The BasicsPopulations rarely have a constant sizeDiagrammatic Life-Tables.What is a population?Assume E = I

t = 0t = 1t = 0t = 1

Overlapping Generations: Discrete BreedingNB: Different age groups have different probabilities of surviving from one time interval to the next, and different age groups produce different numbers of offspringt1t2NB ALL Adults or Females?

Conventional Life-TablesBest studied from Cohort Definea refers to actual numbers counted case specific

Conventional Life-TablesBest studied from Cohort Definea refers to actual numbers counted case specific

Conventional Life-TablesBest studied from Cohort Definea refers to actual numbers counted case specific

Conventional Life-TablesBest studied from Cohort Definea refers to actual numbers counted case specific

Conventional Life-TablesBest studied from Cohort Definea refers to actual numbers counted case specific

Conventional Life-TablesBest studied from Cohort Definea refers to actual numbers counted case specific

Conventional Life-TablesBest studied from Cohort Definea refers to actual numbers counted case specificm mean number offspring per individual a, Fx / ax

Conventional Life-TablesBest studied from Cohort Definea refers to actual numbers counted case specificm mean number offspring per individual a, Fx / ax

lxmx = R0 = Fx / a0 = Basic Reproductive rateR0 = mean number of offspring produced per original individual by the end of the cohort

It indicates the mean number of offspring produced (on average) by an individual over the course of its life, AND, in the case of species with non-overlapping generations, it is also the multiplication factor that converts an original population size into a new population size ONE GENERATION laterR0 is a predictor that can be used to project populations into the future in terms of generations

For populations with overlapping generations, we must tackle the problem in a roundabout mannerFundamental Reproductive Rate (R) = Nt+1 / NtIF Nt = 10, Nt+1 = 20: R = 20 / 10 = 2Populations will increase in size if R >1Populations will decrease in size if R < 1Populations will remain the same size if R = 1R combines birth of new individuals with the survival of existing individualsPopulation size at t+1 = Nt.RPopulation size at t+2 = Nt.R.RPopulation size at t+3 = Nt.R.R.RNt = N0.RtR0 ONLY reflects the birth of new individuals (survival = 0)

lnR = r = lnR0 / T = intrinsic rate of natural increase

Sheet1

xaldqpFmlmxlm

0.001000000.001.001.001.000.000.000.000.000.00

1.0062.000.000.000.450.55285200.004600.000.290.29

2.0034.000.000.000.410.59295800.008700.000.300.59

3.0020.000.000.000.230.78232000.0011600.000.230.70

4.0015.500.000.000.290.71196850.0012700.000.200.79

5.0011.000.000.000.410.59139700.0012700.000.140.70

6.006.500.000.000.690.3182550.0012700.000.080.50

7.002.000.000.000.001.0025400.0012700.000.030.18

8.002.000.000.001.000.0025400.0012700.000.030.20

9.000.000.00--------------

Totals1.28293.935

T3.0672694676

Sheet2

Sheet3

Other statistics that you can calculate from basic life tablesLife Expectancy average length of time that an individual of age x can expect to liveCan also calculate T and L using lx valuesT and L are confusing call them Bob (L) and Margaret (T)

A note on finite and instantaneous ratesThe values of p, q hitherto collected are FINITE rates: units of time those of x expressed in the life-tables (months, days, three-months etc)

They have limited value in comparisons unless same units used

INSTANTANEOUS MORTALITY rates = Loge (FINITE SURVIVAL rates)ALWAYS negativeFinite Mortality Rate = 1 Finite Survival rateFinite Mortality Rate = 1.0 e Instantaneous Mortality RateMUST SPECIFY TIME UNITS

Projecting Populations into the future: Basic Model BuildingKEY PIECES of INFORMATION: p and mRearrange Life TableWHY?

