The Basics

The Basics

Populations rarely have a constant size

Intrinsic Factors

BIRTH

IMMIGRATION

DEATH

EMIGRATION

Extrinsic factors

Predation

Weather

Nt+1 = Nt + B + D + E + I

Populations grow IF (B + I) > (D + E)

Populations shrink IF (D + E) > (B + I)

Diagrammatic Life-Tables….

What is a population?

Assume E = I

AdultsNt

AdultsNt+1

SeedsNt.f

SeedlingsNt.f.g

f

g

e

p

BIR

TH

SU

RV

IVA

L

Nt+1 = (Nt.p) + (Nt.f.g.e)

AdultsM F2.3 2.3

AdultsM F

2.5 2.5

Pods18.25

Eggs200.75

Instar I15.86

Instar II11.42

Instar III8.91

Instar IV6.77

P=0

7.3

11

0.079

0.72

0.78

0.76

0.69

t = 0

t = 1

t = 0

t = 1

AdultsM F5 5

AdultsM F

8.2 8.2

Eggs50

1 mo Nestlings42

3 mo Fledglings29.8

10

0.71

0.1

0.5

0.84

Overlapping Generations: Discrete Breeding

a0 a1 a2 a3 an t1

a0 a1 a2 a3 an t3

a0 a1 a2 a3 an t2

p01 p12

p23

Birth

NB: Different age groups have different probabilities of surviving from one time interval to the next, and different

age groups produce different numbers of offspring

t1

t2

p01 p12

p23

Birth

Birth

NB – ALL Adults or Females?

Conventional Life-Tables Best studied from Cohort – Define

Stage (x) ax lx dx qx px log10ax log10lx log10ax - log10ax+1 Fx mx lxmx

Eggs (0) 44000 1.0000 0.9201 0.9201 0.0799 4.6435 0 1.0975 0 0 0Instar I (1) 3515 0.0799 0.0224 0.2805 0.7195 3.5459 -1.098 0.1430 0 0 0Instar II (2) 2529 0.0575 0.0138 0.2400 0.7600 3.4029 -1.241 0.1192 0 0 0Instar III (3) 1922 0.0437 0.0105 0.2399 0.7601 3.2838 -1.36 0.1191 0 0 0Instar IV (4) 1461 0.0332 0.0037 0.1102 0.8898 3.1647 -1.479 0.0507 0 0 0Adults V (5) 1300 0.0295 -- -- -- 3.1139 -1.53 -- 22617 17.3977 0.51

Subscript x refers to age/stage class

a refers to actual numbers counted – case specific



l refers to proportions wrt t0 – allows comparisons between cases: lx = ax / a0











d refers to standardised mortality, calculated as lx – lx+1: data can be summed














q age specific mortality, calculated as dx / lx: data cannot be summed















p age specific survivorship, calculated as 1 - qx (or ax+1 / ax): cannot be summed




















k killing power – reflects stage specific mortality and can be summed



K





















K

F Total number offspring per age/stage class























K




m mean number offspring per individual a, Fx / ax























K




m mean number offspring per individual a, Fx / ax





lm number of offspring per original individual

REAL DATA

Σ lxmx = R0 = ΣFx / a0 = Basic Reproductive rate

R0 = 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 later



Σ lxmx = R0 = 0.51

N0 . R0 = 44000 . 0.51 = 22400 = NT

Generation time

R0 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 manner

Fundamental Reproductive Rate (R) = Nt+1 / Nt

IF Nt = 10, Nt+1 = 20: R = 20 / 10 = 2

Populations will increase in size if R >1Populations will decrease in size if R < 1

Populations will remain the same size if R = 1

R combines birth of new individuals with the survival of existing individuals

Population size at t+1 = Nt.RPopulation size at t+2 = Nt.R.RPopulation size at t+3 = Nt.R.R.R

Nt = N0.Rt

R0 ONLY reflects the birth of new individuals (survival = 0)

Nt = N0.Rt Overlapping generations

NT = N0.R0Non-overlapping generations

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

NT = N0.RTIF t = T, then

R0 = RT

lnR0 = T.lnR

Can now link R0 and R:

