nonlinear.txt

EXTRA

Remember to look for OFFSLIDE/ONSLIDE/OFF/ON

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

EXTRA Update with information about 2026 student-contributed invasions

Common carp invading Cootes paradise

lampreys from Europe in Great Lakes

youtuber and frog eggs

brown stink bugs

starlings in New York

	2024 update was very cool HREF https://avenue.cllmcmaster.ca/d2l/le/news/595825/849579/view?ou=595825 See examples posted by Julianna

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UNIT 3 Non-linear population models

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

EXTRA

ADD an Ebola or Covid example

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

TSEC Introduction

	In \textbf{linear} population models, per capita rates are independent of
	population size

	In \textbf{non-linear} models, not so much

	Why might per capita birth and death rates change with population size?

	What does this imply about population \textbf{dynamics}?

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

The first law of population dynamics

	If individuals are behaving independently:

		the population-level rate of growth (or decline) is proportional
		to the population size

		the population grows (or declines) exponentially

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

The second law of population dynamics

	Exponential growth (or decline) cannot continue forever

		Size changes would be too extreme:

		$>20000 \times$ after 10 characteristic times

	Something is changing the average rate at which populations we
	observe grow

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

PICSLIDE

FIG webpix/ecoli.jpg

REMARK Often something to do with crowding

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

The third law of population dynamics

	Exponential growth (or decline) cannot continue forever --
	\emph{even on average}

	Environmental variation cannot be the only thing that changes growth
	rates

	Populations are, directly or indirectly, limiting their own growth
	rates

	This is called \textbf{density dependence}

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

Long-term growth rates

	Populations maintain long-term growth rates very close to $r=0$

	This is not possible just by chance, or averaging:

		Population size must affect its own growth rate

		Directly or indirectly

		Long term or short term

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

Changing growth rates

	POLL What is an example of a density-dependent mechanism that
	affects growth rate?

		ANS Predators

			ANS Sometimes. Only if predator number can increase fast enough. 

		ANS Diseases

			ANS Usually can increase fast enough

		ANS Insufficient resources

			ANS Limitation: e.g., oak trees use all the available light

			ANS Destruction: gypsy moths kill all the oak trees

		ANS Not everything that harms the species is density dependent!

			ANS Seasonal and climatic fluctuations don't \emph{regulate}

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

Population regulation

	All the populations we see are \emph{regulated}

		On average, population growth is higher when the population is
		lower

		Maybe with a time delay

	Why is this interesting?

		Lots of populations don't \emph{look like} they are regulated

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

Sometimes regulation is apparent

BC

	Some species seem to fill a niche (mangroves)

	or deplete their own food resources (gypsy moths)

NC

SIDEFIG webpix/mangroves.jpg

SIDEFIG webpix/gm_defoliation.jpg

EC

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

Sometimes regulation is not apparent

BC

	Other species seem like they could easily be more common (pine trees)

		May be controlled by cryptic (hard to see) natural enemies (like
		disease or parasites)

		May be controlled by limitations that occur only at certain times
		(e.g., during regular droughts)

NC

SIDEFIG webpix/forest.jpg

EC

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

Regulation works over the long term

	Not every species is experiencing population regulation at every
	time

	A species that we see now may be expanding into a niche (e.g.,
	because of climate change)

	Some species are controlled by big outbreaks of disease

	Some species have big outbreaks into marginal habitat, and spend
	most of their time contracting back to their ``core" habitat

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

How do we know it's regulation?

	POLL Why don't we believe that population growth is controlled by factors that don't depend on the population itself?

		ANS Because the long-term average value of $r$ has to be
		\emph{very} close to 0

		ANS This is true for \emph{every} population

		ANS This is unlikely to occur by chance

		ANS Thus, it must be through direct or indirect responses to the
		population size

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

SS Population Examples

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

PRESLIDE

Elk

DOUBLEPDF ts/elk.elkplot.Rout

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

EXTRA

CSLIDE Ebola

DOUBLEFIG WA_Ebola_Outbreak/liberia.npc.tsplot.Rout-0.pdf WA_Ebola_Outbreak/liberia.npc.tsplot.Rout-1.pdf

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

PRESLIDE Paramecia

DOUBLEPDF ts/paramecia.plot.Rout

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

PRESLIDE Coronavirus

FIG dd/coronaCA.ON-0.pdf

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

PRESLIDE Coronavirus

FIG dd/coronaCA.ON-1.pdf

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

PRESLIDE Coronavirus

DOUBLEPDF dd/coronaCA.ON

ADD Picked a cutoff date; talk about the testing regime maybe

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

PRESLIDE Gypsy moths

DOUBLEPDF ts/gm10165.plot.Rout

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

Gypsy moths

	What are some factors that limit gypsy-moth populations?

