Transcript Cohort life tables

More about Life Tables
Note that the calculations we’ve done thus far with a life table,
and those we will do today, make no adjustment in lx or mx
as the population grows or declines. What type of growth does
the life table represent?
This represents a serious limitation to population projection
using life tables. There are ‘corrections’, but they are
mathematically complex.
Recognizing the limitations, we move on…
From lx, mx (in Ricklefs’ text fecundity is ‘bx’), and age class
information, you can calculate an approximate (but usually
quite accurate) value for “r”. The method is shown in the
supplement, but we’ll go through our life table to demonstrate
it here. After calculating R0, calculate and sum x lxmx.
survivorship fecundity
Age class
lx
bx
lxbx
x lxbx
0
1.0
0.0
0
0
1
.8
0.2
0.16
0.16
2
.6
0.3
0.18
0.36
3
.4
1.0
0.4
1.2
4
.4
0.6
0.24
0.96
5
.2
0.1
0.02
0.10
6
0
-------R0 = 1.0
 = 2.78
Divide Σxlxmx by R0 . What you have calculated is the
age-weighted mean of the distribution of births by age.
It’s the equivalent of finding the age at which you could
balance a cutout of the distribution on a ruler. In physics
this is called the first moment of the distribution. My
guess...
bx
age
The quotient is an estimate of the generation time for this
species/life table. The generation time is usually called
either T (in Ricklefs’ text) or G (most other texts).
 x lxbx = 2.78
R0 = 1.0
 x lxbx /R0 = 2.78/1.0 = 2.78 = T
Now go back to the basic equation for continuous
exponential growth…
Nt = N0 ert
or
Nt/N0 = ert
if we set the time t as one generation time, we know the
values of most of the terms in the equation...
Since the time is one generation, we know that the ratio
Nt/N0 is the net replacement rate. Thus, the equation
becomes:
Nt/N0 = R0 = erT
Now take the logs of both sides of the equation…
ln (R0) = rT
ln (1.0) = r (2.78)
0 = 2.78r
r
= 0
We already “knew” this result, i.e. that if R0 was 1, then r = 0.
But we also modified the initial life table to show that small
change in survivorship and/or reproduction could lead to
growth or decline in the population. Let’s re-calculate again
for the modified life table (m3 = 0.6 in the original table).
Age class
lx
bx
lxbx
xlxbx
0
1.0
0.0
0
0
1
.8
0.2
0.16
0.16
2
.6
0.3
0.18
0.36
3
.4
1.0
0.4
1.2
4
.4
1.0
0.4
1.6
5
.2
0.1
0.02
0.10
6
0
-------R0 = 1.16
= 3.42
Making the same calculations:
T = 3.42/1.16 = 2.95
r = ln R0 /T = ln (1.16)/2.95
r = 0.148/2.95 = 0.05
An “r” > 0 indicates that this population is growing, but we
knew that because R0 was >1.
Finally (to guild the lily) let’s see what happens if
survivorship and/or reproduction is less than the values
producing an R0 = 1. (by changing l4 and m4.)…
Age class
0
1
2
3
4
5
6
lx
1.0
.8
.6
.4
.3
.2
0
bx
0.0
0.2
0.3
1.0
0.4
0.1
----
lxbx
0
0.16
0.18
0.4
0.12
0.02
-----
R0 = 0.88
x lxbx
0
0.16
0.36
1.2
0.48
0.10
= 2.10
T = 2.10/0.88 = 2.386
r = ln(0.88)/2.386
r = -0.128/2.386 = -.053
R is < 1, and “r” is negative. This population is declining.
The life table uses fixed, constant values for survivorship and
natality. Is there evidence of density-dependence in the life
table?
No!
So, how is a density response evident? There must be a
change in either the birth rate or the death rate as density
changes.
Remember that r = b – d. A change in either will change the
population growth rate.
There are matrix methods to put density dependence into a
form of the life table, changing lx and bx with density , but
they are beyond the scope of this course.
If the change is a general one in the pattern of birth and/or
death rate, the carrying capacity K will change, as well as
the observed rate of increase at any density.
For example, Peregrine falcons exposed to DDT had a much
reduced reproductive success due to egg shell thinning. This
is what it might look like:
Alternatively, excessive harvesting might have its effect on
death rates. This, too, would change the K and the observed
growth rate at any density. This certainly happened to a
variety of whale species.
