structure.txt

UNIT 4: Structured populations

----------------------------------------------------------------------

EXTRA

Found in 2026 and didn't have much time to deal; update as you go.

Go back and look at what's ON and OFF <LINE/SLIDE>, and reclassify

Clean up the logic here. Make sure you know what's in or out for the short version. 

Clarify when you're talking about long-term and year-to-year lambda
* Should be done 2026, check

Replace some content about age distributions and so forth

----------------------------------------------------------------------

TSEC Introduction

	Up until now we've tracked populations with a single state variable
	(population size or population density)

	POLL  What assumption are we making?

		ANS All individuals are similar enough to be counted as if they are the
		same

			ANS Always (continuous time) 

			ANS At census time (discrete time)

	What are some organisms for which this seems like a good
	approximation?

		ANS Short-lived and/or don't grow much

		ANS Dandelions, bacteria, insects

	What are some organisms that don't work so well?

		ANS Trees, people, codfish

----------------------------------------------------------------------

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 webpix/dandy_field.jpg

SIDEFIG webpix/dandy_seeds.jpg

EC

----------------------------------------------------------------------

REPSLIDE 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 \emph{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  What is $f$ in this example? |  What is the fecundity in this example?

		ANS 0.8

	POLL  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 It can change from year to year!

----------------------------------------------------------------------

Long-term growth

	$\lambda$ changes when the population structure changes

	Define $\bar\lambda$ as the long-term average rate of growth.

		$N \approx N_0 \bar\lambda^T $

	Can we calculate $\bar\lambda$ if $p$ is changing?

		ANS We can't take an average over ages until we know the
		population structure

		ANS Not easy to find the population structure until we know $\bar\lambda$
		\ldots

----------------------------------------------------------------------

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  What is the reproductive number?

	ANS If you become an adult you produce (on average)

		COMMENT Blackboard!

		ANS 0.8 adults your first year

		ANS 0.4 adults your second year

	ANS $\R=1.2$

----------------------------------------------------------------------

What does \R\ tell us about $\bar\lambda$?

	\R\ is easier to calculate than $\bar\lambda$

	What is the connection?

		ANS Population increases when $\R>1$, so $\bar\lambda>1$ exactly
		when  $\R>1$

	If $\R=1.2$, then $\bar\lambda$

		ANS $>1$ -- the population is increasing

		ANS $<1.2$ -- the life cycle takes more than 1 time step, so it 
		take more than one time step 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 webpix/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, meaning…

		ANS Average per capita rates do not depend on population size

----------------------------------------------------------------------

OFFSLIDE CSLIDE Complexity

ADD Better art!

FIG webpix/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 webpix/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 webpix/tongass.jpg

EC

----------------------------------------------------------------------

Discrete vs.\ continuous time in unstructured models

	continuous-time models are structurally simpler (and smoother)

	discrete-time models only need to assume everyone's the same sometimes

		ANS At the census time

		ANS More realistic

----------------------------------------------------------------------

DEFHEAD \ldots in structured models

	We no longer assume everyone is the same (we keep track of age or size)

	POLL So it should be mostly about reproduction | When might we prefer a continuous-time model?

		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

	Continuous time with structure gives people headaches

		So we won't do it here, only for the sake of simplicity (and time)

----------------------------------------------------------------------

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}

----------------------------------------------------------------------

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 $N_x(T+1)$

	What are the parameters? | 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)]

----------------------------------------------------------------------

The structured model

WIDEFIG my_images/structure_cc.png

----------------------------------------------------------------------

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 $\lambda_T$

			ANS Multiplicative growth each step approaches a constant

			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  What are the long-term dynamics of such a system?

