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structure.txt
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structure.txt
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UNIT 3: Structured populations
----------------------------------------------------------------------
TSEC Introduction
Up until now we've tracked populations with a single state variable
(population size or population density)
POLL free_text_polls/UVRTWo0NBuHpHs5 What assumption are we making?
ANS All individuals can be counted the same. At least at census
time
ANS Never exactly true
What are some organisms for which this seems like a good
approximation?
ANS Dandelions, bacteria, insects
What are some organisms that don't work so well?
ANS Trees, people, codfish
CHANGE CC: Add: dogs/cats : can reproduce anytime → hard to model
----------------------------------------------------------------------
Structured populations
If we think age or size is important to understanding a population,
we might model it as an {\bf structured} population
Instead of just keeping track of the total number of individuals
in our population \ldots
Keeping track of how many individuals of each age
or size
or developmental stage
----------------------------------------------------------------------
TSS Example: biennial dandelions
Imagine a population of dandelions
Adults produce 80 seeds each year
1% of seeds survive to become adults
50% of first-year adults survive to reproduce again
Second-year adults never survive
Will this population increase or decrease through time?
----------------------------------------------------------------------
How to study this population
Choose a census time
Before reproduction or after
Since we have complete cycle information, either one should work
Figure out how to predict the population at the next census
----------------------------------------------------------------------
Census choices
BC
Before reproduction
All individuals are adults
We want to know how many adults we will see next year
After reproduction
Seeds, one-year-olds and two-year-olds
Two-year-olds have already produced their seeds; once these seeds
are counted, the two-year-olds can be ignored, since they will
not reproduce or survive again
NC
SIDEFIG images/dandy_field.jpg
SIDEFIG images/dandy_seeds.jpg
EC
----------------------------------------------------------------------
RSLIDE Example: biennial dandelions
Imagine a population of dandelions
Adults produce 80 seeds each year
1% of seeds survive to become adults
50% of first-year adults survive to reproduce again
Second-year adults never survive
Will this population increase or decrease through time?
----------------------------------------------------------------------
What determines $\lambda$?
If we have 20 adults before reproduction, how many do we expect to
see next time?
$\lambda = p + f$ is the total number of individuals per individual
after one time step
POLL free_text_polls/RD108ersZU9xUej What is $f$ in this example? What is the fecundity in this example?
ANS 0.8
POLL free_text_polls/QJSfa3XSSQORvvA What is $p$ in this example? What is the survival probability in this example?
ANS 0.5 for 1-year-olds and 0 for 2-year-olds.
ANS We can't take an average, because we don't know the
population structure
----------------------------------------------------------------------
What determines $\R$?
$\R$ is the average total number of offspring produced by an
individual over their lifespan
Can start at any stage, but need to close the loop
POLL free_text_polls/QJSfa3XSSQORvvA What is the reproductive number?
ANS If you become an adult you produce (on average)
ANS 0.8 adults your first year
ANS 0.4 adults your second year
ANS $\R=1.2$
CHANGE CC: Explaining how to calculate R on the board was helpful I think but probably go a little slower
----------------------------------------------------------------------
What does \R\ tell us about $\lambda$?
ANS Population increases when $\R>1$, so $\lambda>1$ exactly
when $\R>1$
If $\R=1.2$, then $\lambda$
ANS $>1$ -- the population is increasing
ANS $<1.2$ -- the life cycle takes more than 1 year, so it should
take more than one year for the population to increase 1.2 times
----------------------------------------------------------------------
TSS Modeling approach
BC
In this unit, we will construct \emph{simple} models of structured
populations
To explore how structure might affect population dynamics
To investigate how to interpret structured data
NC
SIDEFIG images/israelpop.png
EC
----------------------------------------------------------------------
Regulation
\emph{Simple} population models with regulation can have extremely
complicated dynamics
\emph{Structured} population models with regulation can have
insanely complicated dynamics
Here we will focus on understanding structured population models
\emph{without regulation}:
ANS Individuals behave independently, or (equivalently)
ANS Average per capita rates do not depend on population size
----------------------------------------------------------------------
SSLIDE Complexity
FIG images/bifurcation.png
----------------------------------------------------------------------
Age-structured models
BC
The most common approach is to structure by age
In age-structured models we model how many individuals there are in
each ``age class''
Typically, we use age classes of one year
Example: salmon live in the ocean for roughly a fixed number of
years; if we know how old a salmon is, that strongly affects how
likely it is to reproduce
NC
SIDEFIG images/salmon.jpg
EC
----------------------------------------------------------------------
Stage-structured models
BC
In stage-structured models, we model how many individuals there are
in different stages
Ie., newborns, juveniles, adults
More flexible than an age-structured model
Example: forest trees may survive on very little light for a long
time before they have the opportunity to recruit to the sapling
stage
NC
SIDEFIG images/tongass.jpg
EC
----------------------------------------------------------------------
Discrete vs.\ continuous time
Structured models can be done in either discrete or continuous time
Continuous-time models are structurally simpler (and smoother)
POLL free_text_polls/Mu8xWj5Objdg0WJ How do population characteristics affect the choice? How do population characteristics affect the choice between discrete and continuous models?