	Which are likely to be affected by the moths?

		Directly or indirectly, in the short or long term?

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

SEC Continuous-time regulation

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

Build on the linear model

	Our linear population model is:

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

	Per-capita rates are constant

	\textbf{Population-level rates are linear}

	Behaviour is exponential

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

Individual perspective

DOUBLEFIG bd_models/constant.bd.Rout-0.pdf bd_models/constant.bd.Rout-2.pdf

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

CONT Individual perspective

BC

	Per capita rate shows birth and death per individual

	Corresponds to the time plot showing growth on a log scale

		On the log scale we see \emph{multiplicative} or \emph{proportional} change

NC

SIDEFIG bd_models/constant.bd.Rout-0.pdf

SIDEFIG bd_models/constant.bd.Rout-2.pdf

EC

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

Population perspective

DOUBLEFIG bd_models/constant.bd.Rout-1.pdf bd_models/constant.bd.Rout-3.pdf

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

CONT Population perspective

BC

	Total rate shows birth and death for the whole population

	Corresponds to the time plot showing growth on a linear scale

		On the linear scale we see \emph{additive} or \emph{absolute}
		change

NC

SIDEFIG bd_models/constant.bd.Rout-1.pdf

SIDEFIG bd_models/constant.bd.Rout-3.pdf

EC
----------------------------------------------------------------------

Non-linear model

	Population has \emph{per capita} birth rate $b(N)$ and death
	rate $d(N)$

		Per-capita rates change with the population size

	Our non-linear model is: $\bigeq \frac{dN}{dt} = (b(N)-d(N))N$
	$\equiv r(N) N$

		Defines how fast the population is changing at any instant

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

OFFSLIDE Recruitment

	{\bf Recruitment} is when an organism moves from one life stage to
	another:

		Seed \rec seedling \rec sapling \rec tree

		Egg \rec larva \rec pupa \rec moth

	In simple continuous-time population models, recruitment is included
	in birth:

		$b$ is the rate at which adults produce new adults; or seeds
		produce new seeds -- we have to ``close the loop"

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

Birth rates

BC

	When a population is crowded, the birth rate will usually go down

		Resources are limited: space, food, light

	But it may stay the same

	Or even go up

		If individuals shift their resources to reproduction instead of
		survival

NC

SIDEFIG webpix/mussels.jpg

SIDEFIG webpix/fir_wave.jpg

EC
----------------------------------------------------------------------

Death rates

BC

	When a population is crowded, the death rate will often go up

		Individuals are starving, or conflict increases

		But it may stay the same

			if reproduction is limited by competition for breeding sites,
			or by recruitment of juveniles

		Or even go down 

			if organisms go into some sort of ``resting mode"

NC

SIDEFIG webpix/seagulls.jpg

SIDEFIG webpix/spores.png

EC
----------------------------------------------------------------------

Reproductive numbers

	Our model is: $\bigeq \frac{dN}{dt} = (b(N)-d(N))N$
	$\equiv r(N) N$

	Reproductive number now also changes with $N$:

		ANS $\R(N) = b(N)/d(N)$

	When the population is crowded, individuals are stressed and
	the reproductive number will typically go down.

		ANS birth rates go down, death rates go up, or both

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

Carrying capacity

	If a population has $\R(N) > 1$ when it's not crowded

	The population should increase

	Eventually, \R\ will decrease, and eventually cross $\R=1$

	We call the special value of $N$ where $\R(N) = 1$, the
	\textbf{carrying capacity}, $K$

		$\R(K) \equiv 1$

		$b(K) \equiv d(K)$

	When $N=K$:

		ANS Population stays the same, on average

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

Logistic model

	A popular model of density-dependent growth is the logistic model

	Per capita instantaneous growth rate $r$ is a function of $N$

		$r(N) = \rmax(1-N/K)$

		Consistent with various assumptions about $b(N)$ and $d(N)$

	Population increases to $K$ and remans there

		Units of $N$ must match units of $K$

	We don't call this a linear model

		* \emph{population-level} rates are not linear

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

Exponential-rates model

	In this course, we'll mostly use another simple model:

		$b(N) = b_0 \exp(-N/N_b)$

		$d(N) = d_0 \exp(N/N_d)$ 

	This is the simplest model that is smooth and keeps track
	of birth and death rates separately

		Birth rate goes down with characteristic scale $N_b$

		Death rate goes up with characteristic scale $N_d$

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

Exponential-rates vs.\ logistic

	The exponential-rates model is conceptually clearer

		Birth and death rates are clearly defined

	Mathematically nicer

		Rates always stay positive 

	\emph{Looks} a little more scary than the logistic

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

SS A simple, continuous-time model

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

Assumptions

	We model individual-level rates, but individuals are \emph{not}
	independent: my rates depend on the number (or density) of
	individuals in the population

	The population can be censused at any time

	Population size changes continuously

	All individuals are the same all the time

	Population changes deterministically

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

Interpretation

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

		Individuals are giving birth at per-capita rate $b(N)$

		Individuals are dying at per-capita rate $d(N)$

	Population dynamics follow:

		$\bigeq \frac{dN}{dt} = (b(N)-d(N))N \equiv r(N) N$

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

States and state variables

	What variable or variables describe the state of this system?