		ANS Exponential growth or exponential decline

	SHORT Skipping calculations, but you can poke if curious

	HREF http://tinyurl.com/DandelionModel2026 Spreadsheet link

		ADD Update year

----------------------------------------------------------------------

EXTRA

	http://tinyurl.com/Dandelions2023 Spreadsheet link

	http://tinyurl.com/DandelionModel2024 Spreadsheet link

----------------------------------------------------------------------

ONSLIDE 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)

----------------------------------------------------------------------

ONSLIDE 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)

----------------------------------------------------------------------

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 FIG webpix/dandy_field.jpg

----------------------------------------------------------------------

REPSLIDE 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

----------------------------------------------------------------------

REPSLIDE 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

----------------------------------------------------------------------

PSLIDE Dandelion population dynamics

FIG structure/dandy.Rout-1.pdf

----------------------------------------------------------------------

PSLIDE Dandelion age dynamics

FIG structure/dandy.Rout-2.pdf

----------------------------------------------------------------------

Dandelion dynamics

DOUBLEFIG structure/dandy.Rout-1.pdf structure/dandy.Rout-2.pdf

----------------------------------------------------------------------

Squirrel example

CLASS FIG webpix/grey_squirrel.jpg

----------------------------------------------------------------------

QSLIDE Gray squirrel population example

INPUT Life_tables/sq.empty.tab.tex

----------------------------------------------------------------------

Squirrel observations

	POLL  Do you notice anything strange? | What is 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 first year,
		second year, and ``adult" -- not much harder.

			ANS This is what we call stage structure

----------------------------------------------------------------------

ASLIDE Gray squirrel population example

INPUT Life_tables/sq.tab.tex

----------------------------------------------------------------------

Gray squirrel dynamics

FIG structure/squirrels.Rout-0.pdf

----------------------------------------------------------------------

PSLIDE Gray squirrel population dynamics

FIG structure/squirrels.Rout-1.pdf

----------------------------------------------------------------------

PSLIDE Gray squirrel age dynamics

FIG structure/squirrels.Rout-2.pdf

----------------------------------------------------------------------

Gray squirrel dynamics

DOUBLEFIG structure/squirrels.Rout-2.pdf structure/squirrels.Rout-1.pdf

----------------------------------------------------------------------

REPSLIDE The structured model

WIDEFIG my_images/structure_cc.png

----------------------------------------------------------------------

ONSLIDE Salmon example

	What happens when a population has independent cohorts?

		Does not necessarily converge to a SAD

----------------------------------------------------------------------

ONCOMM QSLIDE Salmon example

INPUT Life_tables/salmon.empty.tab.tex

----------------------------------------------------------------------

ONCOMM ASLIDE Salmon example

INPUT Life_tables/salmon.tab.tex

----------------------------------------------------------------------

ONCOMM PSLIDE Salmon example

WIDEFIG webpix/salmon.jpg

----------------------------------------------------------------------

ONCOMM PSLIDE Salmon dynamics

FIG structure/salmon.Rout-0.pdf

----------------------------------------------------------------------

ONCOMM PSLIDE Salmon population dynamics

FIG structure/salmon.Rout-1.pdf

----------------------------------------------------------------------

ONCOMM PSLIDE Salmon age dynamics

FIG structure/salmon.Rout-2.pdf

----------------------------------------------------------------------

ONSLIDE 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  How would I calculate one from the other? 

		ANS How did we calculate $f$ in the dandelion example?

		ANS To get $f_x$ we multiply $m_x$ by one or more survival terms,
		depending on when the census is

		ANS $f_x$ needs to close the loop from one census to the next

----------------------------------------------------------------------

When do we start counting?

	Is the first age class called 0, or 1?

		In this course, we will start from age class 1

		How many times have we seen you (not, how old are you)?

		If we count right {\em after} reproduction, this means we are
		calling newborns age class 1. 

		If we count right {\em before} reproduction

			ANS Age class 1 is about one year old the first time we see them

----------------------------------------------------------------------

REPSLIDE 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?

----------------------------------------------------------------------

ASLIDE Dandelion life table

INPUT Life_tables/pre.tab.tex

----------------------------------------------------------------------

ASLIDE Counting \emph{after} reproduction

INPUT Life_tables/post.tab.tex

COMMENT There are two different approaches to the third age class: if we assume that we count the two-year old adults ($x=3$), we can write $p_2 = 0.5$; $f_3=0$, and get the same answer (with one extra row that has zero contribution).

----------------------------------------------------------------------

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?

		How does \R\ respond to changes in various $p_x$ and $f_x$?

----------------------------------------------------------------------

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  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 many tree populations, most individuals don't survive to get huge,
		but the huge trees may have most of the total reproduction

		ANS In many bird populations, old birds produce well, but not
		enough to outweigh the low probability of surviving to get old.

