Pineda-Krch CMB2011 2lides

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Cycles in finite populations

A reproducible seminar in three acts

Mario Pineda-Krch

October 31, 2011

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Act 1: The ghost of Fermat

(The what, why and how of reproducible research)

Act 2: A tale of two cycles(Demonstrating the existence of quasi-cycles using reproducible

research)

Act 3: Cycles at the edge of existence(Emergence of quasi-cycles in strongly destabilized ecosystems)

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Act 1: The ghost of Fermat

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Irreproducible research

Randall J. LeVeque (2006)

Scientific and mathematical journals are filled with pretty pictures

these days of computational experiments that the reader has nohope of repeating. Even brilliant and well intentionedcomputational scientists often do a poor job of presenting theirwork in a reproducible manner. The methods are often very

vaguely defined, and even if they are carefully defined they wouldnormally have to be implemented from scratch by the reader inorder to test them. Most modern algorithms are so complicatedthat there is little hope of doing this properly.

Pierre de Fermat (1637)It is impossible to separate a cube into two cubes, or a fourthpower into two fourth powers, or in general, any power higher thanthe second, into two like powers. I have discovered a trulymarvellous proof of this, which this margin is too narrow to

contain.4


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What really happened?

Reproducible results = Reproducible research

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Reproducible research: Its not the destination. Its thejourney.

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Reproducible research: The What

LeVeque (2006):The idea of reproducible research in scientific computing is toarchive and make publicly available all of the codes used to createthe figures or tables in a paper in such a way that the reader can

download the codes and run them to reproduce the results.Wikipedia:

Reproducibility is one of the main principles of the scientificmethod, and refers to the ability of a test or experiment to be

accurately reproduced, or replicated, by someone else workingindependently.

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Reproducible research: The Why

American Physical Society

Science is the systematic enterprise of gathering knowledge aboutthe universe and organizing and condensing that knowledge into

testable laws and theories. The success and credibility of scienceare anchored in the willingness of scientists to: Expose their ideasand results to the independent testing and replication by others.This requires the open exchange of data, procedure and materials.

Reproducability = Transparency + Executability

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Journal article

Processed data

Raw data

Computer simulations

Computer code

Algorithms

6

?Autho

r

Reader

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Literate programmingA paradigm for reproducible research in computational sciences

The idea is that you do not

document programs (after thefact), but write documentsthat contain the programs. John Max Skaller

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Literate programming according to Donald Knuth

Prose

ProgramDocumentation

TangleWeave

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Literate programming systems

LP system Document formatting language Programming language Inventor(s) YearWEB TEX Pascal Knuth 1992CWEB TEX C/C++/Java Knuth & Levy 1993FWEB LATEX C/C++/FORTRAN Krommes 1993

noweb TEX/LATEX/HTML/troff agnostic Ramsey 1999Sweave LATEX R Leisch 2002PyLit reStructuredText Python Milde 2005Pweave LATEX/reST/Sphinx/Pandoc Python Pastell 2010

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Literate programming according to RThe evolution of the literate programming paradigm

Prose

CodeDocumentation

StangleSweave

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What really happens

Prose

CodeDocumentation

Results

StangleSweave

Sweave

Execute

Integrate

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This research is reproducible!

These slides are prepared using Sweave.

These slides are executable (look for ). The full project (source, code, results, etc.) will be available

at http://pineda-krch.com.

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http://pineda-krch.com/http://pineda-krch.com/http://pineda-krch.com/


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Implementing reproducible research

Attach code to publish results is good...,

executable manuscripts are better.

Adopt a habit of reproducibility, i.e. make it routine and

require it from others (students, postdocs, colleagues). Keep reproducibility in computational research to the same

rigorous standard as reproducibility in mathematical proofs.

Demand reproducibility in your role as journal editor and

reviewer of manuscripts and grants applications.

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Act 2: A tale of two cycles

17

O ( ) G


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Olaus Magnus (1555) Historia de Gentibus

First known depiction of of population cycles in Olaus Magnus

Historia de Gentibus Septentrionalibus (History of the NorthernPeoples) (1555) shows lemmings falling from the sky with twoweasels with lemmings in their mouths.

