Transcript Document

Demographic Techniques
– Chapter 10
Alligator Gar
1 June 2003
32 days post-hatch
(approx. 90 mm TL)
Life Tables - Mortality
• Mortality is one of the four key parameters that
drive population changes.
• We can use a life table to answer particular
questions about population mortality rates.
– What life stage has the highest mortality?
– Do older organisms die more frequently than
young organisms
• A cohort life table is an age-specific summary of
the mortality rates operating on a cohort of
individuals.
• Cohort – a collective group of individuals
– Fish year class, all mice born in March, tadpoles from a
single frog, freshman year class
Cohort Life Table:
X = age
nx = number alive at time t
lx = proportion of organisms surviving from the start of the life
table to age x (ex: l1 = n1/n0, 0.217 = 25/115; l2=n2/n0,
0.165=19/115)
dx = number dying during the age interval x to x + 1 (ex:d0=n0-n1,
90 = 115-25; d1=n1-n2, 6=25-19)
qx = per capita rate of mortality during the age interval x to x + 1
(ex: q0=d0/n0, 0.78 =90/115; q1=d1/n1)
Formula’s For Cohort Life Table
• X = age group we decide on
• nx = observed
• lx = lx+1 = nx+1/nx (overall survival)
• dx = nx - nx+1 (age specific number dying)
• qx = dx / nx (age specific mortality rate)
Per Capita Rates
• Per capita is a presentation of data as a
proportion of the population.
• Suppose a disease kills 400 ducks:
– If total duck population = 250,000 then the per
capita mortality = 400/250,000 = 0.16%.
– If total duck population = 2,500 then the per
capita mortality = 400/2,500 = 16%.
• Per capita gives us an idea of how the
entire population is affected.
Survivorship Curve
A plot of nx on a Log scale from a starting
cohort of 1,000 individuals.
Three Types of General Curves
 Survival curve
Examples of Each Type:
•Type 1 – Humans
•Type 2 – Birds
•Type 3 - Fish
These curves are
models. Most real
curves are intermediate.
Mortality curve 
Static Life Table
Calculated by taking a cross section of a population at a
specific time:
Per Capita
Cohort Versus Static Life Tables
• Cohort follows an individual cohort
through time and static looks at all of the
individuals currently present.
• The two are equal if and only if the
environment does not change from year to
year and the population is at equilibrium.
– The human cohort life table for 1900 does little
good for predicting life expectancy for today.
How to Collect Life Table Data
1) Survivorship directly observed.
•
•
Follow an individual cohort through time at
close intervals.
Best to have since it generates a cohort life
table directly and does not assume that the
population is stable over time.
Balanus can affect
the survival of
Chthamalus as
determined by
survivorship curve
How to Collect Life Table Data
2) Age at Death Observed.
•
By determining how old individuals were
when they died, we can create a life table.
Based on 584
individuals plus
observed estimated
mortality for age 1 and
2 individuals
How to Collect Life Table Data
3) Age Structure Directly Observed.
•
We can construct a life table based on the
age structure of a population
–
•
Counting rings on a tree or a fish otolith.
Assume a constant age structure, which is
hardly the case.
Aging
• Does mortality increase with age
(senescence)?
This data proves
our simple theory of
senescence is not
correct
Mortality Rate (qx)
– Not for some Mediterranean fruit flies:
Intrinsic Capacity For Increase In Numbers
• By combining reproduction and mortality
estimates, we can determine net
population change (intrinsic capacity for
increase).
• The environment can influence population
mean longevity or survival rate, natality
rate, and growth rate.
– Can be summed with natality and death rate
Fertility Schedule
Population net reproductive rate
bx = natality
0.6% increase each generation
(lx)(bx) = reproductive output for that age class
R0 < 1 population is declining, R0 = 1 population is
stable, R0 > population is increasing.
Population Increase
• If survival and fertility rates do not
change, and no limit is placed on
population growth – at what rate will a
population increase?
• It seems we need to know age-specific
survival rates, age-specific fertility rates,
and age structure
– If all females in U.S. were >50 years old, no
new young would be produced.
• However, the age structure does not need
to be known!
Stable Age Distribution
• Given constant schedule of natality and
mortality rates, a population will
eventually reach a stable age distribution,
and will remain at this age distribution
indefinitely.
• Stable age distribution:
–
–
–
–
60% age 0
25% age 1
10% age 2
4% age 3
• Although the absolute number will
change, the proportion of each age class
remains constant!
For Example
This animal lives three
years, produces two young
at exactly one year, and
one young at exactly year
two, and no young year
three, then dies at end of
year 3.
If a population starts
with one individual at
age 0, the age
distribution quickly
becomes stable: 60%
age 0, 25% age 1,
10% age 2, and 4%
age 3.
Stable Age Distribution
When a population has reached the stable age
distribution, it will increase in numbers according to:
dN = rN Written in integral form  Nt = N0ert
dt
Nt = number of individuals at time t
N0 = number of individuals at time 0
e = 2.71828 (a constant)
r = intrinsic capacity for increase for the
particular environmental conditions
t = time
This equation describes the curve of geometric
increase in an expanding population (or geometric
decrease to zero if r is negative).
For Example:
N0ert = Nt
Any population on a
fixed age schedule of
natality and mortality
will change
geometrically.