ANS Populations with continuous reproduction (e.g. bacteria), may be
better suited to continuous-time models
ANS Populations with \textbf{synchronous} reproduction (e.g., moths) may
be better suited to discrete-time models
Adding age structure is conceptually simpler with discrete time
ANS So we'll do that.
----------------------------------------------------------------------
TSEC Constructing a model
This section will focus on \textbf{linear, discrete-time,
age-structured} models
State variables: how many individuals of each age at any given time
Parameters: $p$ and $f$ \emph{for each age that we are modeling}
CHANGE CC: Helpful to draw the table of what is happening to each age-class next year but probably a little slower for the explanations
----------------------------------------------------------------------
When to count
We will choose a census time that is appropriate for our
study
Before reproduction, to have the fewest number of individuals
After reproduction, to have the most information about the
population processes
Some other time, for convenience in counting
ANS A time when individuals gather together
ANS A time when they are easy to find (insect pupae)
----------------------------------------------------------------------
The conceptual model
Once we choose a census time, we imagine we know the population for
each age $x$ after time step $T$.
We call these values $N_x(T)$
Now we want to calculate the expected number of individuals in each
age class at the next time step
We call these values $N_x(T+1)$
POLL free_text_polls/DHybyQQJegyAYbw What do we need to know? What do we need to know to calculate population for next time?
ANS The survival probability of each age group: $p_x$
ANS The average fecundity of each age group: $f_x$
----------------------------------------------------------------------
Closing the loop
$f_x$ and $p_x$ must close the loop back to the census time, so we
can use them to simulate our model:
$f_x$ has units [new indiv (at census time)]/[age $x$ indiv
(at census time)]
$p_x$ has units [age $x+1$ indiv (at census time)]/[age $x$ indiv
(at census time)]
----------------------------------------------------------------------
ASLIDE The structured model
WIDEFIG images/structure_cc.png
CHANGE Put this in the goddam lecture notes, morph!:e
----------------------------------------------------------------------
SS Model dynamics
----------------------------------------------------------------------
Short-term dynamics
This model's short-term dynamics will depend on parameters
\ldots
It is more likely to go up if fecundities and survival
probabilities are high
\ldots and starting conditions
If we start with mostly very old or very young individuals, it
might go down; with lots of reproductive adults it might go up
----------------------------------------------------------------------
Long-term dynamics
If a population follows a model like this, it will tend to reach
a \textbf{stable age distribution}:
the \emph{proportion} of individuals in each age class is
constant
a stable value of $\lambda$
if the proportions are constant, then we can average over
$f_x$ and $p_x$, and the system will behave like our simple
model
POLL free_text_polls/DtbBUtry5ts5XRz What are the long-term dynamics of such a system?
ANS Exponential growth or exponential decline
----------------------------------------------------------------------
Exception
Populations with \textbf{independent cohorts} do not tend to reach a
stable age distribution
A \textbf{cohort} is a group that enters the population at the
same time
We say my cohort and your cohort interact if my children
might be in the same cohort as your children
or my grandchildren might be in the same cohort as your
great-grandchlidren
\ldots
As long as all cohorts interact (none are independent), then the
unregulated model leads to a stable age distribution (SAD)
----------------------------------------------------------------------
Independent cohorts
Some populations might have independent cohorts:
If salmon reproduce \emph{exactly} every four years, then:
the 2015 cohort would have offspring in 2019, 2023, 2027,
2031, \ldots
the 2016 cohort would have offspring in 2020, 2024, 2028,
2032, \ldots
in theory, they could remain independent -- distribution would
not converge
Examples could include 17-year locusts, century plants, \ldots
----------------------------------------------------------------------
TSEC Life tables
People often study structured models using \textbf{life tables}
A life table is made \emph{from the perspective of a particular
census time}
It contains the information necessary to project to the next census:
How many survivors do we expect at the next census for each
individual we see at this census? ($p_x$ in our model)
How many offspring do we expect at the next census for each
individual we see at this census? ($f_x$ in our model)
----------------------------------------------------------------------
Cumulative survivorship
The first key to understanding how much each organism will
contribute to the population is {\bf survivorship}
In the field, we estimate the probability of survival from age $x$
to age $x+1$: $p_x$
This is the probability you will be \emph{counted} at age $x+1$,
given that you were \emph{counted} at age $x$.