		ANS The same as before: population size (or density)

		ANS We are still assuming that's all we need to know

			ANS In other words, that all individuals are the same.

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

Parameters

	POLL What quantities describe the rules for this system?

		ANS $b_0$ [1/time]

		ANS $d_0$ [1/time]

		ANS $N_b$ [indiv] (or [indiv/area])

		ANS $N_d$ [indiv] (or [indiv/area])

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

Characteristic \emph{scale}

	A characteristic scale for density dependence is analogous to a
	characteristic time

		It is a \emph{parameter} with the same units as a state variable, and
		sets a standard by which we measure that variable

		Nothing (with units) is big or small until it's compared to something
		with the same units

	For example: $b(N) = b_0 \exp(-N/N_b)$

		$N_b$ is the characteristic scale of density-dependence in birth
		rate

		When $N \ll N_b$, density dependence is linear (and relatively
		small)

		When $N \gg N_b$, density dependence is exponential, and has very
		large effect

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

Model

	Dynamics: 

		$\bigeq \frac{dN}{dt} = (b_0 \exp(-N/N_b) - d_0 \exp(N/N_d))N$

	Exact solution: 

		Insanely complicated

	Behaviour of the solution:

		Pretty easy!

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

Dynamics

	What sort of \textbf{dynamics} do we expect from our conceptual
	model?

		I.e., how will it change through time?

	What will the population do if it starts

		near zero?

		near the equilibrium?

		at a high value?

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

PSLIDE A model

FIG bd_models/birth.bd.Rout-0.pdf 

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

What is the characteristic scale?

BCC

PRESENT CFIG bd_models/birth.bd.Rout-0.pdf 

NCC

	No density dependence in death rate

	Characteristic scale for birth rate is about 70 indiv

EC

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

PSLIDE What will this model do?

FIG bd_models/birth.bd.Rout-0.pdf 

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

PSLIDE What will this model do?

FIG bd_models/birth.bd.Rout-6.pdf 

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

PSLIDE What will this model do?

FIG bd_models/birth.bd.Rout-8.pdf 

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

What will this model do?

BCC

CFIG bd_models/birth.bd.Rout-8.pdf 

NCC

	Increase when population is below equilibrium

	Decrease when population is above equilibrium

	Converge

EC

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

PSLIDE Low starting population example

DOUBLEFIG bd_models/birth.bd.Rout-8.pdf bd_models/birth.bd.Rout-2.pdf

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

PSLIDE High starting population example

DOUBLEFIG bd_models/birth.bd.Rout-8.pdf bd_models/birth.bd.Rout-4.pdf

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

Examples

DOUBLEFIG bd_models/birth.bd.Rout-2.pdf bd_models/birth.bd.Rout-4.pdf

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

SS Simulating model behaviour

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

Simulations

	We will simulate the behaviour of populations in continuous time
	using the program R

	This program generates the pictures in this section by implementing
	our model of how the population changes instantaneously

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

Individual-scale pictures

	We can view graphs of our population assumptions on the
	individual scale

		per-capita birth and death rates

			units [1/time]

		what is each individual doing (on average)?

		corresponds to the dynamics we visualize on a log-scale graph of
		the population

		NOTES See above

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

REPSLIDE What will this model do?

DOUBLEFIG bd_models/birth.bd.Rout-8.pdf bd_models/birth.bd.Rout-2.pdf 

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

Population-scale pictures

	We can view graphs of our population assumptions on the
	population scale

		total birth and death rates

			units [indiv/time]

			or [density/time] = [(indiv/area)/time]

		what is changing at the population level?

		corresponds to the dynamics we visualize on a linear-scale graph
		of the population

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

Population perspective picture

DOUBLEFIG bd_models/birth.bd.Rout-9.pdf bd_models/birth.bd.Rout-3.pdf

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

Decreasing birth rate

BC

	Decreasing birth rate (above) might be a good model for organisms
	that experience density dependence primarily in the recruitment stage

	For example, we might count adult trees, and these might not die
	more at high density -- just fail to recruit younger ones

NC

SIDEFIG webpix/tongass.jpg

EC

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

REPSLIDE Decreasing birth rates

FIG bd_models/birth.bd.Rout-0.pdf

REMARK This is the model we've already seen; we could also imagine …

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

PSLIDE Increasing death rates

FIG bd_models/death.bd.Rout-0.pdf

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

Increasing death rate

BC

	Increasing death rate might be a good model for organisms
	that experience density dependence primarily as adults

	For example, in some environments, mussel density might be regulated
	primarily by adult crowding. 

