structure_prelim.txt

UNIT 3: Structured populations

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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
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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

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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?

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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

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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

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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?

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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

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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

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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

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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

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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

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SSLIDE Complexity

FIG images/bifurcation.png

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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

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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

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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.

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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

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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)

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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: $f_x$

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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)]

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ASLIDE The structured model

WIDEFIG images/structure_cc.png

CHANGE Put this in the goddam lecture notes, morph!:e

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SS Model dynamics

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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

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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

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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)

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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

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