To understand how individuals contribute to the population, we are
also interested in the overall probability that individuals survive
to age $x$: $\ell_x$.
ANS $\ell_x = p_1 \times \ldots p_{x-1}$
ANS $\ell_x$ measures the probability that an
individual survives to be counted at age $x$, given that it is
ever counted at all (ie., it survives to its first census)
CHANGE CC: add the definition in the notes and explain more l1 = 1 (needed to explain 2 times)
----------------------------------------------------------------------
Calculating \R
We calculate \R\ by figuring out the estimated contribution at each
age group, \emph{per individual who was ever counted}
We figure out expected contribution given you were ever counted by
multiplying:
ANS $f_x \times \ell_x$
----------------------------------------------------------------------
SS Examples
----------------------------------------------------------------------
Dandelion example
CLASS HIGHFIG images/dandy_field.jpg
----------------------------------------------------------------------
RSLIDE Example: biennial dandelions
Adults produce 80 seeds each
1% of seeds survive to become adults
50% of first-year adults survive to reproduce again
Second-year adults never survive
What does the life table look like?
----------------------------------------------------------------------
QSLIDE Dandelion life table
INPUT life_tables/dandy.skeleton.tab.tex
----------------------------------------------------------------------
RSLIDE Dandelion life table
INPUT life_tables/dandy.empty.tab.tex
----------------------------------------------------------------------
ASLIDE Dandelion life table
INPUT life_tables/dandy.tab.tex
----------------------------------------------------------------------
Dandelion dynamics
FIG structure/dandy.Rout-0.pdf
----------------------------------------------------------------------
RSLIDE Dandelion dynamics
FIG structure/dandy.Rout-1.pdf
----------------------------------------------------------------------
RSLIDE Dandelion dynamics
FIG structure/dandy.Rout-2.pdf
----------------------------------------------------------------------
Dandelion dynamics
DOUBLEFIG structure/dandy.Rout-1.pdf structure/dandy.Rout-2.pdf
----------------------------------------------------------------------
Squirrel example
CLASS HIGHFIG images/grey_squirrel.jpg
----------------------------------------------------------------------
QSLIDE Gray squirrel population example
INPUT life_tables/sq.empty.tab.tex
----------------------------------------------------------------------
Squirrel observations
POLL free_text_polls/GJ5F4CLnWReepvU Do you notice anything strange about the squirrel life table?
ANS Older age groups seem to be grouped for fecundity.
ANS Strange pattern in survivorship; do we really believe
nobody survives past the last year?
ANS Might be better to use a model where they keep track of 1
year, 2 year, and ``adult" -- not much harder.
----------------------------------------------------------------------
ASLIDE Gray squirrel population example
INPUT life_tables/sq.tab.tex
----------------------------------------------------------------------
RSLIDE Gray squirrel dynamics
FIG structure/squirrels.Rout-0.pdf
----------------------------------------------------------------------
RSLIDE Gray squirrel dynamics
FIG structure/squirrels.Rout-2.pdf
----------------------------------------------------------------------
RSLIDE Gray squirrel dynamics
FIG structure/squirrels.Rout-1.pdf
----------------------------------------------------------------------
Gray squirrel dynamics
DOUBLEFIG structure/squirrels.Rout-2.pdf structure/squirrels.Rout-1.pdf
----------------------------------------------------------------------
RSLIDE The structured model
WIDEFIG images/structure_cc.png
----------------------------------------------------------------------
Salmon example
What happens when a population has independent cohorts?