NC

SIDEFIG webpix/mussels.jpg

EC

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

Individual perspective

DOUBLEFIG bd_models/death.bd.Rout-0.pdf bd_models/death.bd.Rout-2.pdf

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

Population perspective

DOUBLEFIG bd_models/death.bd.Rout-1.pdf bd_models/death.bd.Rout-3.pdf

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

Decreasing death rate

BC

	Some organisms (such as many types of bacteria) slow down their
	metabolisms under density dependence, so that death rate
	\emph{decreases}

	VPOLL How is this consistent with density dependence? How can decreasing death rate at high density be consistent with density dependence?

		ANS Birth rate must decrease even faster

NC

SIDEFIG webpix/spores.png

EC
----------------------------------------------------------------------

Individual perspective

DOUBLEFIG bd_models/slow.bd.Rout-0.pdf bd_models/slow.bd.Rout-2.pdf

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

Population perspective

DOUBLEFIG bd_models/slow.bd.Rout-1.pdf bd_models/slow.bd.Rout-3.pdf

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

Other examples

	There are two other possible scenarios for density dependence

		For fun, you can try to think of what they are

	But all of these examples have similar behaviour

		Increase from low density

		Decrease from high density

		Approach carrying capacity

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

PRESLIDE Maximum growth rates

	POLL When does a population in this model have the fastest
	\emph{per-capita} growth rate? | When is per capita growth fastest?

		NOANS When density is low. 

		NOANS This is an assumption.

	POLL When does a population in this model have the fastest \emph{total}
	growth rate? | When is total growth fastest?

		NOANS Intermediate between low density and the carrying capacity.

		NOANS This is something we learn by making a dynamical model!

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

REPSLIDE Individual perspective

DOUBLEFIG bd_models/birth.bd.Rout-0.pdf bd_models/birth.bd.Rout-2.pdf

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

REPSLIDE Population perspective

DOUBLEFIG bd_models/birth.bd.Rout-1.pdf bd_models/birth.bd.Rout-3.pdf

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

Maximum growth rates

	When does a population in this model have the fastest
	\emph{per-capita} growth rate?

		ANS When density is low. 

		ANS This is an assumption.

	When does a population in this model have the fastest \emph{total}
	growth rate?

		ANS Intermediate between low density and the carrying capacity.

		ANS This is something we learn from the model

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

REPSLIDE Individual perspective

DOUBLEFIG bd_models/birth.bd.Rout-0.pdf bd_models/birth.bd.Rout-2.pdf

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

REPSLIDE Population perspective

DOUBLEFIG bd_models/birth.bd.Rout-1.pdf bd_models/birth.bd.Rout-3.pdf

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

TSS Equilibria

	We define \textbf{equilibrium} as when the population is not changing

	Our simple model is $\bigeq \frac{dN}{dt} = (b(N)-d(N))N$

	In this simple model, when does equilibrium occur?

		ANS $(b(N)-d(N))N = 0$

		ANS $b(N) = d(N)$ (the carrying capacity)

		ANS $N=0$ (the population is absent)

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

Stable and unstable equilibria

	Aren't equilibria always stable?

		If we are at an equilibrium we expect to stay there

		(in our simplified model, at least)

	An equilibrium is defined as stable if we expect to approach the
	equilibrium \emph{when we are near it.}

	An equilibrium is defined as unstable if we expect to move away from
	the equilibrium \emph{when we are near it.}

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

REPSLIDE Population perspective

DOUBLEFIG bd_models/birth.bd.Rout-1.pdf bd_models/birth.bd.Rout-3.pdf

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

What kind of equilibrium?

	How can we tell an equilibrium is stable?

		If population is just below the equilibrium:

			ANS It should increase ($b>d$)

		If population is just above the equilibrium:

			ANS It should decrease ($d>b$)

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

Basic reproductive number

	The reproductive number of a population not affected by crowding is
	called the \textbf{basic reproductive number}

		Written \Ro\ or \Rmax.

	In this model, when $\Ro<1$ the population:

		ANS Always decreases

	When $\Ro>1$ the population:

		ANS Increases when it is small

		ANS Eventually $\R$ will decrease

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

CONT Basic reproductive number

	POLL What is \Ro\ in our current model? | What is R0?

		ANS $\Ro=b(0)/d(0)$

	How do we interpret $b(0)$ and $d(0)$?

