Post on 16-Jan-2016
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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