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.