T = Σxlxmx / R0

T can be calculated from the cohort life tables

X = age class

x a l d q p F m lm xlm0.00 1000000.00 1.00 1.00 1.00 0.00 0.00 0.00 0.00 0.001.00 62.00 0.00 0.00 0.45 0.55 285200.00 4600.00 0.29 0.292.00 34.00 0.00 0.00 0.41 0.59 295800.00 8700.00 0.30 0.593.00 20.00 0.00 0.00 0.23 0.78 232000.00 11600.00 0.23 0.704.00 15.50 0.00 0.00 0.29 0.71 196850.00 12700.00 0.20 0.795.00 11.00 0.00 0.00 0.41 0.59 139700.00 12700.00 0.14 0.706.00 6.50 0.00 0.00 0.69 0.31 82550.00 12700.00 0.08 0.507.00 2.00 0.00 0.00 0.00 1.00 25400.00 12700.00 0.03 0.188.00 2.00 0.00 0.00 1.00 0.00 25400.00 12700.00 0.03 0.209.00 0.00 0.00 -- -- -- -- -- -- --

Totals 1.2829 3.935

T 3.06726947

Other statistics that you can calculate from basic life tables

Life Expectancy – average length of time that an individual of age x can expect to live

L average number of surviving individuals in consecutive stage/age classes: (ax + ax+1) / 2

x a l d q p F m lm xlm L T e0 8760 1.000 0.760 0.760 0.240 0 0 0 0 5431.0 6991.0 0.7981 2102 0.240 0.182 0.758 0.242 42040 20 4.799 4.799 1305.5 1560.0 0.7422 509 0.058 0.058 1.000 0.000 12216 24 1.395 2.789 254.5 254.5 0.5003 0 0.000

R0 6.194T 1.225r 1.488R 4.430


R0 6.194T 1.225r 1.488R 4.430

T cumulative L: Σ Lxi

n


R0 6.194T 1.225r 1.488R 4.430

e life expectancy: Tx / ax

NB. Units of e must be the same as those of x

Thus if x is measured in intervals of 3 months, then e must be multiplied by 3 to give life expectancy in terms of months

Can also calculate T and L using lx values

T and L are confusing – call them Bob (L) and Margaret (T)

A note on finite and instantaneous rates

The 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

[Adjusted FINITE] = [Observed FINITE] ts/to

Where ts = Standardised time interval (e.g. 30 days, 1 day, 365 days, 12 months etc)to = Observed time interval

To convert FINITE rates at one scale to (adjusted) finite rates at another:

e.g. convert annual survival (p) = 0.5, to monthly survival

Adjusted = Observed ts/to = 0.5 1/12 = 0.5 0.083 = 0.944

e.g. convert daily survival (p) = 0.99, to annual survival

Adjusted = Observed ts/to = 0.99 365/1 = 0.99 365 = 0.0255

INSTANTANEOUS MORTALITY rates = Loge (FINITE SURVIVAL rates)

ALWAYS negative

Finite Mortality Rate = 1 – Finite Survival rate

Finite Mortality Rate = 1.0 – e Instantaneous Mortality Rate

MUST SPECIFY TIME UNITS

Projecting Populations into the future: Basic Model Building

KEY PIECES of INFORMATION: p and m

Rearrange Life Table

WHY?

m 0.000 20.000 24.000 0p 0.24 0.242 0 0

0 1 2 30 8760 2102 509 0123456

Age Classtime

x a l d q p F m0 8760 1.000 0.760 0.760 0.240 0 01 2102 0.240 0.182 0.758 0.242 42040 202 509 0.058 0.058 1.000 0.000 12216 243 0 0.000

Dealing first with survivorship

m 0.000 20.000 24.000 0p 0.24 0.242 0 0

0 1 2 30 8760 2102 509 01 2102.423456

Age Classtime

m 0.000 20.000 24.000 0p 0.24 0.242 0 0

0 1 2 30 8760 2102 509 01 2102.4 508.684 02 0 508.7808 03 0 0 04 0 0 05 0 0 06 0 0 0

