linear.txt

UNIT 2: Linear population models

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

SEC Constructing models

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

TSS Dynamical models

BC

Tools to link scales

	Models are what we use to link:

		Individual-level to population-level processes

		Short time scales to long time scales

	In both directions

NC

SIDEFIG webpix/touching.jpg

SIDEFIG webpix/ew_measles.png

EC

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

Assumptions

BC

	Models are always simplifications of reality

		``The map is not the territory''

		``All models are wrong, but some are useful''

	Models are useful for:

		linking assumptions to outcomes

		identifying where assumptions are broken

NC

SIDEFIG webpix/flat.png

EC

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

Dynamical models

	\textbf{Dynamical models} describe rules for how a system changes at
	each point in time

	We will see what these assumptions about how the system
	\emph{changes} lead to conclusions about what the system \emph{does}
	over longer time periods

	ADD should be “how these”

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

States and state variables

	Our dynamic models imagine that a system has a \textbf{state} at any
	given time, described by one or more \textbf{state variables}

	These are the things that follow our rules and change

	Examples:

		Dandelions: state is population size, described by one state
		variable (the number of individuals)

		Bacteria: state is population density, described by one state
		variable (the number of individuals per ml)

		Pine trees: state is amount of wood, described by one state
		variable (tons per hectare)

	Limiting the number of state variables is key to simple models

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

Parameters

	\textbf{Parameters} are the quantities that describe how the
	rules for our system work

	Examples:

		Birth rate, death rate, fecundity, survival probability

	Typically \emph{remain constant} while we are simulating a particular
	scenario

	\emph{Vary} when we compare different scenarios

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

How do populations change?

	I survey a population of elk in northern Ontario in 2019, and again in 2023.
	I get a different answer the second time.

	POLL  What are some reasons why this answer might change?

		ANS Birth

		ANS Death

		ANS Immigration and emigration

		ANS Sampling (ie., my counts are not perfectly correct)

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

Censusing and intermediate variables

	Often, our population models will imagine that the population is
	\textbf{censused} (counted) at particular periods of time

	Calculations of what happens between census times may be part of how
	we make our population model, without showing up in the main model
	itself

		For example, our moth and dandelion examples

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

Linear population models

	We will focus mostly on births and deaths

	Births and deaths are done by individuals

		We model the rate of each individual (per capita rates)

		Total rate is the per capita rate multiplied by population size

	If per capita rates are constant, we say that our population
	\emph{models} are \textbf{linear}

		Linear models do not usually correspond to linear growth!

		What behaviour do we expect from a linear model?

			ANS They usually correspond to exponential growth

			ANS \ldots or exponential decline

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

SS Examples

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

REPSLIDE Gypsy moths

BC

	A pest species that feeds on deciduous trees

	Introduced to N. America from Europe ~ 150 years ago

	Capable of wide-scale defoliation

NC

SIDEFIG webpix/gm_caterpillar.jpg

SIDEFIG webpix/gm_defoliation.jpg

EC

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

REPSLIDE Gypsy moth populations

DOUBLEPDF ts/gm10165.simple.Rout

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

Moth example

BC

	POLL State variable | What might be a state variable in this system?

		ANS Number of moths/ha

	Parameters 

		ANS Number of eggs
		
		ANS sex ratio
		
		ANS larval survival, pupal survival, adult survival

		ANS Time step

	Census time

		ANS Annually; use the same time (and stage) each year

NC

SIDEFIG webpix/cool_moth.jpg

EC


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

Bacteria

BC

	State variables

		ANS Number of bacteria/ml

	POLL Parameters | What might be a parameter in this system?

		ANS Division rate, death rate, washout rate

	Census time

		ANS Always!

NC

SIDEFIG webpix/violet_bacteria.jpg

EC

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

Dandelions

BC

	State variables

		ANS Number of dandelions in a field

		POLL Are there intermediate variables?

			ANS Number of seeds

	Parameters 

		ANS Seed production, survival to adulthood, adult survival

	Census time

		ANS Annually, before reproduction

		ANS When new and returning individuals are most similar

NC

SIDEFIG webpix/dandy_flower.jpg

SIDEFIG webpix/dandy_seeds.jpg

EC

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

SS A simple discrete-time model

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

Assumptions

	If we have $N$ individuals after $T$ time steps, what determines how
	many individuals we have after $T+1$ time steps?

