Quantifying population growth

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

Age and stage specificDifferential

reproductionDifferential survival

Tool for quantifying population growth

Birth DeathImmigration

Emigration

Population

growth

+ - -=Original populati

on+

Life tables

Life history features

Reproductiv

e output

Raw coun

t data

Raw coun

t data

Converse = survival rate

px = 1 - qx

Population characteristics

Makes explicit the dependence of r on the reproduction of individuals (R0) and the length of generation (Tc)

c

o

T

Rrln

t

t

N

NR

1

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

1. R0 = the basic reproductive rate

2. Tc = cohort generation time

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

1. R0 = the basic reproductive rate

2. Tc = cohort generation time

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

Can use life tables to determine characteristics

about the population:

Non-overlapping generations

Overlapping generations

BUT still useful to characterise a population in terms of its potential…especially when trying to make comparisons

e.g. between populations of the same species in different environments to see which environment is more favourable for the

species.

More on r (the intrinsic growth rate)

r = intrinsic rate of natural increase that the population

has the potential to

achieve. time

Constant survival rates

Constant rates of reproduction

00.10.20.30.40.50.60.70.80.9

1

lx

Stable age

structure

Stable r

Rarely the case!

Life tables – merits and shortcomings

Population growth potential POPULATION

PROJECTION

More general and useful method of analysing and

interpreting the survival and

fecundity schedules of

population with overlapping generations

Life tables useful – particularly for organisms with

simple life histories

Life tables less informative for organisms with

complex life histories

Overlapping generations

Non-overlapping generations

Projecting population changeProjecting doesn’t forecast what WILL happen

Projects forward to what WOULD happen if fecundity and survival schedules remained the same over time

Key pieces of information are for projecting:

p = survival rate

m = fecundity (individuals produced per surviving individual)

x ax lx dx qx px Fx mx

lxmx

xlxmx

0 87601.00

00.76

00.76

00.24

0 0 0 0.00 0.00

1 21020.24

00.18

20.75

80.24

24204

0 20 4.80 4.80

2 5090.05

80.05

81.00

00.00

01221

6 24 1.39 2.79

3 00.00

0           0.00 0.00

Certain steps must be followed to project data…

Projecting population change

mx 0 20 24 0

px 0.240 0.242 0.000 0.000

x 0 1 2 3

1 8760 2102 509 0

2 54256.000 2102.000 509.000 0.000

3 54256.000 13018.963 509.000 0.000

4 272595.251 13018.963 3152.546 0.000

5 336040.358 65410.413 3152.546 0.000

x ax lx dx qx px Fx mx

lxmx

xlxmx

0 87601.00

00.76

00.76

00.24

0 0 0 0.00 0.00

1 21020.24

00.18

20.75

80.24

24204

0 20 4.80 4.80

2 5090.05

80.05

81.00

00.00

01221

6 24 1.39 2.79

3 00.00

0           0.00 0.00

1. Rearrange data (Copy < Paste Special < Transpose)

Copy formula down

mx 0 20 24 0

px 0.240 0.242 0.000 0.000

x 0 1 2 3

1 8760 2102 509 0

2   2102.000 509.000 0.000

3   0.000 509.000 0.000

4   0.000 0.000 0.000

5   0.000 0.000 0.000

Projecting population change2. Calculate survivorship

for x1 to xn

Based on the px values (survival

rate) we can calculate the size

of each age class at t+1 based on the

size of the age class at time t

X1=Xt*px

=

*

=

*

=

*

Make sure that references to the p-values are absolute ($ signs) and that references to the X-values are

relative (no $ signs)

mx 0 20 24 0

px 0.240 0.242 0.000 0.000

x 0 1 2 3

1 8760 2102 509 0

2 54256.000 2102.000 509.000 0.000

3 54256.000 13018.963 509.000 0.000

4 272595.251 13018.963 3152.546 0.000

5 336040.358 65410.413 3152.546 0.000

Copy formula down

Projecting population change

3. Add fecundity to x0 column

Based on the mx values (fecundity) we can calculate

the number of new individuals

produced per individual at each

age

X0 = X1*mx + X1*mx+ X1*mx

* * *

Make sure that references to the m-values are absolute ($ signs) and that references to the X-values are

relative (no $ signs)

Based on the mx values (fecundity

rate) we can calculate the

number of individuals

produced for each individual in each

age class

mx 0 20 24 0    

px 0.240 0.242 0.000 0.000    

x 0 1 2 3 Nt R

1 8760 2102 509 0 11371 11371

2 54256.000 2102.000 509.000 0.00056867.0

00 5.001

3 54256.000 13018.963 509.000 0.00067783.9

63 1.192

4272595.25

1 13018.963 3152.546 0.000288766.