This geometric change
will dictate a fixed and
unchanging age
distribution – the stable
age distribution.
Generation Time
• Generation time – mean period elapsing
between the production or ‘birth’ of
parents and the production or ‘birth’ of
offspring.
• We can calculate generation time from a
life table:
Gc =
lxbxx
Ro
Calculating r from a life table:
Gc = 4.0/3.0 = 1.33 years
 lxbxx = 4.0
r=
loge(R0)
G
=
loge(3.0)
1.33
= 0.824 per individual per year
r > 0 population increasing, r = 0 population stable,
r < 0 population decreasing
lx = proportion of original individuals surviving to
each age class.
bx = number of offspring produced per individual for the
given age class (often refers to females only)
R0 = net reproductive output (lxbx)
> 1 pop increasing, = 1 pop stable, < 1 pop decreasing
Gives us a multiplier to see how much the population
increases each generation
Gc = generation time
this is an approximation because not all births
occur at once.
r
= the populations intrinsic capacity for increase
each r is for a specific set of environmental conditions
environmental conditions may affect
survival/reproduction
> 0 pop increasing, = 0 pop stable, < 0 pop decreasing
Temperature and moisture
effects on r value for a
wheat beetle (Store wheat
in cool dry place).
Comparison of r value’s
for two species of wheat
beetle.
Species With a High r
• Are not necessarily more common
• However, these species can recover more
quickly from disturbances
Increasing r: r = R0/G
1) Reduction in age at first reproduction
–
Basically reduce generation time
Age at First Breeding
# for r = 0.76
1
15
2
32
3
67
4
141 (actual)
5
297
6
564
2) Increase the number of progeny in each
reproductive event
–
Increases R0
3) Increase the number of reproductive events
–
Increase in longevity essentially increases R0
About r
• r is an oversimplification of nature
– We do not find populations with a stable age
distribution or with constant age-specific
mortality and fertility rates
• Actual population increases we observed
varies in more complex ways than the
theoretical r
• However, the importance of r lies mostly in
its use as a model for comparison with the
actual rates of increase we see in nature
– Can be used to assess environmental quality
Reproductive Value
• Reproductive value – the contribution to
the future population that an individual
will make
• In a stable population reproductive value
at age x =
w
Vx = 
t=x
ltbt
lx
w
or

Vx = bx + t=x+1
t and x are age and w is age of last reproduction
ltbt
lx
Females begin breeding
Males protect harems
Present progeny
Vx = bx +
Residual reproductive value
= number of progeny that
on average will be
ltbt
produced in the rest of an
individuals lifetime
l
x
If the population is growing (not stable), then this
value must be discounted because the value of one
progeny is less in a larger population.
Reproductive value is important in the evolution of
life-history traits because natural selection acts more
strongly on age classes with high reproductive
values (cancer in humans).
Predation has more of an effect if acting on
individuals with high reproductive value.
Age Distributions
• Stable age distribution – age-specific
fertility and mortality are fixed and the
population grows exponentially.
• Stationary age distribution – when the
fertility rate exactly equals the mortality
rate and the population does not change
in size over time.
• Populations are almost never stable so we
never find a stable age distribution or a
stationary age distribution.
Relationships
Natality Rates
Environmental
factors
Age
Composition
Mortality Rates
Rate of increase
or decrease of
the population
Age Distribution
• Increasing populations have a predominance of
young organisms, whereas stable or declining
populations do not
Populations of
vole grown in
the lab.
Judging the Status of the 1995
Human Population
Age structure can differ strongly year to year in plant and
animal species (dominant fish year classes).
Neither has an age
structure
representative of a
stable age distribution.
Reproductive Strategies
• Big-bang reproduction (semelparity) –
reproduce once in a lifetime
– Salmon – spawn once and die
• Repeated reproduction (iteroparity) –
reproduce more than once in a lifetime
– Oak tree – may drop thousands of acorns for
200 years or more
• Why have different life cycles evolved?
Reproduction Tradeoffs
At high levels of
reproductive effort, a
small increase in effort
is more beneficial for
big-bang reproduction
than for repeated
reproduction
• The key demographic effect of big-bang
reproduction is higher reproductive rates.
– Plants that reproduce only once produce 2 – 5 as many
seeds as closely related species that reproduce
repeatedly
– Big-bang reproducers usually have a similar r as similar
species that are repeat reproducers
• Repeated reproduction is favored when
– Adult survival rates are high
– Juvenile survival is highly variable
• Repeated reproduction spreads the risk of
reproducing over a longer time period and thus
acts as an adaptation that thwarts environmental
fluctuations.
– If conditions are bad this year, then reproduce next year.
Repeated reproduction may be an evolutionary
response to uncertain survival from zygote to
adult stages.
Long Life Span
Short Life Span
Steady Reproductive
Success
?
Possible
Variable Reproductive
Success
Possible
Not Possible
Summary
• Age-specific natality and mortality rates for any
population can be summarized quantitatively in
fertility schedules and in life tables.
• The intrinsic capacity for increase (r) summarizes
the natality and mortality schedules and
forecasts the rate of population growth implicit in
these schedules.
• The age structure of a populations is determined
by these rates of natality and mortality.
• These quantitative methods are useful for
comparing the life history consequences of
particular natality and mortality schedules in
populations.