Age Classtime

Copy Formula Down and Across

Table quickly fills up with 0s

m 0.000 20.000 24.000 0p 0.24 0.242 0 0

0 1 2 30 8760 2102 509 01 2102.4 508.684 02 0 508.7808 03 0 0 04 0 0 05 0 0 06 0 0 0

Age Classtime

m 0.000 20.000 24.000 0p 0.24 0.242 0 0

0 1 2 30 8760 2102 509 01 54256.42 2102.4 508.684 02 13021.54 508.7808 03 0 3151.212641 04 0 0 05 0 0 06 0 0 0

Age Classtime

54256.42

Adding Fecundity

Copy Down

m 0.000 20.000 24.000 0p 0.24 0.242 0 0

0 1 2 30 8760 2102 509 01 54256.416 2102.4 508.684 02 272641.536 13021.54 508.7808 03 1384308.48 65433.969 3151.212641 04 7024721.18 332234.03 15835.02041 05 35648276.9 1685933.1 80400.6363 06 180903629 8555586.5 407995.8059 0

Age Classtime

m 0.000 20.000 24.000 0p 0.24 0.242 0 0

0 1 2 3 Total R0 8760 2102 509 0 11371.00 5.001 54256.416 2102.4 508.684 0 56867.50 5.032 272641.536 13021.5398 508.7808 0 286171.86 5.083 1384308.476 65433.9686 3151.212641 0 1452893.66 5.074 7024721.176 332234.034 15835.02041 0 7372790.23 5.075 35648276.91 1685933.08 80400.6363 0 37414610.63 5.076 180903628.5 8555586.46 407995.8059 0 189867210.79 5.077 918028263.1 43416870.8 2070451.923 0 963515585.86 5.078 4658700849 220326783 10506882.74 0 4889534514.59 5.079 23641422030 1118088204 53319081.52 0 24812829315.48 5.07

10 1.19973E+11 5673941287 270577345.3 0 125917200664.89 5.0711 6.08823E+11 2.8793E+10 1373093792 0 638989662230.89

Age Classtime

NB – R eventually stabilises

R = (Nt+1) / Nt

Converting NUMBERS of each age class to PROPORTIONS (of the TOTAL) generates the age-structure of the population. NOTE, when R stabilises, so too does the age-structure, and this is known as the stable-age distribution of the population, and proportions represent TERMS (cx)

time0 1 2 3

0 0.7704 0.1849 0.0448 01 0.9541 0.0370 0.0089 02 0.9527 0.0455 0.0018 03 0.9528 0.0450 0.0022 04 0.9528 0.0451 0.0021 05 0.9528 0.0451 0.0021 06 0.9528 0.0451 0.0021 07 0.9528 0.0451 0.0021 08 0.9528 0.0451 0.0021 09 0.9528 0.0451 0.0021 010 0.9528 0.0451 0.0021 011 0.9528 0.0451 0.0021 0

Age Class

Because the terms of the stable age distribution are fixed at constant R, we can partition r (lnR) into birth and death per individual

Nt+1 = Nt.(Survival Rate) + Nt.(Survival Rate).(Birth Rate)

Nt+1 = Nt.(Survival Rate).(1 + Birth Rate)

m 0.000 20.000 24.000 0p 0.24 0.242 0 0

0 1 2 3 Total R Births Birth Rate Survivors Survival Rate0 8760 2102 509 0 11371.00 5.00 8760 3.3550 26111 54256.416 2102.4 508.684 0 56867.50 5.03 54256 20.7793 2611 0.22962 272641.536 13021.53984 508.7808 0 286171.86 5.08 272642 20.1504 13530 0.23793 1384308.476 65433.96864 3151.21264 0 1452893.66 5.07 1384308 20.1838 68585 0.23974 7024721.176 332234.0343 15835.0204 0 7372790.23 5.07 7024721 20.1820 348069 0.23965 35648276.91 1685933.082 80400.6363 0 37414610.63 5.07 35648277 20.1821 1766334 0.23966 180903628.5 8555586.459 407995.806 0 189867210.79 5.07 180903629 20.1821 8963582 0.23967 918028263.1 43416870.85 2070451.92 0 963515585.86 5.07 918028263 20.1821 45487323 0.23968 4658700849 220326783.1 10506882.7 0 4889534514.59 5.07 4658700849 20.1821 230833666 0.23969 23641422030 1118088204 53319081.5 0 24812829315.48 5.07 23641422030 20.1821 1171407285 0.2396