760 4.260

5336040.35

8 65410.413 3152.546 0.000404603.

317 1.401

Projecting population change

4. Calculate R (fundamental reproductive rate)

=sum(all x-values)

• Calculate Nt (size of each generation)

• Calculate R (fundamental reproductive rate)

Copy ALL formulas down until R stabilises….

72 1.62E+30 1.67E+29 1.74E+28 0.00E+00 1.80E+30 2.324

73 3.76E+30 3.89E+29 4.05E+28 0.00E+00 4.19E+30 2.324

74 8.75E+30 9.03E+29 9.41E+28 0.00E+00 9.74E+30 2.324

75 2.03E+31 2.10E+30 2.19E+29 0.00E+00 2.26E+31 2.324

76 4.72E+31 4.88E+30 5.08E+29 0.00E+00 5.26E+31 2.324

77 1.10E+32 1.13E+31 1.18E+30 0.00E+00 1.22E+32 2.324

78 2.55E+32 2.63E+31 2.74E+30 0.00E+00 2.84E+32 2.324

79 5.92E+32 6.12E+31 6.38E+30 0.00E+00 6.60E+32 2.324

80 1.38E+33 1.42E+32 1.48E+31 0.00E+00 1.53E+33 2.324

81 3.20E+33 3.30E+32 3.44E+31 0.00E+00 3.56E+33 2.324

mx 0 20 24 0    

px 0.240 0.242 0.000 0.000    

x 0 1 2 3 Nt R

1 8760 2102 509 0 11371 11371

2 54256.000 2102.000 509.000 0.00056867.00

0 5.001

3 54256.000 13018.963 509.000 0.00067783.96

3 1.192

4 272595.251 13018.963 3152.546 0.000288766.7

60 4.260

5 336040.358 65410.413 3152.546 0.000404603.3

17 1.401

Eventually R stabilises….in this case it

did so after ≈ 72 generations

72 1.62E+30 1.67E+29 1.74E+28 0.00E+00 1.80E+30 2.324

73 3.76E+30 3.89E+29 4.05E+28 0.00E+00 4.19E+30 2.324

74 8.75E+30 9.03E+29 9.41E+28 0.00E+00 9.74E+30 2.324

75 2.03E+31 2.10E+30 2.19E+29 0.00E+00 2.26E+31 2.324

76 4.72E+31 4.88E+30 5.08E+29 0.00E+00 5.26E+31 2.324

77 1.10E+32 1.13E+31 1.18E+30 0.00E+00 1.22E+32 2.324

78 2.55E+32 2.63E+31 2.74E+30 0.00E+00 2.84E+32 2.324

79 5.92E+32 6.12E+31 6.38E+30 0.00E+00 6.60E+32 2.324

80 1.38E+33 1.42E+32 1.48E+31 0.00E+00 1.53E+33 2.324

81 3.20E+33 3.30E+32 3.44E+31 0.00E+00 3.56E+33 2.324

NOTE: when R stabilises, so too does the age-structure, and this is known as the stable-age distribution of the population.

The proportions of the stable-age distribution are termed Cx

To calculate Cx: - Where R is stable, calculate the proportions of each age class in the generation

Nt R

Cx

Projecting population changeWith a constant r (lnR) and a constant stable age distribution (Cx) we can now calculate the fecundity and survival rates of the population…

r (lnR)

time

Constant survival rates

Constant rates of reproduction

Stable r

00.10.20.30.40.50.60.70.80.9

1

lx

Stable age

structure

Projecting population change

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

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

Must calculate Birth Rate and Survival Rate

Birth rate (B) = number of births/number of reproducing individuals

Survival rate (S) = No Survivors at time t, divided by total population size at time t-1

So, at stable-age

distribution:

B = 8.77

S = 0.24

Individual contributions Natural selection favours those individuals that make the greatest proportionate contribution to the future of the population to which they belong

