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Demographic Techniques – Chapter 10
Alligator GarAlligator Gar1 June 20031 June 2003
32 days post-hatch32 days post-hatch(approx. 90 mm TL)(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
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)
Cohort Life Table:
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 CurveA plot of nx on a Log scale from a starting cohort of 1,000 individuals.
Three Types of General Curves
Examples of Each Type:
•Type 1 – Humans
•Type 2 – Birds
•Type 3 - Fish
These curves are models. Most real curves are intermediate.
Survival curve
Mortality curve
Static Life Table
Per Capita
Calculated by taking a cross section of a population at a specific time:
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)?– Not for some Mediterranean fruit flies:
Mo
rtal
ity
Rat
e (q
x)
This data proves our simple theory of senescence is not correct
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
bx = natality
(lx)(bx) = reproductive output for that age class
R0 < 1 population is declining, R0 = 1 population is stable, R0 > population is increasing.
Population net reproductive rate
Fertility Schedule
0.6% increase each generation
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 ExampleThis 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
rN=dNdt
Written in integral form Nt = N0ert
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
When a population has reached the stable age distribution, it will increase in numbers according to:
This equation describes the curve of geometric increase in an expanding population (or geometric decrease to zero if r is negative).
For Example:
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.
N0ert = Nt
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
lxbxx = 4.0
Gc = 4.0/3.0 = 1.33 years
1.33r =
loge(R0)
G
loge(3.0)= = 0.824 per individual per year
Calculating r from a life table:
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
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
Age at First Breeding # for r = 0.76
1 15
2 32
3 67
4 141 (actual)
5 297
6 564
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 =
Vx =ltbt
lx
t=x
w
t and x are age and w is age of last reproduction
t=x+1
w
Vx =ltbt
lx
bx +or
Males protect haremsFemales begin breeding
Vx =ltbt
lx
bx +
Present progeny Residual reproductive value= number of progeny that on average will be produced in the rest of an individuals lifetime
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
Environmental factors
Mortality Rates
Natality Rates
Age Composition
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.
Long Life Span Short Life Span
Steady Reproductive Success
? Possible
Variable Reproductive Success
Possible Not Possible
Repeated reproduction may be an evolutionary response to uncertain survival from zygote to adult stages.
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.