		ANS Nothing actually grows or dies when $N=0$

		ANS We think of $b(0)$, and $d(0)$ (and $\Ro$) as limits

			ANS What are the values when density is very low?

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

Invasion

	We say a species can ``invade" a system if its rate of change is
	positive when the population is small.

	In other words, population can invade if the extinction equilibrium
	is not stable

	In this conceptual model, this is the same as saying $b(0) > d(0)$

	Which is the same as saying $\Ro>1$

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

Invasion examples

	POLL What are some examples of biological invasions?

		NOANS

		NOANS

		NOANS

		NOANS

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

EXTRA Invasion examples

	kudzu

	covid

	rabbits in Australia

		there was an attempt to control them with disease

	starlings were released in at least one place just for fun

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

Describing equilibria

	When $\Ro>1$, the population invades

		* Zero equilibrium is unstable, carrying capacity equilibrium is
		stable

	When $\Ro<1$, the population fails to invade

		* Zero equilibrium is stable, carrying capacity equilibrium does
		not exist

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

Individual perspective

DOUBLEFIG bd_models/fail.bd.Rout-0.pdf bd_models/fail.bd.Rout-2.pdf

	When $\Ro<1$ population always decreases

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

Population perspective

DOUBLEFIG bd_models/fail.bd.Rout-1.pdf bd_models/fail.bd.Rout-3.pdf

	When $\Ro<1$ population always decreases

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

REPSLIDE Individual perspective

DOUBLEFIG bd_models/birth.bd.Rout-0.pdf bd_models/birth.bd.Rout-2.pdf

	When $\Ro>1$ population increases when it is small

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

REPSLIDE Population perspective

DOUBLEFIG bd_models/birth.bd.Rout-1.pdf bd_models/birth.bd.Rout-3.pdf

	When $\Ro>1$ population increases when it is small

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

DEFHEAD \Ro\ and thresholds

	A population with $\Ro<1$ in general cannot survive in an area

	As conditions get worse for a species in a
	particular area, or along a particular gradient:

		It will suddenly disappear at the population level

		Even while it can still survive and reproduce at an individual
		level

	This is why there are no white spruce trees in Cootes Paradise

	And no malaria in the mainland United States

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

OFFSLIDE Characteristic times

	Just like in the simple model, an equilibrium will have a
	characteristic time

	If I'm close to an equilibrium, how long would it take:

		to go the distance to the equilibrium at my current ``speed''

		to actually get $e$ times closer, or $e$ times farther

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

OFFSLIDE PSLIDE Characteristic times

DOUBLEFIG bd_models/birth.bd.Rout-2.pdf bd_models/birth.bd.Rout-3.pdf

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

Smooth dynamics

	Populations following this model change \emph{smoothly}

		Equations tell how the population will change at each instant

	They have no memory

		Birth rate and death rate are determined by population size alone

	Cycling is impossible

		ANS If I went from A to B, I can't go from B to A by
		following the same rules

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

REPSLIDE Paramecia

DOUBLEPDF ts/paramecia.plot.Rout

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

REPSLIDE Elk

DOUBLEPDF ts/elk.elkplot.Rout

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

Dynamics of real-world populations

	Initial exponential growth and leveling off frequently observed

	Exponential approach to equilibrium hard to observe

		Real populations are subject to {\bf stochastic} (random) effects 

		Real populations are subject to changing conditions

	Some species seem to cycle predictably

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

Summary

	Continuous-time regulation in simple models makes useful predictions:

		Threshold value for populations to survive

		Greatest population-level growth at intermediate density

		Greatest individual-level growth at low density

	Cannot explain complicated dynamics

		More mechanisms are needed

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

SEC Discrete-time regulation

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

TSS A simple, discrete-time model

	We extend our discrete-time model from the previous unit:

		$N_{T+1} = (p+f)N_T \equiv \lambda N_T$

		$t_{T+1} = t_T + \Delta t$ (does not change)

	To:

		$N_{T+1} = (p(N_T)+f(N_T))N_T \equiv \lambda(N_T) N_T$

	This means:

		ANS $p$ and $f$ can change when $N$ changes

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

Assumptions

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

	All individuals are the same at the time of census

	Population changes deterministically

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

Specific assumptions

	For our examples, we will assume:

		$f(N)=f_0 \exp(-N/N_f)$

		$p(N) = p_0 \exp(-N/N_p)$

	This is the simplest model that is smooth and keeps track
	of birth and death rates separately

		Fecundity goes down with characteristic scale $N_f$

		Survival goes down with characteristic scale $N_p$

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

States and state variables

	What variable or variables describe the state of this system?

		The same as before: population size (or density)

		We are still assuming that's all we need to know

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

Parameters

	What quantities describe the rules for this system?