Here’s a neat experiment showing density dependence:
Eisenberg, working in a pond only a few miles from here,
showed that egg production in snails is density-dependent.
He established 3 kinds of populations, each in a separate
enclosure, along the edges of the pond:
1. high density (an experimental treatment produced
by increasing egg density by 5x)
2. low density (this experimental treatment produced
by reducing the density of eggs to 1/5 of control)
3. control (natural egg density)
The snails grew from these eggs, and he measured egg
production in the three treatments…
There were naturally about 1000 eggs within each enclosed
area. Eggs were moved from low density (LD) areas into
high density (HD) areas, bringing the total number of eggs
at HD to ~5000, and reducing the number in LD areas to
~200.
C
LD
Diagram of the
pond with fenced
enclosures along
the shore.
HD
After the snails grew, egg production at high density (5000
snails/m2) was reduced to about 10% of that in treatments at
low density (200 snails/m2). That difference in fecundity
brought the numbers of eggs in all three treatments to very
similar levels – density compensation.
In snails, egg production depends on food quality. They scrape
food from the surfaces of plants in the pond.
plants
protein
egg production
Eisenberg added a 6 oz. package of frozen spinach to a few
groups. Their egg production was about 10x controls, even
though the quantitative increase in amount of plant was
miniscule. Food quality!! Density-dependence!!
Remember again the critical problems with the exponential
growth model…
Basically, they involve a lack of realism.
Natural populations are limited by both biological and
physical conditions in the environment.
Physical conditions include climatic extremes (seasonality,
storms, etc.), drought, and other factors.
Biological conditions such as competition and predation also
limit population growth. They are the subject of much of the
second half of the course.
Whatever the causes, there is evidence of density dependence
even in the growth of the U.S. population…
Evidence that growth slows as carrying capacity is
approached:
1) Human population growth in the United States…
as population size has increased, the intrinsic rate of
increase has slowed...
However, more recent censuses indicate that there has been
a change since that 1910 census. Growth hasn’t continued to
slow as much as the graph plotting effective r indicated.
2) Evidence of the impact of habitat quality in limiting
population growth. Remember that r = b - d. Anything
that causes birth rate to decline or death rate to increase
will decrease little r. Anything that leads to a lower value
for K should cause a population to respond in life history
values.
In a population of white-tailed deer in New York State,
habitat quality and statistics indicating birth rate clearly
indicate the linkage…
Region
% pregnant
Western (best)
94
Central (intermediate) 87
Adirondack (worst)
79
embryos
1.71
1.37
1.06
Some other examples:
In Drosophila lifespan declines at high density, but there is a
decrease in fecundity beginning at much lower densities…
In song sparrows on Mendarte Island off the BC coast…
Males are territorial. As population
density increases, there are more
males that don’t hold territories and
can’t breed successfully.
The number of young fledged per
female decreases with density,
and…
The percentage of juveniles
(fledged) decreases as the autumn
adult population increases.
Finally, as plant density increases, the size of individual
plants decreases. Since the number of seeds produced by a
plant (a preliminary measure of its fitness) is basically
proportional to its biomass (size), decreasing plant size
indicates a key density response that will tend to limit the
plant’s population size. This is for flax…
The equation for the logistic model can be manipulated to
demonstrate other aspects of the predictions…
dN/dt = rN (1 - N/K)
1/N dN/dt = r (1 - N/K)
now we have an equation for growth rate per head. Note that
it is a linear equation, where the per head growth rate drops
from r to 0 as the population grows toward K.
The way in which dN/dt changes with increasing population
size is also important. It dictates the population size that
can be most effectively harvested, i.e. when the population
will grow most rapidly to replace harvested individuals…
dN/dt = rN (1 - N/K) = rN - rN2/K
this is the equation of an inverted parabola, with the
maximum value of dN/dt indicated by a 0 in the derivative
of this equation with respect to N…
0 = r - 2rN/K
r = 2rN/K
K = 2N
N = K/2
Review summarizing density-dependence thus far:
• For most natural species density-dependence is evident in
birth and/or death rates
• As N increases, typically birth rates decrease and death
rates increase
• These changes are reflected in the growth rate, and keep
population size close to K
All this is described as negative density-dependence, since
the rate of population growth decreases as population size
increases.