		A fixed proportion $p$ of the population (on
		average) survives to be counted at time step $T+1$

		Each individual creates (on average) $f$ new individuals
		that will be counted at time step $T+1$

	How many individuals do we expect in the next time step?

		ANS $N_{T+1} = (pN_T+fN_T) = (p+f)N_T$

	Diagram

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

CONT Assumptions

BC

	Individuals are \textbf{independent}: what I do does not depend on
	how many other individuals are around

	The population is censused at regular time intervals $\Delta
	t$

		Usually $\Delta t$ = 1\yr

	All individuals are the same at the time of census

	Population changes deterministically

NC

SIDEFIG webpix/flat.png

EC

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

Definitions

	$p$ is the \textbf{survival probability}

	$f$ is the \textbf{fecundity}

	$\lambda \equiv p + f$ is the \textbf{finite rate of increase}

		... associated with the time step $\Delta t$

		($\Delta t$ has units of time)

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

Model

	Dynamics: 

		$N_{T+1} = \lambda N_T$ 

		$t_{T+1} = t_T + \Delta t$

	Solution: 

		$N_T = N_0 \lambda^T$

		$t_T = T \Delta t$

	POLL  How does $N$ behave in this model? | How does N behave in this model?

		ANS Increases exponentially (geometrically) when $\lambda>1$

		ANS Decreases exponentially when $\lambda<1$

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

PSLIDE Example

BC

SIDEFIG webpix/dandy_field.jpg

SIDEFIG webpix/spreadsheet.png

NC

	HREF http://tinyurl.com/DandelionModel2020 Spreadsheet (see resource page)

EC

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

Interpretation

	Assumptions are simplifications based on reality

	We can understand why populations change exponentially sometimes

	We can look for \emph{reasons} when they don't

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

Examples

BC

	Moths

		$p=0$, so $\lambda=f$. 

			Moths are \textbf{semelparous} (reproduce once); they have
			an \textbf{annual} population

	Dandelions

		If $p>0$, then the dandelions are \textbf{iteroparous}; they are
		a \textbf{perennial} population

NC

SIDEFIG webpix/cool_moth.jpg

SIDEFIG webpix/dandy_flower.jpg

EC

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

SS A simple continuous-time model

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

Assumptions

	If we have $N$ individuals at time $t$, how does the population
	change?

		Individuals are giving birth at per-capita rate $b$

		Individuals are dying at per-capita rate $d$

	How we describe the population dynamics?

		ANS $\ds \frac{dN}{dt} = (b-d)N $

		ANS That's what calculus is \emph{for} -- describing
		instantaneous rates of change

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

CONT Assumptions

	Individuals are \textbf{independent}: what I do does not depend on
	how many other individuals are around

	The population can be censused at any time

	Population size changes continuously

	All individuals are the same all the time

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


Definitions

	$b$ is the \textbf{birth rate}

	$d$ is the \textbf{death rate}

	$r \equiv b-d$ is the {\bf instantaneous rate of increase}.

	These quantities have true units:

		ANS 1/[time]

			ANS $\equiv$ (indiv/[time])/indiv

	COMMENT With units, we don't need to mess with “associated with a time
	period”

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

Model

	Dynamics: 

		$\ds \frac{dN}{dt} = rN $

	Solution: 

		$N(t) = N_0 \exp(rt)$

	Behaviour

		ANS Increases exponentially when $r>0$

		ANS Decreases exponentially when $r<0$

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

Bacteria

	Conceptually, this is just as simple as the dandelions or the moths

		In fact, simpler

	On the computer, it's a little more complicated to simulate

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

CONT Bacteria

DOUBLEFIG exponential/growth.Rout-0.pdf exponential/growth.Rout-1.pdf

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

Summary

	We can construct simple, conceptual models and make them into
	dynamic models

	If we assume that \emph{individuals} behave independently, then

		we expect \emph{populations} to grow (or decline) exponentially

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

SEC Units and scaling

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

Units are our friends

BC

	Keep track of units at all times

	Use units to confirm that your answers make sense

		Or to find quick ways of getting the answer

	What is $3\dd \cdot 4 \esp/\dd$?