		ANS $f_0$ [1]

		ANS $p_0$ [1]

		ANS $N_f$ [indiv] (or [indiv/area])

		ANS $N_p$ [indiv] (or [indiv/area])

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

What is \Ro?

	\R\ is the fecundity multiplied by the lifespan

		ANS Lifespan = $1/\mu = 1/(1-p)$

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

	\Ro\ is \R\ in the limit where density is low

		ANS $f_0/(1-p_0)$

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

Behaviours

	When $\Ro<1$ population always declines

	When $\Ro>1$, population can show:

		Smooth behaviour (like the continuous-time model)

		Damped oscillations (like delayed models)

		Two-year cycles (high \rec low \rec high \rec low)

		All \emph{kinds} of other stuff

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

TSS Simulating this system

	This system can be simulated very easily by following the rule for
	$N_{T+1}$ as a function of $N_T$

	We can even do it in the spreadsheet if we have time

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

PRESLIDE What dynamics do we expect?

FIG compensation/stablePer.plots.Rout-0.pdf

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

PRESLIDE What dynamics do we expect?

FIG compensation/stablePer.plots.Rout-1.pdf

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

We expect simple dynamics

DOUBLEPDF compensation/stablePer.plots.Rout

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

PSLIDE What dynamics do we get?

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

REPSLIDE Simple dynamics

DOUBLEPDF compensation/stablePer.plots.Rout

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

Damped oscillations

DOUBLEPDF compensation/dampedPer.plots.Rout

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

Persistent oscillations

DOUBLEPDF compensation/twoPer.plots.Rout

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

Lots of other behaviours

DOUBLEPDF compensation/wildPer.plots.Rout

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

TSS Interpreting complex behaviour

BC

	In a simple cycle:

		Low populations this year mean high populations next year

		and vice versa

NC

HIDEFIG compensation/twoPer.plots.Rout-0.pdf

HIDEFIG compensation/twoPer.plots.Rout-1.pdf

EC

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

REPSLIDE Interpreting complex behaviour 

DOUBLEPDF compensation/twoPer.plots.Rout

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

Complex behaviour in our simulations

	In our simple models, as $N_T$ increases, what happens to $\lambda$?

		ANS We assume it goes down

	POLL In our simple models, as $N_T$ increases, what happens to next year's population? | What happens to next year's population as N_T increases? goes up; goes down; up then down; it depends

		ANS $\Nprime = \lambda(N) N_T$

		ANS It's not obvious! $\lambda$ goes down, but $N$ goes up.

		ANS In this model, $\Nprime$ always goes down eventually, but
		other models may differ

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

Response to population increase

DOUBLEFIG compensation/stablePer.plots.Rout-2.pdf compensation/stablePer.plots.Rout-3.pdf

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

Turnover

BC

	When $N_T$ is small, $\Nprime$ increases with $N$.

	Complex behaviour arises when the relationship between $N_T$ and
	\Nprime\ \textbf{turns over} below the equilibrium value

		A small population this year leads to a large population next year

NC

SIDEFIG compensation/stablePer.plots.Rout-3.pdf

EC

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

Simple dynamics

DOUBLEFIG compensation/stablePer.plots.Rout-3.pdf compensation/stablePer.plots.Rout-1.pdf

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

Damped oscillations

DOUBLEFIG compensation/dampedPer.plots.Rout-3.pdf compensation/dampedPer.plots.Rout-1.pdf

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

Persistent oscillations

DOUBLEFIG compensation/twoPer.plots.Rout-3.pdf compensation/twoPer.plots.Rout-1.pdf

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

Complex behaviour in our conceptual model

	Biologically, when might we expect \Nprime\ to ``turn over''?

		ANS If resources are \emph{depleted}

		ANS If there is a \emph{delayed} effect of individuals' not
		having enough resources

	When should the mapping \emph{not} turn over?

		ANS When competition does not lead to depletion

		ANS When effects of competition are immediate

		ANS When dominant individuals are not affected by crowding

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

Scramble competition

	{\bf Scramble} competition refers to the case where all
	individuals in a crowded population are gathering resources at
	similar rates: as the density goes up there is less resource for
	everyone, and everyone does less well

		If there is any kind of delay, scramble competition can lead to
		turning over

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

Contest competition

	{\bf Contest} competition refers to a case where some individuals
		successfully control key resources and do well no matter how
		large the population is

		Contest competition doesn't usually lead to turning over, even
		with delay

	How does contest competition square with regulation?

		ANS Regulation means that $\lambda$ has to go down with $N_T$ \ldots

		ANS \emph{not} that \Nprime\ has to.

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

PSLIDE Contest competition

BC

	How does contest competition square with regulation?

		Regulation means that $\lambda$ has to go down with $N_T$ \ldots

		\emph{not} that \Nprime\ has to.