Does not necessarily converge to a SAD
CHANGE CC: add the solution of salmon table
----------------------------------------------------------------------
QSLIDE Salmon example
INPUT life_tables/salmon.empty.tab.tex
----------------------------------------------------------------------
ASLIDE Salmon example
INPUT life_tables/salmon.tab.tex
----------------------------------------------------------------------
RSLIDE Salmon example
WIDEFIG images/salmon.jpg
----------------------------------------------------------------------
RSLIDE Salmon dynamics
FIG structure/salmon.Rout-0.pdf
----------------------------------------------------------------------
RSLIDE Salmon dynamics
FIG structure/salmon.Rout-2.pdf
----------------------------------------------------------------------
RSLIDE Salmon dynamics
FIG structure/salmon.Rout-1.pdf
----------------------------------------------------------------------
Salmon dynamics
DOUBLEFIG structure/salmon.Rout-2.pdf structure/salmon.Rout-1.pdf
----------------------------------------------------------------------
SS Calculation details
----------------------------------------------------------------------
DEFHEAD $f_x$ vs.~$m_x$
Here we focus on $f_x$ -- the number of offspring seen at the
next census (next year) per organism of age $x$ seen at this census
An alternative perspective is $m_x$: the total number of offspring
per reproducing individual of age $x$
POLL free_text_polls/JTcb09r4ByRS4uH What is the relationship? What is the relationship between f and m?
ANS To get $f_x$ we multiply $m_x$ by one or more survival terms,
depending on when the census is
ANS Need to close the loop from one census to the next
CHANGE CC: ask the poll more sharply
----------------------------------------------------------------------
When do we start counting?
Is the first age class called 0, or 1?
In this course, we will start from age class 1
If we count right {\em after} reproduction, this means we are
calling newborns age class 1. Don't get confused.
----------------------------------------------------------------------
RSLIDE Example: biennial dandelions
Adults produce 80 seeds each ($m_x$)
1% of seeds survive to become adults
50% of first-year adults survive to reproduce again
Second-year adults never survive
What does the life table look like?
CHANGE CC: It was cool that you re-do the life table on the board (life cycle+calculation with explanation at the same time)
----------------------------------------------------------------------
ASLIDE Dandelion life table
INPUT life_tables/pre.tab.tex
----------------------------------------------------------------------
ASLIDE Counting \emph{after} reproduction
INPUT life_tables/post.tab.tex
REMARK Explain two-line and three-line versions. Take your time
----------------------------------------------------------------------
Calculating \R
The reproductive number \R\ gives the average lifetime reproduction
of an individual, and is a valuable summary of the information in
the life table
$\R = \sum_x{\ell_x f_x}$
If $\R>1$ in the long (or medium) term, the population will
increase
If $\R$ is persistently $<1$, the population is in trouble
We can ask (for example):
Which ages have a large \emph{contribution} to \R?
POLL free_text_polls/pPsDVfbJqwXXMvk Which values of $p_x$ and $f_x$ is \R\ sensitive to? Which values of p_x and f_x is R sensitive to?
ANS The $p$s for young individuals affect all the $\ell$s.
CHANGE CC: p1 f1 better answers to the poll: add the fecundity for the young
----------------------------------------------------------------------
The effect of old individuals
Estimating the effects of old individuals on a population can be
difficult, because both $f$ and $\ell$ can be extreme
The contribution of an age class to $\R$ is $\ell_x f_x$
POLL free_text_polls/Liv5rLUGya1qW91 Extreme how? How are these values extreme?
ANS In most populations $\ell$ can be very small for large $x$
ANS In many populations, $f$ can be very large for large $x$
Reproductive potential of old individuals \emph{may} or \emph{may
not} be important
ANS In tree populations, most trees don't survive to get huge,
but the huge trees may have most of the total reproduction
ANS In bird populations, old birds produce fairly well, but not
nearly enough to outweigh the low probability of being old.
CHANGE CC: for birds need to add "most bird population ..." + probably add the Canadian fish example you were thinking about
----------------------------------------------------------------------
CSLIDE Old individuals
BC
SIDEFIG images/big_tree.jpg
NC
HIDEFIG images/stork.jpg
EC
----------------------------------------------------------------------
TSS Measuring growth rates
In a constant population, each age class replaces itself:
$\R = \sum_x \ell_x f_x = 1$
In an exponentially changing population, each year's {\bf cohort} is
a factor of $\lambda$ bigger (or smaller) than the previous one
$\lambda$ is the finite rate of increase, like before
Looking back in time, the cohort $x$ years ago is $\lambda^{-x}$ as
large as the current one
The existing cohorts need to make the next one:
$\sum_x \ell_x f_x \lambda^{-x} = 1$
----------------------------------------------------------------------
The Euler equation
If the life table doesn't change, then $\lambda$ is given by $\sum_x
\ell_x f_x \lambda^{-x} = 1$
We basically ask, if the population has the structure we would
expect from growing at rate $\lambda$, would it continue to grow at
rate $\lambda$.