----------------------------------------------------------------------

CSLIDE Old individuals

BC

SIDEFIG webpix/big_tree.jpg

NC

HIDEFIG webpix/stork.jpg

EC

----------------------------------------------------------------------

SS Measuring growth rates

----------------------------------------------------------------------

SHORTCOMMENT Calculating $\bar\lambda$

	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 $\bar\lambda$ bigger (or smaller) than the previous one

		$\bar\lambda$ is the finite rate of increase, like before

	Looking back in time, the cohort $x$ years ago is $\bar\lambda^{-x}$ as
	large as the current one

	The existing cohorts need to make the next one:

		$\sum_x \ell_x f_x \bar\lambda^{-x} = 1$

----------------------------------------------------------------------

ONSLIDE The Euler equation

	If the life table doesn't change, then $\bar\lambda$ is given by $\sum_x
	\ell_x f_x \bar\lambda^{-x} = 1$

	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?

	Example from spreadsheet

----------------------------------------------------------------------

DEFHEAD $\bar\lambda$ and \R

	If the life table doesn't change, then $\bar\lambda$ is given by $\sum_x
	\ell_x f_x \bar\lambda^{-x} = 1$

		What's the relationship between $\bar\lambda$ and \R?

	When $\bar\lambda=1$, the left hand side is just \R.

		If $\R>1$, the population more than replaces itself when
		$\bar\lambda=1$.  We must make $\bar\lambda>1$ to decrease LHS and
		balance.

		If $\R<1$, the population fails to replace itself when
		$\bar\lambda=1$.  We must make $\bar\lambda<1$ to increase LHS and
		balance.

	So \R\ and $\bar\lambda$ tell the same story about whether the population
	is increasing

----------------------------------------------------------------------

Time scales

	$\bar\lambda$ gives the number of individuals per individual \emph{every
	year}

		In the long term

	$\R$ gives the number of individuals per individual \emph{over a
	lifetime}

	POLL  What relationship do we expect for an annual population (life span = census interval)? |  What relationship do we expect for an annual population? R>lam; R<lam; R=lam; R between lam and 1; lam between R and 1.

		ANS $\R=\lambda$; each organism observed reproduces \R\ offspring
		on average, all in one time step

	POLL  For a longer-lived population? |  What relationship do we expect for a longer-lived population? R>lam; R<lam; R=lam; R between lam and 1; lam between R and 1.

		ANS The \R\ offspring are produced slowly, so population changes
		slowly

			ANS $\bar\lambda$ should be closer to 1 than \R\ is.

			ANS But on the same side (same answer about whether population
			is growing)

----------------------------------------------------------------------

Studying population growth

	$\bar\lambda$ and \R\ give related information about your population

	\R\ is easier to calculate, and more generally useful

	But $\bar\lambda$ 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

----------------------------------------------------------------------

Growth and decline

	If we think a particular period of growth or decline is important,
	we might want to study how factors affect $\bar\lambda$

		Complicated, but well-developed, theory

		In a growing population, what happens early in life is more
		important to $\bar\lambda$ than to \R.

		In a declining population, what happens late in life is more
		important to $\bar\lambda$ than to \R.

	POLL Which phase is likely to be more important to ecology and evolution?

		ANS The two phases (growth and decline) will be roughly balanced

		ANS Because otherwise the population would go to zero or infinity

----------------------------------------------------------------------

SEC Life-table patterns

----------------------------------------------------------------------

SS Survivorship

----------------------------------------------------------------------

Patterns of survivorship

	POLL  What sort of patterns do you expect to see in $p_x$? | What sort of patterns do you expect to see in survival probabilities?

		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

----------------------------------------------------------------------

Starting off

	Recall: we always start from age \emph{class} 1

		If we count newborns, we still call them class 1.  