10 1.19973E+11 5673941287 270577345 0 125917200664.89 5.07 119972682032 20.1821 5944518633 0.239611 6.08823E+11 28793443688 1373093792 0 638989662230.89 608823124752 20.1821 30166537479 0.2396

timeAge Class

No Births = No a0

Calculating Birth Rate First

m 0.000 20.000 24.000 0p 0.24 0.242 0 0


10 1.19973E+11 5673941287 270577345 0 125917200664.89 5.07 119972682032 20.1821 5944518633 0.239611 6.08823E+11 28793443688 1373093792 0 638989662230.89 608823124752 20.1821 30166537479 0.2396

timeAge Class

Divide by No Individuals producing them: Σax1

n

e.g. B = 35648277 / (1685933 + 80401 + 0) = 20.1821

m 0.000 20.000 24.000 0p 0.24 0.242 0 0


10 1.19973E+11 5673941287 270577345 0 125917200664.89 5.07 119972682032 20.1821 5944518633 0.239611 6.08823E+11 28793443688 1373093792 0 638989662230.89 608823124752 20.1821 30166537479 0.2396

timeAge Class

Calculating Survival Rate

Σax1

n

Survivors: Total number of individuals at time t, older than 0:

m 0.000 20.000 24.000 0p 0.24 0.242 0 0


10 1.19973E+11 5673941287 270577345 0 125917200664.89 5.07 119972682032 20.1821 5944518633 0.239611 6.08823E+11 28793443688 1373093792 0 638989662230.89 608823124752 20.1821 30166537479 0.2396

timeAge Class

Survival Rate: No Survivors at time t, divided by total population size at time t-1

e.g. Survival Rate (t4) = No survivors (t4) / total population size (t3)

S = 348069 / 1452894 = 0.2396

Nt+1 = Nt.(Survival Rate).(1 + Birth Rate)

Nt+1 / Nt = R = er = (Survival Rate).(1 + Birth Rate)

m 0.000 20.000 24.000 0p 0.24 0.242 0 0


10 1.19973E+11 5673941287 270577345 0 125917200664.89 5.07 119972682032 20.1821 5944518633 0.239611 6.08823E+11 28793443688 1373093792 0 638989662230.89 608823124752 20.1821 30166537479 0.2396

timeAge Class

B = 20.1821S = 0.2396

At Stable-Age

R = 0.2396 x (20.1821 + 1) = 5.07

Annual Survival Rate for an individual in the population is in the range p0, p1, p2, but NOT the average

Annual Birth Rate for an individual in the population is between m1 and m2, but NOT the average

NOTE

Reproductive Value (vx) – a measure of present and future contributions by the

different age classes of a population to R

vx 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

vx* = [(vx+1.lx+1) / (lx.R)]

vx* = residual reproductive value

vx = mx + vx*This expression can

ONLY be used to calculate vx* IF the time

intervals used in the life-table are equal.

To calculate vx* work backwards in the life-table, because vx* = 0 in the last year of life

x a l m v* v0 8760 1.000 01 2102 0.240 202 509 0.058 24 0.000 24.0003 0 0.000

x a l m v* v0 8760 1.000 01 2102 0.240 20 1.145 21.1452 509 0.058 24 0.000 24.0003 0 0.000

x a l m v* v0 8760 1.000 0 1.000 1.0001 2102 0.240 20 1.145 21.1452 509 0.058 24 0.000 24.0003 0 0.000

Copy upwards

STATIC LIFE TABLES………

Documents

The Basics