What is a population? Localised group of individuals of the same species Within a given area where...

What is a population?

Localised group of individuals of the same species

Within a given area where the scale of the area is study-dependent

e.g. population of aphids on a leaf

e.g. population of orchids in a 10km2 area of the Peninsula

e.g. population of baboons on the Cape Peninsula

Population Biology

Pop

ula

tion

siz

e

Time (t)

Describe

Explain (Influencing factors)Birth Mortality Death EmigrationIntrinsic

Extrinsic Environment

Weather

0

50

100

150

200

250

Year

1

Year

2

Year

3

Year

4

Year

5

Year

6

Year

7

Year

8

Year

9

Year

10

Year

11

Year

12

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

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

N = population sizet = time period (eg. Days, months, years…depends on study organism)

Nt+1 = Nt + B + I – D - E

Quantifying population growth

Birth DeathImmigration

Emigration

Population

growth

+ - -=Original populati

on+

Dependent on organisms life cycle

• Overlapping and non-overlapping

Life cycles

Overlapping generations

Non-overlapping (discrete) generations

Population growth potential

Life cycles – discrete generations

Adults2

Adults1

Pods

Eggs

Instar I

Instar II

Instar III

Instar IV

www.kidfish.bc.ca/caddis_cycle.htm

Generation 1

Generation 2

Replace

Often seasonal

ly determi

ned

R2 R2 R2

R1 R1 R1

Tim

e

1

R1 R1 R1

Tim

e

2

R3 R3 R3R3 R3 R3R1 R1 R1

Tim

e

3

R2 R2 R2

Life cycles – overlapping generations

Individuals of different ages reproducing at the same

time

Differential survivalDifferential reproduction

Life cyclesOverlapping generationsNon-overlapping generations

Frequency of reproduction

Semelparous Iteroparous

Population

growth potentia

l

Population Life Tables

Reproductive phaseGrowth phase Post-Reproductive phase

Multiple Reproductive Events

One individ

ual

E.g. most birds & mammals

ITEROPAROUS: multiple reproductive events over extended portions of their lives

Semelparous vs Iteroparous Life Cycles

Reproductive phaseGrowth phase Post-Reproductive phase

Single Reproductive Event

One individ

ual

E.g. most invertebrates

SEMELPAROUS: only one reproductive event in their lifetime

Year 1

Year 1Year 2Year 3Year 4


Dependent on organisms life cycle:Generation overlap & Semel/Iteroparous


Emigration

Population

growth


on+

Age and stage specific

Differential reproduction

Differential survival

AdultsNt

AdultsNt+1

SeedsNt.f

SeedlingsNt.f.g

Survival to maturity (s)

BIR

TH

SU

RV

IVA

L

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

AdultsM F

AdultsM F

Pods

Eggs

Instar I

Instar II

Instar III

Instar IV

P=0

7.3

11

0.079

0.72

0.78

0.76

0.69

Fecundity (f)

Germ

inate

(g

) Su

rvival (p

)

Different ages and stage classes have

different probabilities of survival and

different probabilities of successful reproduction


Dependent on organisms life cycle:Generation overlap & Semel/Iteroparous

Age and stage specificDifferential

reproductionDifferential survival

Tool for quantifying population growth


Emigration

Population

growth


on+

Frequency of reproductive

events


Emigration

Population

growth


on+

Number of young

produced in each

reproductive event

Length of each

generation

LIFE TABLES a simple method for keeping track of births,

deaths, and reproductive output in a population of interest


Life tables

2 ways of constructing Life tables

STATIC LIFE TABLE

compares population size from

different cohorts, across the entire

range of ages, at a single point in time

COHORT LIFE TABLE

N of Age 1

N of Age 2

N of Age 3

Time 1

0

50

100

150

200

250

Adults Sub Adults Juveniles Infants

Num

ber in

eac

h ag

e cl

ass

Snapshot in time

Used to estimate

population growth

Static tables make two important assumptions:1) the population has a stable age structure (i.e. the proportion of

individuals in each age class does not change from generation to generation)