18

El (1924) B i i h J l f E i l Bi l


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Elton (1924) British Journal of Experimental Biology

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K d ll l (1998) E l L


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Kendall et al. (1998) Ecology Letters

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Fl t ti l ti


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Fluctuating populations

1750 1800 1850 1900

0

20000

60000

Lynx

1850 1860 1870 1880 1890 1900 1910

6000

10000

140

00

18000

Populationsize

Otter

1750 1800 1850 1900

0

500

1500

2500 Wolverine

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E l i i l l ti d i i i l


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Explaining complex population dynamics using simplemodels

Classical Lotka-Volterra Exponential growth in prey, linear (Type 1) functional response

in predator Structurally unstable, mainly of historical interest

Lotka-Volterra Logistic growth in prey, linear (Type 1) functional response in

predator Does not cycle

Rosenzweig-MacArthur Logistic growth in prey, non-linear (Type 2) functional

response in predator (i.e. satiation) Cycles

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R s ig M A th d t d l (RMPP)


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Rosenzweig-MacArthur predator-prey model (RMPP)

dNdt = rN

1 NK

a1 + wNNP

dPdt = c

a1 + wNNP gP

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The deterministic RMPP in R


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The deterministic RMPP in R

> ode.rmpp


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The stochastic RMPP in R

> ssa.rmpp


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Structure of simulation model

rmpp.R

ssa.rmpp.Rode.rmpp.R

ssa_rmpp.c

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Run a simulation in the stable node region


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Set the parameters

> alpha beta gamma alpha; beta; gamma

[1] 0.5

[1] 1

[1] 1.2

Set equilibrium population size

> eq tmax n < - 1

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Run the simulation

> fn.node fn.node

[1] "../results/rmpp-alpha0.5-beta1-gamma1.2-eq1000-1.RData"

Loading the data> load(fn.node)

Looking at the contents of the data file

> ls()

[1] "alpha" "beta" "d.lynx" "d.otter" "d.wolveri[6] "eq" "fn" "fn.node" "gamma" "i"

[11] "n" "ode.res" "ode.rmpp" "parms" "rmpp"

[16] "ssa.res" "ssa.rmpp" "start.time" "tmax" "x0"

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Plot results


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Plot results> layout(matrix(c(1,2), ncol=2))

> ymax ymin plot(N~time, ode.res, type='l', lwd=3, col=rgb(0,1,0,.75),

xlab="Time", ylab="Population size", xlim=c(0, 250), ylim=c(ymin, ymax), bty='n')

> points(P~time, ode.res, type='l', lwd=3, col=rgb(1,0,0,.75))> title('Deterministic')

> plot(N~time, ssa.res, type='l', lwd=3, col=rgb(0,1,0,.75),

xlab="Time", ylab="", xlim=c(0, 250), ylim=c(ymin, ymax), bty='n')> points(P~time, ssa.res, type='l', lwd=3, col=rgb(1,0,0,.75))

> title('Stochastic')

0 50 100 150 200 250

400

600

800

1000

1200

1400

Time

Population

size

Deterministic

0 50 100 150 200 250

400

600

800

1000

1200

1400

Time

Stochastic

Figure: Time series for RMPP model in stable node region ( = 0.5, = 1, = 1.2).

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Run a simulation in the limit cycle region


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Run a simulation in the limit cycle region

0 50 100 150 200 250

0

500

1000

2000

3000

Time

Population

size

Deterministic

0 50 100 150 200 250

0

500

1000

2000

3000

Time

Stochastic

Figure: Time series for RMPP model in limit cycle region ( = 0.5,

= 1, = 3.5).

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Run a simulation in the stable focus region


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Run a simulation in the stable focus region

0 50 100 150 200 250

500

1000

1500

Time

Population

size

Deterministic

0 50 100 150 200 250

500

1000

1500

Time

Stochastic

Figure: Time series for RMPP model in stable focus region = 0.5,

= 1, = 2.5).

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Detecting periodic fluctuations in simulated time


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Detecting periodic fluctuations in simulated time

series

100 150 200 250 300 350

600

800

1000

1200

Node

100 150 200 250 300 350

600

1000

1400

18

00

Population

size

Focus

100 150 200 250 300 350

0

500

1500

2500

Limit cycle

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Autocorrelation Function (ACF)


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( C )Detecting periodic fluctuations

0 10 20 30 40

0.0

0.2

0.4

0.6

0.8

1.0

ACF

Node

0 5 10 15 20 25 30

0.5

0.0

0.5

1.0

Lag

Focus

0 5 10 15 20 25 30

0.5

0.0

0.5

1.0

Limit cycle

Evidence of periodicity if ACF(T) > 2/tm where T is the lag ofthe dominant frequency (first maximum), tm is the number of datapoints in the time series.