NC

SIDEFIG compensation/monotonic.plots.Rout-2.pdf

SIDEFIG compensation/monotonic.plots.Rout-3.pdf

EC

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

Contest regulation

DOUBLEFIG compensation/monotonic.plots.Rout-2.pdf compensation/monotonic.plots.Rout-3.pdf

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

Songbirds

BC

	Some songbird populations are limited primarily by competition for
	breeding sites, whereas others are limited primarily by competition
	for insects to eat

		ANS Food can be depleted

			ANS Making scramble competition and turnover more likely

		ANS Nest sites can be occupied, but they don't go away

			ANS More like contest competition, less likely to have turnover

NC

SIDEFIG webpix/swallows.jpg

EC

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

Plants

BC

	Some plant populations are limited by water, and some by light

	POLL Which is more likely to work out as a scramble? Light; water

		ANS Light is very likely to work out as a ``contest'' -- the
		taller individuals will win and do OK

		ANS Water works as a scramble in some environments, and a contest
		in others

NC

SIDEFIG webpix/poppies.jpg

EC

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

Complex behaviour from a simple model

BC

	It's interesting that we can get complicated behaviour from such
	a dead-simple model	

	Complex dynamics may have simple causes

	People always tend to look for specific reasons, but sometimes the
	changes we observe are just natural dynamics

NC

SIDEFIG compensation/wild.plots.Rout-1.pdf

EC

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

Complex behaviour in real populations

	We can plot $\lambda$ and \Nprime\ vs.\ $N$ for real population data

	We expect $\lambda$ to decrease (on average)

	We're curious about \Nprime.

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

REPSLIDE Paramecia

DOUBLEPDF ts/paramecia.plot.Rout

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

Paramecia

DOUBLEFIG ts/paramecia.plot.Rout-3.pdf ts/paramecia.plot.Rout-2.pdf

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

REPSLIDE Elk

DOUBLEPDF ts/elk.elkplot.Rout

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

Elk

DOUBLEFIG ts/elk.elkplot.Rout-3.pdf ts/elk.elkplot.Rout-2.pdf

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

Real populations

	It's hard to find examples of turnover from real population data.

	So how do we explain real population cycles?

		Regulation may happen on a longer time scale

		May be hard to see because of ``noise'' -- i.e., other sources of
		variation

		Cycles may be due to more complicated mechanisms

	Time-delayed population models are cool!

		But not covered in class this year

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

EXTRA Time-delayed population models

delay.txt

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

SEC Small populations

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

Example

BC

	POLL What would happen if I released one butterfly into a new, highly suitable habitat? | What happens to one happy butterfly?

		NOANS

	What about two butterflies?

		NOANS

NC

SIDEFIG webpix/viceroy.jpg

EC

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

Small populations

	Population success (reproductive number) may be lower for very small
	populations

		We've already assumed reproductive numbers are low for very large
		populations

	Small populations are likely to be harder to predict

		More affected by stochasticity

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

TSS Allee effects

	Effects which cause small populations to have low per-capita growth
	rates are called Allee effects

		Equivalent to saying that medium-sized populations have larger
		per-capita growth rates than small ones

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

CONT What causes Allee effects?

	POLL Why might growth rates be low when populations are small?

		ANS Individuals may have trouble finding mates

		ANS Individuals in larger populations may protect each other from
		predators (birds) or from weather (plants)

		ANS Individuals in larger populations may hunt co-operatively

		ANS Genetic effects (inbreeding, loss of valuable variation)

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

Types of Allee effect

	Allee effects can affect the (per capita) birth rate

		ANS if the rate is \emph{smaller} when density is low

	... or the (per capita) death rate

		ANS if the rate is \emph{larger} when density is low

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

EXTRA

	Use magic word CUTOFF for exam notes as appropriate

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

Allee effect models

BC

	What will this model do, if the initial population is:

		low, medium or high?

NC

SIDEFIG bd_models/allee.bd.Rout-0.pdf

EC

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

Individual perspective

DOUBLEFIG bd_models/allee.bd.Rout-0.pdf bd_models/allee.bd.Rout-2.pdf

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

Individual perspective

DOUBLEFIG bd_models/allee.bd.Rout-4.pdf bd_models/allee.bd.Rout-6.pdf

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

Population perspective

DOUBLEFIG bd_models/allee.bd.Rout-1.pdf bd_models/allee.bd.Rout-3.pdf

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

Population perspective

DOUBLEFIG bd_models/allee.bd.Rout-5.pdf bd_models/allee.bd.Rout-7.pdf

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

REPSLIDE Population comparison

DOUBLEFIG bd_models/allee.bd.Rout-1.pdf bd_models/birth.bd.Rout-9.pdf

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

Allee effect in death rate

BC

	What is the difference between this example and the previous one?