		ANS $12\esp$

	What is $1\hr \cdot 0.2\cm/\dd$?

		ANS $1\hr \cdot 0.2\cm/\dd$

		ANS $1\hr \cdot 0.2\cm/\dd \cdot \frac{1\dd}{24\hr}$

		ANS 0.0083\cm

NC

SIDEFIG webpix/espresso.jpg

EC

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

Manipulating units

BC

	We can {multiply} quantities with different units by keeping track
	of the units

	We \emph{cannot} {add} quantities with different units (unless they
	can be converted to the same units)

	POLL  How many seconds are there in a day?

		ANS
		$\bigeq \frac{60 \uname{sec}}{\uname{min}} $ 
		$\bigeq  \cdot \frac{60 \uname{min}}{\uname{hr}} $ 
		$\bigeq \cdot \frac{24\uname{hr}}{\uname{day}}$ 

		ANS 86400 \uname{sec}/\uname{day}

	NOTES http://www.alysion.org/dimensional/fun.htm

NC

SIDEFIG webpix/clock.jpg

EC

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

Scaling

	Quantities with units set scales, which can be changed

		If I multiply all the quantities with units of time in my model
		by 10, I should get an answer that looks the same, but
		with a different time scale

		If a multiply all the quantities with units of dandelions in my
		model by 10, I should get an answer that looks the same, but with
		a different number of dandelions

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

PSLIDE Scaling time in bacteria

DOUBLEFIG exponential/growth.Rout-0.pdf exponential/growth.Rout-1.pdf

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

Scaling time in bacteria

DOUBLEFIG exponential/growth.Rout-0.pdf exponential/growth.Rout-2.pdf

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

ASLIDE Scaling population

DOUBLEFIG exponential/growth.Rout-0.pdf exponential/growth.Rout-3.pdf

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

ASLIDE Scaling population

DOUBLEFIG exponential/growth.Rout-0.pdf exponential/growth.Rout-4.pdf

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


Thinking about units 

	POLL  What is $10^{3 \dd}$? | What is 10 to the power of 3d?

		NOANS

	What is $10^{72\hr}$?

		ANS Nonsense! $72\hr$ means \emph{exactly} the same thing as
		$3\dd$ -- there is no way to resolve this to make sense.

	What is $3\dd \cdot 3\dd$?

		ANS $9\dd^2$ -- this \emph{could} make sense

			ANS but you probably asked the wrong question

		ANS \ldots very different from $9\dd$.

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

Unit-ed quantities

	Quantities with units \emph{scale}

		If you change everything with the same units by the same factor,
		you should not change the behaviour of your system

	We typically make sense of quantities with units by comparing them
	to other quantities with the same units, e.g.: 

		birth rate vs.\ death rate

		characteristic time of exponential growth vs.\ observation time

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

Unitless quantities

	Quantities in exponents must be unitless

	Quantities with variable exponents (quantities that can be
	multiplied by themselves over and over) must be unitless

	Quantities that determine \emph{how} a system behaves must have a
	unitless form

		Otherwise, they could be scaled

		Zero works as a unitless quantity:

			0km = 0cm

	What unitless quantities have we already talked about?

		ANS $\lambda$, $f$ and $p$. 

		ANS These all depend on a time period

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

REPSLIDE Moth calculation

	Researchers studying a gypsy moth population make the following
	estimates:

		The average reproductive female lays 600 eggs

		10% of eggs hatch into larvae

		10% of larvae mature into pupae

		50% of pupae mature into adults

		50% of adults survive to reproduce

		All adults die after reproduction

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

Moths

	$600 \uname{egg}/\uname{rF}$

	$\cdot 0.1 \uname{larva}/\uname{egg}$

	$\cdot 0.1 \uname{pupa}/\uname{larva}$

	$\cdot 0.5 \uname{A}/\uname{pupa}$

	$\cdot 0.5 \uname{rA}/\uname{A}$

	POLL  What's the product?