All life history components affect this contribution – ultimately through

fecundity and survival

But necessary to combine these

effects into single currency so that different life

history strategies may be judged Reproductive value (RVx)

Consider the

contribution of each individual

…

1. Sum of the current reproductive output (mx) and the future (residual) reproductive value (Vx

*)

2. Vx* combines expected future survival and expected future

fecundity

3. This is done in a way that takes account of the relative contributions of individual to future generations

4. The life history favoured by natural selection is the one for which the sum of contemporary output and Vx

* is the highest

vx = mx + vx*

vx* = residual reproductive value

Reproductive value

CURRENT

reproductive

output (mx)

FUTURE reproduct

ive output (Vx

*)

CURRENT

reproductive

output (mx)

FUTURE reproduct

ive output (Vx

*)

Reproductive value

0

5

10

15

50 100

150

200

250

300

350

400

Reproductive value

Age (days)

Data for an annual plant

Time of greatest

contribution to future

generations

0

2

4

8

2 4 6 8 10

Reproductive value

Age (years)

Data for a sparrowhawk

0

6

Time of greatest

contribution to future

generations

Reproductive value

To calculate Vx

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

*

*

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

∑

Copy formula upCopy formula up

0.00

10.00

20.00

30.00

0 1 2 3 4

Reproductive value

Age

Age where contribution of an individual to future generations is greatest relative to the contribution of others in the population

Reproductive value

So far…

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

Age and stage specificDifferential

reproductionDifferential survival

Birth DeathImmigration

Emigration

Population

growth

+ - -=Original populati

on+

Birth PulseBirth Flow

Birth flow vs. Birth pulse

reproduction is concentrated at a single point when individuals leave an age class.

BIRTH

Pulse Flow1. constant addition and

leaving of individuals2. reproduction is spread

across an age class

mx

Reproduction

0

10

20

30

0 1 2

Age class

Single Leaving Event

mx

Age class

0

10

20

30

0 1 2

Reproduction

Constant

Addition and

Leaving

Projecting with birth

flow

1. Assume that all reproduction occurs at the mid-point of an age-class.

2. The mx values are appropriate for the end of an x class - not at the

middle - need get average….To get at average of mx and mx+1 = (mx-1

+ mx)/2

Birth flow vs. Birth pulse

Dealing with Birth FLOW: Need to adjust mx values to reflect FLOW rather than PULSE. Then project data as normal with new mx values.

0

10

20

30

0 1 2

mx-

1

mx

mx

Age

Birth flow vs. Birth pulseThis is the basic life table (birth pulse) that we have

constructed so far:

Projections are based on mx

MUST ADJUST mx

HOW?

Need to consider survival from period x-1 through x to x+1:i.e. px = [(lx + lx+1)] / [(lx + lx-1)]

Calculate a new px value

mx* = ((mx-1 + pxmx)/2Use formula to adjust mx with new px value

Need to consider survival from period x-1 through x to x+1:i.e. px = [(lx + lx+1)] / [(lx + lx-1)]

1. Calculate a new px value

mx* = ((mx-1 + pxmx)/22. Use formula to adjust mx with new px value

3. Change formulas for x0 to use mx* instead of mx

4. Copy new formula down until R stabilises

*

Birth flow vs. Birth pulseNOTE:

px values used in the life tables, and

calculations therein, do not

change.

i.e. px (birth-flow) = px (birth-

pulse) and the revised px values above are only

used to calculate mx values.

BIRTH PULSE

BIRTH FLOW

The difference between the mx and mx

* values may appear inconsequential,

but it is not!

Birth flow vs. Birth pulse

See the effect it has on R….

R stabilised after 72 generations at

5.07

R stabilised after 41 generations at

1.26

Both populations are growing (R>1) but the birth pulse population

is growing more quickly R=1.26 vs

R=5.07

Steps of projecting data:1. Rearrange data from rows to columns2. Calculate projected survivorship for x1 to xn

3. Calculate projected fecundity for x1 to xn

4. For each x calculate Nt and Nt+1

5. For each x calculate R (Nt+1/Nt)

6. Copy the formulas down until R stabilises7. Calculate the proportions of each class where R is stable

(ax /Nt) to calculate stable age class distribution (cx)

Projecting population change