On the left-side each cohort started as $\lambda$ times smaller than the
one after it
Then got multiplied by $\ell_x$.
Under this assumption, is the next generation $\lambda$ times bigger
again?
CHANGE MK: I think it may be useful to have a numeric life table example to show in parallel with these equations.
CHANGE CC: you went very fast using different values of lambda, then erasing the results of the calculation, then quickly do another one... Should go slower (you realized at the end of the explanation that you went fast), a little too fast to be clear
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DEFHEAD $\lambda$ and \R
If the life table doesn't change, then $\lambda$ is given by $\sum_x
\ell_x f_x \lambda^{-x} = 1$
What's the relationship between $\lambda$ and \R?
When $\lambda=1$, the left hand side is just \R.
If $\R>1$, the population more than replaces itself when
$\lambda=1$. We must make $\lambda>1$ to decrease LHS and
balance.
If $\R<1$, the population fails to replace itself when
$\lambda=1$. We must make $\lambda<1$ to increase LHS and
balance.
So \R\ and $\lambda$ tell the same story about whether the population
is increasing
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Time scales
$\lambda$ gives the number of individuals per individual \emph{every
year}
$\R$ gives the number of individuals per individual \emph{over a
lifetime}
POLL free_text_polls/eOupL6vFaV2QAsn What relationship do we expect for an annual population (individuals die every year)? What relationship do we expect for an annual population?
ANS $\R=\lambda$; each organism observed reproduces \R\ offspring
on average, all in one time step
POLL free_text_polls/98SeJLcR0TmWqcT For a long-lived population? What relationship do we expect for a long-lived population?
ANS The \R\ offspring are produced slowly, so population changes
slowly
ANS $\lambda$ should be closer to 1 than \R\ is.
ANS But on the same side (same answer about whether population
is growing)
CHANGE CC: change 1st poll into multiple choices and for the answers of 2nd poll: don’t suppose that the pop is growing
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Studying population growth
$\lambda$ and \R\ give similar information about your population
\R\ is easier to calculate, and more generally useful
But $\lambda$ gives the actual rate of growth
More useful in cases where we expect the life table to be
constant with exponential growth or decline for a long time
CHANGE MK: In tutorial I showed how lambda changes per generation as the age distribution is approaching its stable distribution, which seemed to help students. Maybe include a quick calculation?
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Growth and decline
If we think a particular period of growth or decline is important,
we might want to study how factors affect $\lambda$
Complicated, but well-developed, theory
In a growing population, what happens early in life is more
important to $\lambda$ than to \R.
In a declining population, what happens late in life is more
important to $\lambda$ than to \R.
A common error is to assume that periods of exponential
\emph{growth} are more important to ecology and evolution the
periods of exponential \emph{decline}. In the long term, these
should balance.
ANS Because otherwise the population would go to zero or infinity
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SEC Life-table patterns
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SS Survivorship
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Patterns of survivorship
POLL free_text_polls/Y3w5PwZhoobOKoN What sort of patterns do you expect to see in $p_x$?
ANS Younger individuals usually have lower survivorship
ANS Older individuals often have lower survivorship
What about $\ell_x$?
ANS It goes down
ANS But sometimes faster and sometimes slower
ANS Best understood on a log scale
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Starting off
Recall: we always start from age \emph{class} 1
If we count newborns, we still call them class 1.
POLL free_text_polls/hIXTOy066y90aYM What is $\ell_1$ when we count before reproduction? What is l_1 when we count before reproduction?
ANS 1
ANS $\ell_1$ is the probability you're counted at age class 1,
\emph{given} that you're counted at age class 1.
ANS We don't count individuals that we don't count
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RSLIDE Constant survivorship
FIG age/const.Rout-0.pdf
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RSLIDE Constant survivorship
FIG age/const.Rout-2.pdf
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RSLIDE Constant survivorship
FIG age/const.Rout-1.pdf
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Constant survivorship
DOUBLEFIG age/const.Rout-0.pdf age/const.Rout-1.pdf
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DEFHEAD ``Types'' of survivorship
There is a history of defining survivorship as:
Type I, II or III depending on whether it increases, stays
constant or decreases with age {\em (don't memorize this, just be
aware in case you encounter it later in life)}.
Real populations tend to be more complicated
Most common pattern is: high mortality at high and low ages, with