More about Life Tables

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More about Life Tables Note that the
calculations weve 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
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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 well 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
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Divide Sxlxmx by R0 . What you have calculated
is the age-weighted mean of the distribution of
births by age. Its 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
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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...
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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
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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. Lets
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
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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 gt 0 indicates that this
population is growing, but we knew that because
R0 was gt1. Finally (to guild the lily) lets see
what happens if survivorship and/or reproduction
is less than the values producing an R0 1. (by
changing l4 and m4.)
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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
.3 0.4 0.12 0.48 5
.2 0.1 0.02 0.10 6
0 ---- ----- R0 0.88 ?
2.10 T 2.10/0.88 2.386 r
ln(0.88)/2.386 r -0.128/2.386
-.053 R is lt 1, and r is negative. This
population is declining.
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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.
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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
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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.
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Heres 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
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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
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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!!
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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.
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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...
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However, more recent censuses indicate that there
has been a change since that 1910 census. Growth
hasnt continued to slow as much as the graph
plotting effective r indicated.
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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 embryos Western
(best) 94 1.71 Central (intermediate) 87
1.37 Adirondack (worst) 79 1.06
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Some other examples In Drosophila lifespan
declines at high density, but there is a decrease
in fecundity beginning at much lower densities
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In song sparrows on Mendarte Island off the BC
coast
Males are territorial. As population density
increases, there are more males that dont hold
territories and cant 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.
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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
plants population size. This is for flax
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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.
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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
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• 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

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• 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 density-dependence, particularly when a
  population is small
• 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.

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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
density- dependence 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?
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Florida panthers (Felis concolor) are highly
endangered. In the 1980s the total population was
lt50 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.
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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.
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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?
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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
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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
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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.
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Here is a figure that shows you how the
oscillations change with the product rt
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Reviewing the other assumptions of the logistic
model 1) The population structure is
homogeneousthere 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 constantthe
quality of the environment does not change
over time. So is r. Not only are their life
histories identical, but they dont change
with density or over time. 4) There are no time
lags.
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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
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To show you how well the logistic can fit real
data, heres a classic growth curve for E. coli
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And heres 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?