	POLL  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

----------------------------------------------------------------------

PSLIDE Constant survivorship

FIG age/const.Rout-0.pdf

----------------------------------------------------------------------

PSLIDE Constant survivorship

FIG age/const.Rout-2.pdf

----------------------------------------------------------------------

PSLIDE Constant survivorship

FIG age/const.Rout-1.pdf

----------------------------------------------------------------------

Constant survivorship

DOUBLEFIG age/const.Rout-0.pdf age/const.Rout-1.pdf

----------------------------------------------------------------------

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
	less mortality between

----------------------------------------------------------------------

PSLIDE Changing survivorship

FIG age/arch.Rout-0.pdf

----------------------------------------------------------------------

PSLIDE Changing survivorship

FIG age/arch.Rout-1.pdf

----------------------------------------------------------------------

Changing survivorship

DOUBLEFIG age/arch.Rout-0.pdf age/arch.Rout-1.pdf

----------------------------------------------------------------------

TSS Fecundity

	Just as in our simple population growth models, we define fecundity
	as the expected number of offspring at the next census produced by
	an individual observed at this census

		Parent must survive from counting to reproduction

		Parent must give birth

		Offspring must survive from birth to counting

	Remember to think clearly about sex when necessary

		Are we tracking females, or everyone?

----------------------------------------------------------------------

Fecundity patterns

	$f_x$ is the average number of new individuals \emph{counted} next
	census per individual in age class $x$ \emph{counted} this census

	Fecundity often goes up early in life and then remains constant

		ANS Most birds, many large mammals

	It may also go up and then come down

		ANS people

	It may also go up and up as organisms get older

		ANS Many fish, many trees

----------------------------------------------------------------------

SHORTSLIDE Age distributions

	Not covered this year

	HLINK http://www.gapminder.org/population/tool/

	HLINK https://en.wikipedia.org/wiki/Population_pyramid

----------------------------------------------------------------------

SEC Age distributions

----------------------------------------------------------------------

LONGSLIDE Age distributions

	HLINK http://www.gapminder.org/population/tool/

	HLINK https://en.wikipedia.org/wiki/Population_pyramid

----------------------------------------------------------------------

LONGSLIDE CSLIDE Age distributions

WIDEFIG webpix/israelpop.png

----------------------------------------------------------------------

LONGSLIDE CSLIDE Age distributions

WIDEFIG webpix/cypruspop.png

----------------------------------------------------------------------

LONGSLIDE CSLIDE Age distributions

WIDEFIG webpix/iranpop.png

----------------------------------------------------------------------

ONCOMM CSLIDE Age distributions

WIDEFIG webpix/canadapop.jpg

----------------------------------------------------------------------

LONGSLIDE CSLIDE Age distributions

WIDEFIG webpix/bahrainpop.jpg

----------------------------------------------------------------------

LONGSLIDE CSLIDE Age distributions

WIDEFIG webpix/cambodiapop.png

----------------------------------------------------------------------

ONSLIDE Learning from the model

	If a population has constant size (ie., the same number of
	individuals are born every year), what determines the proportion of
	individuals in each age class?

		ANS Distribution should be proportional to $\ell_x$

	What if population is growing?

		ANS We expect proportionally more individuals in younger age
		classes

			ANS Number of births in more recent cohorts is larger

----------------------------------------------------------------------

ONSLIDE Stable age distribution

	If a population has reached a SAD, and is increasing at rate
	$\lambda$ (given by the Euler equation):

		the $x$ year old cohort, born $x$ years ago originally had a
		size $\lambda^{-x}$ relative to the current one

		a proportion $\ell_x$ of this cohort has survived

		thus, the relative size of cohort $x$ is $\lambda^{-x} \ell_x$

		SAD depends only on survival distribution $\ell_x$ and $\lambda$.

----------------------------------------------------------------------

ONSLIDE Patterns

	Populations tend to be bottom-heavy (more individuals at lower age
	classes)

		Many individuals born, few survive to older age classes

	If population is growing, this increases the lower classes further

		More individuals born more recently

	If population is \emph{declining}, this shifts the age
	distribution in the opposite direction 

		Results can be complicated

		Declining populations may be bottom-heavy, top-heavy or just
		jumbled

----------------------------------------------------------------------

ONSLIDE University cohorts

	McMaster accepts only first-year students.  For any given stage
	(e.g., end of third year) the same proportion drop out each year

	What can you say about the relative size of the classes if:

		The same number of students is admitted each year?

			ANS The lower classes are larger

		POLL  More students are admitted each year? |  If more students are admitted each year, lower classes will be: larger, smaller, it depends.