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Marginal distribution


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gDistinguishing between quasi-cycles and noisy limit cycles

Node

Frequency

600 800 1000 1200

0

500

1000

150

0

2000

2500

3000

Focus

Prey population size

600 1000 1400 1800

0

50

100

150

200

Limit cycle

0 500 1500 2500

0

50

100

150

Evidence of limit cycle: bimodal distribution

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Detecting periodic fluctuations in natural time


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series

1750 1800 1850 1900

0

20000

60000

Lynx

1850 1860 1870 1880 1890 1900 1910

6000

10000

14000

18000

Population

size

Otter

1750 1800 1850 1900

0

500

1500

2500 Wolverine

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Lynx timeseries


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1750 1800 1850 1900

0

20000

60000

Year

Populationsize

qq

q qqq

qqqqqqqqq

qqq

q

qqqqqqqqqqqq

qqqq

q

qqqq

q

qqqqqqqqqqq

q

qq

q

qqqqq

qq

qqqqqqq

qqqq

2.0 1.5 1.0 0.5

4

5

6

7

8

9

10

log10(Cycle frequency)

log

10(Power)

0 5 10 15 200.2

0.2

0.4

0.6

0.8

1.0

Lag

ACF

Population size

F

requency

0 20000 40000 60000 80000

0

20

40

60

80

100

38

Otter timeseries


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1850 1870 1890 1910

6000

10000

14

000

18000

Year

Populationsize

q q q q

qqqq

qqqqqqqq

q

qq

qq

qqq

q

qq

q

qqqq

1.5 1.0 0.5

4

5

6

7

8

9

10


log

10(Power)

0 5 10 15

0.2

0.2

0.40

.6

0.8

1.0

Lag

ACF

Population size

F

requency

5000 10000 15000 20000

0

5

10

15

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Wolverine timeseries


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1750 1800 1850 1900

0

500

1000

2000

Year

Populationsize

qq

q

q

q

qqqqq

q

q

qqq

qqqqqq

q

q

q

qqqqq

qqqqqq

q

qqqq

qqq

qqqq

qqqq

q

qqqq

q

qq

q

qqqq

2.0 1.5 1.0 0.5

4

5

6

7

8

9

10


log10(Power)

0 5 10 15 20

0.2

0.2

0.4

0.6

0.8

1.0

Lag

ACF

Population size

F

requency

0 500 1000 2000

0

5

10

15

20

25

30

40

Conclusions


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Time series ACF Power spectrum Marginal distribution Stochastic behaviourSimulated

node none uniform normal noisy equilibriumfocus periodic periodic normal quasi-cycles

limit cycle periodic periodic non-normal noisy limit cycleReal

lynx periodic periodic non-normal noisy limit cycleotter none uniform normal noisy equilibriumwolverine periodic ambiguous normal quasi-cycles

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Quasi-cycles are not new


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Nisbet & Gurney, 1976

...population cycles of rather large amplitude may occur when anunderdamped control system is subject to random environmentalnoise, and we suggest that this mechanism is worthy ofconsideration when analyzing real populate data.

McKane & Newman, 1998We expect that this resonance mechanism will occur in otherstochastic systems in which the mean field theory shows dampedoscillations, for instance, biochemical reactions in microscopicsystems.

Pineda-Krch, 2007...the existence of quasi-cycles should generalize to anydeterministic model that exhibits a stable focus and is perturbed bynoise.

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Parameter space for RMPP model


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0.0 0.2 0.4 0.6 0.8 1.0

0

2

4

6

8

10

Saddle point (unstable)

Nod

eFocus

Limit cycle

q

q

q

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Act 3: Cycles at the edge of

existence

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Lotka Volterra predator-prey (LVPP) model


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dNdt = rN

1 NK

cNP

dPdt = bNP mP

45

Simplifying the model


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To simplify the LVPP model the number of parameters are reducedby defining x

N/K, y

Pa/r and t

rt the model can be

rescaled todxdt

= x(1 x y)

dy

dt

= y

caKr

x

g

caK

.

