nonlinear.txt

EXTRA

Common carp invading Cootes paradise

lampreys from Europe in Great Lakes

youtuber and frog eggs

brown stink bugs

starlings in New York

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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 But! Unless the amount of predators increases as well, they might
			do \emph{worse} at controlling the population

		ANS Diseases

			ANS Usually able to increase in proportion to a population

		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 This picture may be updated automatically

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

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

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

REGSLIDE 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.

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

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 $K$

	Not a linear model, because \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, but it really isn't

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

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 very
		large (virtually no births)

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

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

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 limited
	by adult crowding. Although juvenile mussels tend to have a hard
	time, this might not be density dependent

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 a 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?

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

		Thanks to the contributors

	ADD Update

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

EXTRA

NOANS

NOANS

NOANS

NOANS

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

Different behaviours

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

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

REGSLIDE 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

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

REGSLIDE 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 the delayed model)

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

ADD CUTOFF 2020 Jan 21 (Tue): what does this mean?

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

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 won't necessarily stick
around)

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

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