	What will this model do, if the initial population is:

		low, medium or high?

NC

SIDEFIG bd_models/death_allee.bd.Rout-0.pdf

EC

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

Individual perspective

DOUBLEFIG bd_models/death_allee.bd.Rout-0.pdf bd_models/death_allee.bd.Rout-2.pdf

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

More reproductive numbers

	The reproductive number \R\ means the average lifetime number of
	offspring per individual

		Should be unitless, so we consider offspring at the same stage as
		the individual.

	We can apply \R\ in general for any set of conditions, or we can
	distinguish:

		the \textbf{basic reproductive number \Ro}: \R\ in the limit when
		the population is small, and

		the \textbf{maximal reproductive number \Rmax}: \R\ at whatever
		level is the peak

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

Invasion

	We previously said that when $\Ro<1$, the population always went
	extinct

		A population that can't invade can never replace itself on average

	When Allee effects are present, it's no longer true that a species
	that can't invade can't persist

		ANS If $\Ro<1$ population can't invade, but if $\Rmax>1$ it can
		still persist

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

REPSLIDE \Ro\ and \Rmax

FIG bd_models/allee.bd.Rout-0.pdf

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

Weak Allee effects

BC

	If birth rates go down or death rates go up at low density, we
	consider this an Allee effect

	If $\Ro < 1$ we say it's a \textbf{strong} Allee effect

		ANS Population can't invade

	If $\Ro > 1$ we say it's a \textbf{weak} Allee effect

NC

SIDEFIG bd_models/weak.bd.Rout-0.pdf

EC

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

Individual perspective

DOUBLEFIG bd_models/weak.bd.Rout-0.pdf bd_models/weak.bd.Rout-2.pdf

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

Allee effect summary

	Population may go extinct if it drops below a certain threshold

	POLL How come the population is there in the first place if there's an Allee
	effect?

		ANS Maybe it's a weak effect

		ANS Maybe conditions have changed (it used to be a weak effect,
		or no effect)

		ANS Maybe a large initial group established by chance

		ANS Maybe the population arrived recently and hasn't had time to go
		extinct yet.

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

TSS Stochastic effects

	The world is complicated and biological populations are not
	perfectly predictable

	Real populations don't go smoothly to equilibria, instead they
	bounce around (or sometimes do other wild stuff)

	We divide stochastic (or random) effects into demographic and
	environmental stochasticity

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

Example

	Female butterflies of a certain species lay 200 eggs on average, of
	which:

		Half are female

		50% hatch successfully into larvae

		10% of larvae successfully pupate

		60% of pupae become adults

		Half of adult females successfully reproduce

	A single gravid (pregnant) female butterfly is blown away by a freak
	storm, and lands by chance on a suitable island with no butterflies
	What do you expect to happen?

		ANS $\lambda=1.5$

		REGULAR ANS(remember not to multiply by the sex ratio
		twice!)

		ANS Almost anything can happen 

		COMMENT Try this calculation with units 

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

Butterfly example

	Depending on unknown conditions, especially in that first season,
	all of those probabilities could change dramatically

	Even if we knew the \emph{probabilities}, that would not guarantee an
	exact result

		ANS Population could be lucky or unlucky

	What if $\lambda<1$?

		ANS The population would go extinct eventually, even if it's lucky

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

Demographic stochasticity

	{\bf Demographic} stochasticity is stochasticity that operates at
	the level of individuals

		Individuals don't increase gradually, they die or give birth

		Individuals don't produce 1.2 offspring: they produce 0, 1, 2 or
		3 \ldots

	Even if we control conditions perfectly, we can't exactly predict
	the dynamics of small populations

	Demographic stochasticity averages out in large populations

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

Environmental stochasticity

	{\bf Environmental} stochasticity is stochasticity that operates at
	the level of the population

		E.g,. weather, pollution

	Environmental stochasticity can have large effects on any population

		ANS A bad year is bad for everyone

	But small populations are the ones in danger of going extinct

		ANS Large populations can average out over \emph{time}

		ANS If the ``mean" value of $R_0$ is greater than 1, large
		population should survive the ups and downs

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

Simulations

BC

	We can simulate stochastic systems very easily

	But if we do the same simulation twice, we can get different answers

	Adds realism

		But harder to interpret

NC

HIDEFIG compensation/mothtest.Rout-1.pdf

HIDEFIG compensation/mothtest.Rout-2.pdf

EC

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

Summary

	Stochasticity is very important in real populations, but hard to
	study

		Mathematical analysis is very difficult

		Simulations are useful, but hard to interpret

			Each time you simulate, you get a different answer

	Ecologists need to learn to recognize and communicate our
	uncertainty about the future