			ANS The lower classes are larger (even more so)

		POLL  Fewer students are admitted each year? |  If fewer students are admitted each year, lower classes will be: larger, smaller, it depends.

			ANS Anything could happen (drop outs and size change are
			operating in different directions)

----------------------------------------------------------------------

ONCOMM REPSLIDE Changing survivorship

DOUBLEFIG age/arch.Rout-0.pdf age/arch.Rout-1.pdf

----------------------------------------------------------------------

ONSLIDE Age distributions

FIG age/arch_sad.Rout-0.pdf

----------------------------------------------------------------------

ONSLIDE Age distributions

FIG age/arch_sad.Rout-2.pdf

----------------------------------------------------------------------

ONSLIDE Age distributions

FIG age/arch_sad.Rout-1.pdf

----------------------------------------------------------------------

ONCOMM REPSLIDE The Euler equation

	If the life table doesn't change, then $\bar\lambda$ is given by $\sum_x
	\ell_x f_x \bar\lambda^{-x} = 1$

	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?

----------------------------------------------------------------------

ONCOMM CSLIDE Age distributions

WIDEFIG webpix/canadapop.jpg

----------------------------------------------------------------------

SEC Other structured models

----------------------------------------------------------------------

Forest example

BC

	Forests have obvious population structure

	They also seem to remain stable for long periods of time

	Populations are presumably \emph{regulated} at some time scale

NC

SIDEFIG webpix/larches.jpg

EC

----------------------------------------------------------------------

Forest size classes

	When we go to an apparently stable forest ecosystem, it seems to be
	dominated by large trees, not small ones.  What up?

	POLL  How is it possible that these systems are dominated by large trees?

		ANS Large trees are larger

		ANS Population may be declining

		ANS Trees may spend longer in some size classes than in others

		ANS Life table may not be constant (smaller trees may recruit at
		certain times and places)

----------------------------------------------------------------------

TSS Stage structure

	Stage structure works just like age structure, except that what
	stage you are in is not strictly predicted by how old you are

		Age-structured models need fecundity, and survival probability

		ANS In stage-structured models survival is typically broken into:

			ANS Survival into same stage

			ANS Survival with recruitment (ie., to the next larger class
			of individuals)

		More complicated models are also possible

----------------------------------------------------------------------

Advantages

	Stage structured models don't need a maximum age

	Nor one box for every single age class

----------------------------------------------------------------------

Unregulated growth

	What happens if you have a constant stage table (no regulation)?

		Fecundity, and survival and recruitment probabilities are constant

	Similar to constant life table

		Can calculate \R\ and $\bar\lambda$ (will be consistent with each
		other)

		Can calculate a stable stage distribution

	Unregulated growth cannot persist

----------------------------------------------------------------------

Summary

	If the life table remains constant (no regulation or stochasticity):

		Reach a stable age (or stage) distribution

		Grow or decline with a constant $\lambda$

		Factors behind age distribution can be understood

----------------------------------------------------------------------

TSS Regulated growth

	Our models in this unit assume that individuals are independent

	In this case, we expect populations to grow (or decline)
	exponentially

	We do not expect that the long-term average value of \R\ or
	$\bar\lambda$ will be exactly 1.

		ANS So we need regulation in real life

----------------------------------------------------------------------

The value of simple models

	There is a lot of mathematical study of unregulated, age-structured
	populations, but it should be taken with a grain of salt

		ANS We know that real populations are regulated

		ANS Populations can't increase or decrease exponentially for very
		long

	Understanding this behaviour is helpful:

		interpreting age structures in real populations

		beginning to understand more complicated systems

----------------------------------------------------------------------

Regulation and structure

	We expect real populations to be regulated

	Under regulation, the value of $\bar\lambda$ under regulation
	\emph{could} be exactly 1

	There is also likely to be substantial variation from year to year,
	due to changing conditions and other random-seeming forces

----------------------------------------------------------------------

Dynamics

	We expect to see smooth behaviour in many cases

	Cycles and complex behaviour should arise for reasons similar to our
	unstructured models: 

		Delays in the system

		Strong population response to density

	Age distribution will be determined by:

		$\ell_x$, and

		whether the population has been growing or declining recently