		ANS $1.5\uname{rA}/\uname{rF}$

		ANS Not enough information to make a prediction!

		ANS Need to multiply by something with units rF/rA to close the
		loop

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

Closing the loop

	Once we close the loop, it doesn't matter where we start:

		Reproductive adults to reproductive adults

		Larvae to larvae

		Pupae to pupae is common in real studies

			ANS Pupae are easy to count

			ANS Egg masses, too (depending on species)

	If we don't close the loop, we can't correctly move from step to step

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

Calculating $\lambda$

	$\lambda \equiv p + f$ is the \textbf{finite rate of increase}

	If $N_{T+1} = \lambda N_T$, what are the units of $\lambda$?

		ANS We multiply by $\lambda$ over and over

		ANS Therefore $\lambda$ must be unitless

	Therefore $p$ and $f$ must be unitless

		example, rA/rA; seed/seed

		to do it right, we close the loop

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

SEC Key parameters

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

TSS Discrete-time model

	$N_{T+1} = \lambda N_T$ 

	$\lambda \equiv p + f$

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

Calculating fecundity

	Fecundity $f$ in our model must be unitless

	Multiply:

		Probability of surviving from census to reproduction

		Expected number of offspring when reproducing (maternity)

		Probability of offspring surviving to census

	Need to end where we started

	Diagram

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

Calculating survival

	Survival $p$ must be unitless

	Multiply:

		Probability of surviving from census to reproduction

		Probability of surviving the reproduction period

		Probability of surviving until the next census

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

Finite rate of increase

	Population increases when $\lambda>1$

	So $\lambda$ must be unitless

	But it is \emph{associated with} the time step $\Delta t$

		Potentially confusing. It is often better to
		use $\R$ or $r$ (see below).

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

Reproductive number 

	The reproductive number \R\ measures the average number of offspring
	produced by a single individual over the course of its lifetime

	POLL  The population will increase when \R\ldots:| The population will increase when R … .

		ANS $\R>1$

	POLL  What are the units of \R?  |What are the units of R?

		ANS \R\ must be unitless

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

Lifespan

	In this model world, how long do individuals live, on average?

	If $p$ is the proportion of individuals that survive, then the
	proportion that die is:

		ANS $\mu = 1-p$

	How many time steps do you expect to survive, on average?  

		ANS $1/\mu$ 

			ANS Roughly makes sense, and is also right (trust me)

			ANS \ldots\ or ask me

		ANS Average lifetime is $1/\mu * \Delta t$

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

Calculating \R

	\R\ is fecundity multiplied by lifespan

	$\R = f/\mu = f/(1-p)$

	Both $f$ and $1/(1-p)$ are unitless (but associated with the time step)!

		ANS Offspring per time steps

		ANS Life span in time steps

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

Comparison

BCC

\emph{Lifetime reproduction}

	$\R = f/\mu = f/(1-p)$

	Unitless

	Population behaviour depends on the \textbf{comparison} $\R:1$

		Equivalent to $f:\mu$

NCC

\emph{Reproduction over one time step}

	$\lambda = f+p = f+(1-\mu)$

	Unitless

	Population behaviour depends on the comparison $\lambda:1$

		Equivalent to $f:\mu$

EC

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

Is the population increasing?

	What does $\lambda$ tell us about whether the population is
	increasing?

		ANS Population is increasing each time step when $\lambda>1$

	What does $\R$ tell us about whether the population is increasing?

		ANS Population is increasing when $\R>1$.  Each individual is (on
		average) more than replacing itself over its lifetime

	Therefore, these two criteria must be the same!

		ANS Both come down to $f>\mu$.  

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

SS Continuous-time model

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

Calculating birth rate

	The birth rate $b$ in the continuous-time model is new individuals
	per individual per unit time

		An instaneous rate

		Units of [1/time] -- implies what assumption?

			ANS New individuals are cancelling with old individuals in the equation

			ANS New individuals are being treated the same as old individuals

			ANS Not very realistic -- a potential problem with our model world

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

Calculating death rate

	The death rate $d$ in the continuous-time model is deaths per
	individual per unit time

		An instaneous rate

		Units of [1/time]

	What happens to the individuals?