However…
There are certain conditions that can lead to positive densitydependence, particularly when a population is small:
1) When a population is very small, mate finding may be
difficult. Therefore, the birth rate at low density is far
lower than the simple logistic model projects. This is
called the Allee effect. Very small (frequently rare,
endangered species) populations dwindle toward or to
extinction.
2) Small populations may also suffer from genetic
inbreeding. Inbreeding exposes the presence of
deleterious mutations in offspring, reducing their fitness.
Fitness reduction means that these individuals are likely
to produce fewer offspring, and the population will
decline. Increasing population size decreases inbreeding
and leads to a higher growth rate – a positive densitydependence – but only in the increase from very small
population size that causes high inbreeding. Then
negative density-dependence takes over again.
This happened in Florida panthers and may be happening in
the African cheetah. Evidence?
Florida panthers (Felis concolor) are highly endangered. In the
1980s the total population was <50 individuals with an
effective population size of ~25. They occurred in two main
groups:
One group in the Everglades,
and the other further north, but
with limited contact between
them.
In these small populations inbreeding occurred. Molecular
evidence of the inbreeding became apparent from low
genetic variation in microsatellite loci.
Inbreeding also exposed a number of deleterious traits
(appearing in homozygous recessives):
A kinked tail
A ‘cowlick’ of fur over the shoulders
Males are cryptorchid (only one testicle descends
A high proportion of defective sperm (also kinked ‘tails’)
Genetic rescue, introducing some Texas panthers (same
species but different subspecies) creates the likelihood of
outbreeding and has been relatively successful since 2000.
In the case of the cheetah, humans have only had limited
impact on a long-term loss of genetic diversity. Cheetahs
today are remarkable in the low genetic diversity evident in
comparisons of protein and nuclear markers.
What has this meant for cheetahs?
One more complication…
The basic logistic model predicts smooth growth to an
equilibrium population size, K. There are reasons to question
the simplicity of that model.
One source of complication: it assumes that each individual
contributes to further population growth beginning at birth.
Is that realistic? No!
Correcting that assumption means adding time lags to the
model. There are 2 basic forms of lag…
a gestation time lag - the time between conception
and birth
a maturation time lag - the time between birth and
reproductive maturity
Time lags lead to oscillations in population size…
As a population approaches K, time lags cause a
reproductive response appropriate for an
earlier, smaller population
Depending on the amount of growth that occurs
during the lag time, the result can be a
population with damped oscillations that
eventually lead to a stable equilibrium, or
one that undergoes continuing, pernicious
oscillations until extinction.
Since the result depends on growth during lag,
high “r” leads to high amplitude oscillation
long lag leads to high amplitude oscillation
You can generate oscillating population dynamics using
Populus.
Try the following parameters to get damped oscillation:
N0 = 5, K = 500, r = 0.2, T = 2
and try these values to get “pernicious” oscillations that
would lead to likely extinction:
N0 = 5, K = 500, r = 0.5, T = 5
if you were to increase the r in the last set of parameters to
1, you get a single, large oscillation.
Here is a figure that shows you how the oscillations change
with the product rτ:
Reviewing the other assumptions of the logistic model:
1) The population structure is homogeneous…there is no
age structure or individual variation in size and resource
use.
2) All have the same birth and death rates. Earlier this
was described as all individuals having identical life
histories. This also obviates sex.
3) K is a constant…the quality of the environment does not
change over time. So is r. Not only are their life histories
identical, but they don’t change with density or over time.
4) There are no time lags.
How useful is the logistic in describing the growth of real
populations?
It works reasonable well for some species with simple life
cycles, generally those with short lifespans as well.
It also works for additional species when growth occurs
under strictly controlled laboratory conditions.
In the “real” world, however, most species have:
1)either high r or long enough time lags to produce, if
not cycles, growth which does not fit the simple
logistic well.
2)exposure to a changing environment, and thus a
changing K.
3)apparently stochastic dynamics
To show you how well the logistic can fit real data, here’s a
classic growth curve for E. coli:
And here’s one that shows you how dramatically a change in
temperature can alter the carrying capacity for a growing
population. These are growth curves for Drosophila
melanogaster. How could the growth curve look logistic if
the temperature keeps varying?