2) the population size is stationary , or nearly stationary

Static Life Tables

00.10.20.30.40.50.60.70.80.9

1

Eggs

Insta

r I

Insta

r II

Insta

r III

Insta

r IV

Insta

r VPu

pae

Adult

s

lx

Cohort 2

00.10.20.30.40.50.60.70.80.9

1

Eggs

Insta

r I

Insta

r II

Insta

r III

Insta

r IV

Insta

r VPu

pae

Adult

s

lx

Cohort 3

00.10.20.30.40.50.60.70.80.9

1

Eggs

Insta

r I

Insta

r II

Insta

r III

Insta

r IV

Insta

r VPu

pae

Adult

s

lx

Cohort 4

00.10.20.30.40.50.60.70.80.9

1

Eggs

Insta

r I

Insta

r II

Insta

r III

Insta

r IV

Insta

r VPu

pae

Adult

s

lx

Cohort 1

0500

1000150020002500300035004000

Coho

rt 1

Coho

rt 2

Coho

rt 3

Coho

rt 4

Coho

rt 5

Coho

rt 6

Coho

rt 7

Coho

rt 8

Pop

ula

tion

siz

e (

n)

Life tables


STATIC LIFE TABLE

compares population size from

different cohorts, across the entire

range of ages, at a single point in time

COHORT LIFE TABLE

follows a group of same-aged

individuals from birth (or fertilized eggs)

throughout their lives

Less accurate than cohort tables

Note: For organisms that have separate sexes, life tables frequently follow only female individuals.

Age 1birth Age 1death

Time (t)

Considers differential

probabilities at each life stage

Cohort Life TablesSimplest form:

• annual life cycle• Semelparous (only one breeding season in its life time)• no overlap of generations

Animal with

www.kidfish.bc.ca/caddis_cycle.htm

To make a life table for this simple life history, we need only count (or estimate) the population size at each life history stage and the number of eggs produced by the adults.

Age classificatio

n

From this raw data we can calculate several LIFE HISTORY FEATURES

Cohort Life Tables

COUNT DATA

On

e

gen

era

tion

Cohort Life Tables

Age classificatio

nProporti

on of original cohort

surviving to each stage

lx

Calculated life history features

Calculate by: divide the number of

individuals living at the beginning of each age (ax)

by the initial number of eggs (a0)

COUNT DATA

This data is STANDARDIZED therefore comparable between

populations

...Raw data is NOT

Age classificatio

nProporti



lx


Cohort Life Tables

Calculate by:lx - lx+1

COUNT DATA

ADVANTAGE:

Proportions can be added together to get a measure of

mortality for different stage groups

DISADVANTAGE: > ax = > lx and dx values ; Therefore dx does not indicate the stage where mortality is most INTENSE

Age classificatio

nProporti



lx


Cohort Life Tables

qx is the fraction of the population

dying at each stage

ADVANTAGE: qx does indicate the stage where mortality is most INTENSE

Calculate by:dx/lx

Stage specific

COUNT DATA

∑CANNOT

DISADVANTAGE:

K

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

log

Cohort Life Tables

Combining advantages of dx (can be summed) and qx (indicates mortality intensity) is K (killing power)


Cohort Life Tables

Age specific

Age classificatio

nProporti



lx

Assessing the populations reproductive output

COUNT DATA

COUNT DATA


Cohort Life Tables

Age specific

Age classificatio

nProporti



lx


COUNT DATA

COUNT DATA

mx is the eggs produced per

surviving individual at each age or individual fecundity

Calculate by:Fx/ax


Cohort Life Tables

Age specific

Age classificatio

nProporti



lx


COUNT DATA

COUNT DATA

The number eggs

produced per original

individual at each age

(lxmx)

Calculate by:lx*mx

Cohort Life Tables

Age classificati

onProporti



lx


COUNT DATA

COUNT DATA


Age specific

∑ lxmx = R0

individuals produced for every individual in every

generation

basic reproductive rate

If only females in the life table then: individuals

produced for every female in every generation

lxmx is an important value to

consider in population studies

R0 is the population’s replacement rate:

If R0 = 1.0…no population

growth

If R0 < 1.0…the population is declining

If R0 > 1.0…the population is increasing

Calculating population features from life tables

Life history features

Reproductiv

e output

Raw coun

t data

Raw coun

t data

1. R0 – the basic reproductive rate

2. Tc = cohort generation time

3. ex = life expectancy4. r = intrinsic growth rate



3. ex = life expectancy4. r = intrinsic growth rate ∑ lxmx

Can use life tables to determine characteristics

about the population:

Cohort generation time (Tc)Cohort generation time (Tc) can be defined as the average length of time

between when an individual is born and the birth of its offspring. Tc is quite easy to obtain from our first example…

But Tc is less obvious for more complex life cycles – must be calculated

xx

xxc

ml

mlxT

.

