svalbard reindeer data from monitoring...
• Crowding:
– results in less food for individuals and their offspring
– aggravates social strife
– promotes the spread of disease
– attracts the attention of predators
700
600
500
• These factors act to slow and eventually halt
population growth.
Nt
400
300
200
1976
1980
1984
1988
1992
1996
2000
Year
4
Sigmoid growth
Behavior of the Logistic Equation
• We know populations cannot grow
indefinitely (INCLUDING HUMANS!)
• Resources are limited so a leveling off
must occur
• Generally, growth results in an S-shaped
population curve called Sigmoid growth
curve
Carry capacity, K
• The logistic equation (sigmoid growth) 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 of population growth.
Sigmoid growth and K
• The maximum population size the
habitat can support
dN
= rN
dt
–Below K, populations increase in size
–Above K, populations decrease in size
Models exponential growth
Now we need to add a new term,
dN
K−N
= rN (
)
dt
K
When N approaches K
then, (K-N)/K approaches 0
K is the carrying capacity
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 logistic equation:
dN/dt = r0N(1 - N/K)
Decelerating
Accelerating
When N is very small (K-N)/K is near
1.0, growth is near exponential
Reduction in growth rate
due to crowding
Fig. 14.17
5
Density-dependent
Density-independent
• Having an influence on the individuals in a
population that varies with the degree of
crowding within the population
• Having an influence on the individuals in a
population that does not vary with the degree of
crowding within the population.
Density Dependence in Animals
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:
• 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 juvenile survival on density
– 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:
– temperature, precipitation, catastrophic events
Density-dependent factors
can control the size of
natural populations
-Song sparrows on
Mandarte Island, B.C.
Population size
Fig. 14.20
Population size
Fig. 14.20
6
Density Dependence in Plants
• 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
• 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
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 agespecific survival and fecundity.
• Life tables summarize demographic data.
• Analyses of life table data permit determination of
population growth rates and stable age distributions.
Summary 2
• Populations have potential for explosive growth, but
all are eventually regulated by scarcity of resources
and other density-dependent 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 processes.
7