		ANS The units cancel: individuals show up in the deaths part and in the
		population part.

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

Instaneous rate of increase

	Population increases when $r=b-d > 0$

	$r$ is not unitless, units are:

		ANS [1/time]

	So how can $r=0$ be a criterion? | How can r=0 be a criterion?

		ANS Because 0$\times$anything is unitless!

		ANS 0km = 0cm!

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

Calculating \R

	The mean lifespan is $L = 1/d$

		Equivalent to the characteristic time for the death process

	\R\ is the average number of births expected over that time frame:

		$\R = bL = b/d$

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

Comparison

BCC

\emph{Lifetime reproduction}

	$\R = bL = b/d$

	Unitless

	Population behaviour depends on the comparison $\R:1$

		Equivalent to $b:d$

NCC

\emph{Instantaneous change}

	$r = b-d $

	Units [1/t] (a rate)

	Population behaviour depends on the comparison $r:0$

		Equivalent to $b:d$

EC

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

Is the population increasing?

	What does $r$ tell us about whether the population is
	increasing?

		ANS Population is increasing at any instant in time when
		$r>0$

	What does $\R$ tell us about whether the population is increasing?

		ANS Population is increasing when $\R>1$.  Each individual is (on
		average) more than replacing itself over its lifetime

	Therefore, these two criteria must be the same!

		ANS Both come down to $b>d$.  

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

TSS Links

	After one time step in a discrete-time model

		$N_0 \rightarrow N_0 \lambda$

		$t \rightarrow t + \Delta t$

	In a continuous model

		$N_0 \rightarrow N_0 \exp(r\Delta t)$ in the same time period

	To link them, we
	set:

		$\lambda = \exp(r\Delta t)$

	In the other direction:

		ANS $r = \log_e(\lambda)/\Delta t$

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

Characteristic time

	We can now find characteristic times of exponential change:

		$T_c = 1/r$ for exponential growth when $r>0$

		$T_c = -1/r$ for exponential decline when $r<0$

	Rule of thumb: population changes by a factor of 20 after 3
	characteristic times

		$\exp(3) = 20.1$

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

SEC Growth and regulation

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

PRESLIDE Long-term growth rate

BC

	What is the long-term average exponential growth rate (using either
	$r$ or $\lambda$) of:

		A successful population?

			NOANS

		An unsuccessful population?

			NOANS

NC

SIDEFIG webpix/humans.jpg

SIDEFIG webpix/wolves.jpg

EC

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

Example: Human population growth

	In the last 50,000 years, the population of \textbf{modern humans} has
	increased from about 1000 to about 7 billion

	What value of $r$ does this correspond to?  If we use a time step of
	20-year generations, what value of $\lambda$ does it correspond to?

		ANS $N(t) = N(0) \exp(rt)$

			ANS $r = \log_e(N/N(0))/t$

			ANS $r = \log_e(7000000000/1000)/50000\uname{yr}$ $=
			0.0003/\uname{yr}$

		ANS $N_T = N_0 \lambda^T$

			ANS $T = {t/\Delta t} = 50000\uname{yr}/20\uname{yr} = 2500$

			ANS $\lambda = (N_T/N_0)^{1/T}$

			ANS $\lambda = (7000000000/1000)^{1/2500}$ $= 1.006$

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

Long-term growth rate

	What is the long-term average exponential growth rate (using either
	$r$ or $\lambda$) of:

		A successful population?

			ANS Very close to $r=0$ or $\lambda=1$

			ANS But a little larger

		An unsuccessful population?

			ANS \emph{Probably} very close to $r=0$ or $\lambda=1$ 

			ANS But a little smaller

			ANS If more than a little, it would probably be gone by now!

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

Summary

	We can make simple model worlds where populations are composed of
	individuals that reproduce and die independently

		Discrete or continuous time

	We can do structured closed-loop calculations and predict how these
	populations will change

	If individuals are independent, we expect populations to change
	exponentially through time

		ANS The rate at which the population changes is proportional to the size
		of the population