Calculate Tc:1. Calculate the length of time to offspring production for each

age class2. Add all the lengths of time to offspring production for the

entire cohort 3. Calculate the total offspring produced by the survivors4. Divide by lengths of time to offspring production/the total offspring

produced by the survivors

semelparous annual life cycle

(Tc =1 year)

1872.03

TOTAL 610.32

Tc = 3.1

BIRTH

DEATHOFFSPRINGGeneration time


Life history features

Reproductiv

e output

Raw coun

t data

Raw coun

t data









Life expectancy (ex)

Life expectancy = the probability of living ‘x’ amount of time beyond a given age.

Most commonly quoted as the life expectancy at birth, e.g., life expectancy for South Africans females = 50 yrs, and for South African males = 55 years (http://www.who.int/countries/zaf/en/)

Note: time unit depends on organims being studied)

We can also calculate the mean length of life beyond any given age for the population.

Age 1

Age 2

Age 3

Time still to live (probability)

Time still to live

Time still to live

Death

Death

Death

Any Age

Time still to live (probability)Deat

h

2

1

xxx

aaL

1. Calculate Lx - number of surviving individuals in consecutive stage/age classes

2. Calculate Tx - the total number of living individuals at age ‘x’3. Calculate ex

Life expectancy (ex)

2

1

xxx

aaL

nxxxx LLLT .....1

xxa

Txe

Calculating ex:

NB. Units of e must be the

same as those of x

Thus if x is measured in intervals of 3

months, then ex must be

multiplied by 3 to give life

expectancy in terms of months










Non-overlapping generations


c

o

T

Rrln

HOW??

Intrinsic growth rate (r)

R0 converts the initial population size (N0) to the new size one generation later (NT)

NT=N0.R0

If R0 remains constant from generation to generation, then we can also use it to predict population size several generations into the future.

N0 N1

1 generation

N2 N3 Nn

2 generation

s

3 generation

s

n generation

s

Constant R0

N0 NT

1 generation

= ∑ lxmx R0 considers birth of new individuals

Basic reproductive rate (R0)

Non-Overlapping generations

Population size at t+1 = N0.R N1 = N0.R1


If R= 1.0…no population growth

If R < 1.0…the population is declining

If R > 1.0…the population is increasing

Fundamental Reproductive Rate (R)Consider birth of new individuals + survival of existing individuals

Nt = 10Nt+1 =

20

t

t

N

NR

1

R=20/10

R=2

N2 = N0.R2N3 = N0.R3

Population size at t+2 = N0.R.R

Population size at t+3 = N0.R.R.R

Nt = N0.Rt

Rearrange


As for R0

takes generation time into account

c

o

T

Rrln

Used to project population growth in population models

r = average rate of increase/individual

NT=N0.R0

Non-Overlapping generations

Nt = N0.Rt



NT = N0.RTIF t = T, then

R0 = RT

lnR0 = T.lnRlnR0/T = lnRBut lnR = r

Can now link R0 and R

Combine

Life tables


COHORT LIFE TABLE

follows a group of same-aged

individuals from birth (or fertilized eggs)

throughout their lives

STATIC LIFE TABLES

is made from mortality data

collected from a specified time period

Problems:1. Most organisms have complex life histories

(overlapping generations)2. Not always possible or feasible to follow a single cohort

from birth to death

Finite and instantaneous rates

The values of p, q hitherto collected are FINITE rates…their units of time = units of time for x (months, days, three-months etc)

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

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

They have limited value in comparisons unless same

units used

Finite SURVIVAL rates

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

Finite and instantaneous rates

What is a population? Localised group of individuals of the same species Within a given area where...

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