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Weather

Extrinsic factors

Intrinsic Factors

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

BIRTH

IMMIGRATION

DEATH

EMIGRATION

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

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

The Basics

What is a population?

Populations rarely have a constant size

Diagrammatic Life-Tables….

Assume E = I

18.25

Adults

M F

2.5 2.5

7.3

11

Eggs

200.75

Adults

Nt

0.079

f

Instar I

15.86

Seeds

Nt.f

P=0

0.72

SURVIVAL

g

p

BIRTH

Instar II

11.42

Seedlings

Nt.f.g

0.78

e

Instar III

8.91

Adults

Nt+1

Adults

M F

2.3 2.3

0.76

0.69

Instar IV

6.77

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

t = 0

t = 0

t = 1

t = 1

M F

5 5

10

Birth

Birth

Birth

Eggs

50

0.84

a0

a1

a2

a3

an

1 mo Nestlings

42

t1

0.5

p23

p23

p12

p12

p01

p01

0.71

a0

a1

a2

a3

an

3 mo Fledglings

29.8

t2

0.1

Adults

M F

8.2 8.2

a0

a1

a2

a3

an

t3

Overlapping Generations: Discrete Breeding

t1

t2

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

NB – ALL Adults or Females?

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

Subscript x refers to age/stage class

Conventional Life-Tables

Best studied from Cohort – Define

a refers to actual numbers counted – case specific

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

Subscript x refers to age/stage class

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

Conventional Life-Tables

Best studied from Cohort – Define

a refers to actual numbers counted – case specific

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

Subscript x refers to age/stage class

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

Conventional Life-Tables

Best studied from Cohort – Define

a refers to actual numbers counted – case specific

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

Subscript x refers to age/stage class

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

Conventional Life-Tables

Best studied from Cohort – Define

a refers to actual numbers counted – case specific

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

Subscript x refers to age/stage class

K

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

Conventional Life-Tables

Best studied from Cohort – Define

a refers to actual numbers counted – case specific

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

Subscript x refers to age/stage class

K

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

F Total number offspring per age/stage class

Conventional Life-Tables

Best studied from Cohort – Define

a refers to actual numbers counted – case specific

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

Subscript x refers to age/stage class

K

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

F Total number offspring per age/stage class

Conventional Life-Tables

Best studied from Cohort – Define

a refers to actual numbers counted – case specific

m mean number offspring per individual a, Fx / ax

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

REAL DATA

Subscript x refers to age/stage class

K

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

F Total number offspring per age/stage class

lm number of offspring per original individual

Conventional Life-Tables

Best studied from Cohort – Define

a refers to actual numbers counted – case specific

m mean number offspring per individual a, Fx / ax

Σ lxmx = R0 = 0.51

N0 . R0 = 44000 . 0.51 = 22400 = NT

Generation time

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-overlappinggenerations, it is also the multiplication factor that converts an original population size into a new population size – ONE GENERATION later

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 >1

Populations 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

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

Population size at t+1 = Nt.R

Population size at t+2 = Nt.R.R

Population size at t+3 = Nt.R.R.R

Nt = N0.Rt

N the problem in a roundabout mannert = N0.Rt

Overlapping generations

NT = N0.R0

Non-overlapping generations

NT = N0.RT

IF 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

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

L the problem in a roundabout manner average number of surviving individuals in consecutive stage/age classes: (ax + ax+1) / 2

n

T cumulative L: ΣLx

i

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

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

Can also calculate T and L using lx values

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

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

[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

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

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

Adjusted = Observed ts/to

Adjusted = Observed ts/to

= 0.5 1/12

= 0.99 365/1

= 0.5 0.083

= 0.99 365

= 0.944

= 0.0255

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

INSTANTANEOUS rates at another: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

Dealing first with survivorship rates at another:

Copy Formula Down and Across

Table quickly fills up with 0s

Projecting Populations into the future: Basic Model Building

KEY PIECES of INFORMATION: p and m

Rearrange Life Table

WHY?

R = (N rates at another:t+1) / Nt

NB – R eventually stabilises

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)

Calculating Birth Rate First rates at another:

n

Divide by No Individuals producing them: Σax

No Births = No a0

1

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

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)

Calculating Survival Rate rates at another:

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

n

Σax

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

1

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

S = 348069 / 1452894 = 0.2396

At Stable-Age rates at another:

B = 20.1821

S = 0.2396

NOTE

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

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

This expression can ONLY be used to calculate v rates at another:x* IF the time intervals used in the life-table are equal.

vx = mx + vx*

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

vx* = residual reproductive value

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

Copy upwards

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

STATIC LIFE TABLES……… rates at another:

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