Sheet1

xaldqpFmlmxlmLTe

087601.0000.7600.7600.24000005431.2006604.3390.754

121020.2400.1820.7580.24242048204.84.81112.1701173.1390.558

21220.0580.0581.0000.0002927241.3922.78460.97060.9700.500

300.000-0.240

m0.00020.00024.0000

p0.240.24200

timeAge Class

0123

0876021025090

12102.4508.6840

20508.78080

3000

4000

5000

6000

7000

8000

9000

10000

11000

Sheet2

Sheet3

Because the terms of the stable age distribution are fixed at constant R, we can partition r (lnR) into birth and death per individualNt+1 = Nt.(Survival Rate) + Nt.(Survival Rate).(Birth Rate)Nt+1 = Nt.(Survival Rate).(1 + Birth Rate)

Nt+1 = Nt.(Survival Rate).(1 + Birth Rate)Nt+1 / Nt = R = er = (Survival Rate).(1 + Birth Rate)R = 0.2396 x (20.1821 + 1) = 5.07

Sheet1

xaldqpFmlmxlmLTe

087601.0000.7600.7600.24000005431.2006604.3390.754

121020.2400.1820.7580.24242048204.84.81112.1701173.1390.558

21220.0580.0581.0000.0002927241.3922.78460.97060.9700.500

300.000-0.240

m0.00020.00024.0000

p0.240.24200

timeAge Class

0123TotalR

087602102122010984.005.18

154256.4162102.4508.684056867.505.03

2272641.53613021.53984508.78080286171.865.08

31384308.4761907265433.968643151.2126412801452893.665.07

47024721.17557657332234.03428577315835.0204108807372790.235.07

535648276.91389931685933.0821383880400.636297157037414610.635.07

6180903628.5277768555586.45933584407995.8058774870189867210.795.07

7918028263.08914943416870.84666632070451.923159270963515585.865.07

84658700848.70535220326783.14139610506882.744893304889534514.595.07

923641422030.27091118088203.6892853319081.5202178024812829315.485.07

10119972682032.3285673941287.26502270577345.2928070125917200664.895.07

11608823124751.60828793443687.75871373093791.518130638989662230.89

timeAge Class

0123TotalR

00.79750.19140.011101.001.00

10.95410.03700.008901.001.00

20.95270.04550.001801.001.00

30.95280.04500.002201.001.00

40.95280.04510.002101.001.00

50.95280.04510.002101.001.00

60.95280.04510.002101.001.00

70.95280.04510.002101.001.00

80.95280.04510.002101.001.00

90.95280.04510.002101.001.00

100.95280.04510.002101.001.00

110.95280.04510.002101.00

Sheet2

m0.00020.00024.0000

p0.240.24200

timeAge Class

0123TotalRBirthsBirth RateSurvivorsSurvival Rate

087602102509011371.005.0087603.35502611

154256.4162102.4508.684056867.505.035425620.779326110.2296

2272641.53613021.53984508.78080286171.865.0827264220.1504135300.2379

31384308.4761907265433.968643151.2126412801452893.665.07138430820.1838685850.2397

47024721.17557657332234.03428577315835.0204108807372790.235.07702472120.18203480690.2396

535648276.91389931685933.0821383880400.636297157037414610.635.073564827720.182117663340.2396

6180903628.5277768555586.45933584407995.8058774870189867210.795.0718090362920.182189635820.2396

7918028263.08914943416870.84666632070451.923159270963515585.865.0791802826320.1821454873230.2396

84658700848.70535220326783.14139610506882.744893304889534514.595.07465870084920.18212308336660.2396

923641422030.27091118088203.6892853319081.5202178024812829315.485.072364142203020.182111714072850.2396

10119972682032.3285673941287.26502270577345.2928070125917200664.895.0711997268203220.182159445186330.2396

11608823124751.60828793443687.75871373093791.518130638989662230.8960882312475220.1821301665374790.2396

Sheet3

Reproductive Value (vx) a measure of present and future contributions by the different age classes of a population to Rvx is calculated as the number of offspring produced by an individual age x and older, divided by the number of individuals age x right now

STATIC LIFE TABLES