Setting (g/(caK)) and (caK)/r, this further simplifies todxdt

= x(1 x y)

dydt

= y(x )

where > 0 and > 0.

46

Stable coexistence equilibrium


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q

0.001 0.200 0.400 0.600 0.800 1.000

0

1

2

3

4

5

6

7

8

9

10

Node

Focus

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Structure of simulation model


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lvpp.R

ssa.lvpp.Rode.lvpp.R

ssa_lvpp.c

(The R code is here..., but the output is suppressed) 48

Simulations


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Set the parameter combinations> alpha.vec beta.vec eq.vec tmax n for (beta in beta.vec) {

for (alpha in alpha.vec) {

for (eq in eq.vec) {file.names


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0.1

0.1

0.2

0.20.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.001 0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900 1.000

0

1

2

3

4

5

6

7

8

9

10

0.95

50

Quasi-cycle rate


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0.1 0.1

0.2

0.2

0.30.4

0.5

0.6

0.7

0.8

0.9

1

0.001 0.200 0.400 0.600 0.800 1.000

0

1

2

3

4

5

6

7

8

9

10

0.95

eq=500

0.1

0.1

0.2

0.2

0.3

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.001 0.200 0.400 0.600 0.800 1.000

0

1

2

3

4

5

6

7

8

9

10

0.95

eq=1000

0.1

0.1

0.2

0.2

0.3

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.001 0.200 0.400 0.600 0.800 1.000

0

1

2

3

4

5

6

7

8

9

10

0.95

eq=5000

0.1 0.1

0.2

0.2

0.30.4

0.5

0.60.7

0.8

0.9

1

0.001 0.200 0.400 0.600 0.800 1.000

0

1

2

3

4

5

6

7

8

9

10

0.95

eq=10000

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Extinction rate, eq=500


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0

.1

0.1

0.2

0.2

0.3

0.3

0.4

0.4

0.5

0.6

0.7

0.8

0.9

1

0.001 0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900 1.000

0

1

2

3

4

5

6

7

8

9

10

0.95

52

Conclusions


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In non-cyclic models quasi-cycles arise only in part of theregion exhibition damped oscillations

These results are consistent for population sizes across 3orders of magnitude

For small population sizes quasi-cycles arise close to the pointof extinction

Potential for using quasi-cyclicity for quantifying extinctionrisks in PVA

How common are limit cycles in natural populations?

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AcknowledgementsAct 1:


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Act 1:

The Tria Project (Genome Canada, Genome Alberta, GenomeBC)

Lewis Research Group

Act 2:

Rob J. Hyndman (Time Series Data Library) for permission touse the otter and wolverine time series

James S. McDonnell Foundation

NSERC

PIMS

Act 3:

The Tria Project (Genome Canada, Genome Alberta, GenomeBC)

National Institute for Mathematical and Biological Synthesis(NIMBioS)

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Background


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Act 1: Pineda-Krch. 2011. The Joy of Sweave Abeginners guide to reproducible research withSweave. http://pineda-krch.com/2011/01/17/the-joy-of-sweave/

Act 2: Pineda-Krch, Blok, Dieckmann, Doebeli. 2007. A

tale of two cycles distinguishing quasi-cycles andlimit cycles in finite predator-prey populations. Oikos116: 5364.

Act 3: Pineda-Krch. 2011. Cycles at the edge of existence:

Emergence of quasi-cycles in strongly destabilizedecosystems. Submitted

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Session information
http://pineda-krch.com/2011/01/17/the-joy-of-sweave/http://pineda-krch.com/2011/01/17/the-joy-of-sweave/http://pineda-krch.com/2011/01/17/the-joy-of-sweave/http://pineda-krch.com/2011/01/17/the-joy-of-sweave/


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R version 2.13.1 (2011-07-08), x86_64-apple-darwin9.8.0

Locale: C

Base packages: base, datasets, grDevices, graphics, methods,stats, utils

Other packages: odesolve 0.5-20, spatial 7.3-3

Loaded via a namespace (and not attached): tools 2.13.1

The R code in this vignette took 21 seconds to execute on Mon

Oct 31 23:40:05 2011.

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Pineda-Krch CMB2011 2lides