The document summarizes a reproducible seminar on population cycles in finite populations presented in three acts. Act 1 introduces the concept of reproducible research and how it allows results to be independently verified. Act 2 demonstrates the existence of population cycles using a predator-prey model. Act 3 examines how cycles can emerge in destabilized ecosystems. The seminar is presented using literate programming techniques to integrate executable code, results, and documentation.

1. Cycles in finite populations
—
A reproducible seminar in three acts
Mario Pineda-Krch
October 31, 2011
1

2. 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)
2

3. Act 1: The ghost of Fermat
3

4. Irreproducible research
Randall J. LeVeque (2006)
“Scientific and mathematical journals are filled with pretty pictures
these days of computational experiments that the reader has no
hope of repeating. Even brilliant and well intentioned
computational scientists often do a poor job of presenting their
work in a reproducible manner. The methods are often very
vaguely defined, and even if they are carefully defined they would
normally have to be implemented from scratch by the reader in
order to test them. Most modern algorithms are so complicated
that there is little hope of doing this properly.”
Pierre de Fermat (1637)
“It is impossible to separate a cube into two cubes, or a fourth
power into two fourth powers, or in general, any power higher than
the second, into two like powers. I have discovered a truly
marvellous proof of this, which this margin is too narrow to
contain.”
4

5. What really happened?
Reproducible results = Reproducible research
5

6. Reproducible research: It’s not the destination. It’s the
journey.
6

7. Reproducible research: The What
LeVeque (2006):
“The idea of reproducible research in scientific computing is to
archive and make publicly available all of the codes used to create
the 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 scientific
method, and refers to the ability of a test or experiment to be
accurately reproduced, or replicated, by someone else working
independently.”

8. Reproducible research: The Why
American Physical Society
“Science is the systematic enterprise of gathering knowledge about
the universe and organizing and condensing that knowledge into
testable laws and theories. The success and credibility of science
are anchored in the willingness of scientists to: Expose their ideas
and results to the independent testing and replication by others.
This requires the open exchange of data, procedure and materials.”
Reproducability = Transparency + Executability
8

9. Journal article
Reader
6
Processed data
Raw data
Computer simulations
Computer code
Author
Algorithms ?
9

10. Literate programming
A paradigm for reproducible research in computational sciences
“The idea is that you do not
document programs (after the
fact), but write documents
that contain the programs.”
— John Max Skaller

11. Literate programming according to Donald Knuth
Prose
Weave Tangle
Documentation Program
11

12. Literate programming systems
LP system Document formatting language Programming language Inventor(s) Year
WEB TEX Pascal Knuth 1992
CWEB TEX C/C++/Java Knuth & Levy 1993
FWEB L TEX
A C/C++/FORTRAN Krommes 1993
noweb TEX/L TEX/HTML/troff
A agnostic Ramsey 1999
Sweave L TEX
A R Leisch 2002
PyLit reStructuredText Python Milde 2005
Pweave L TEX/reST/Sphinx/Pandoc
A Python Pastell 2010
12

13. Literate programming according to R
The evolution of the literate programming paradigm
Prose
Sweave Stangle
Documentation Code
13

14. What really happens
Prose
Sweave Stangle
Sweave
Documentation Code
Execute
Integrate
Results
14

15. 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.
15

16. 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.

17. Act 2: A tale of two cycles
17

18. Olaus Magnus (1555) Historia de Gentibus
First known depiction of of population cycles in Olaus Magnus’
Historia de Gentibus Septentrionalibus (History of the Northern
Peoples) (1555) shows lemmings falling from the sky with two
weasels with lemmings in their mouths.
18

19. Elton (1924) British Journal of Experimental Biology
19

20. Kendall et al. (1998) Ecology Letters
20

21. Fluctuating populations
Lynx
60000
20000
0
1750 1800 1850 1900
Otter
Years
10000 14000 18000
Population size
6000
1850 1860 1870 1880 1890 1900 1910
Wolverine
Years
2500
1500
500
0
1750 1800 1850 1900

22. Explaining complex population dynamics using simple
models
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

23. Rosenzweig-MacArthur predator-prey model (RMPP)
dN N
= rN 1 − K − 1 +awN NP
dt
dP = c 1 +awN NP − gP
dt
23

24. The deterministic RMPP in R
> ode.rmpp <- function(parms = stop('Missing parms!'),
x0 = stop('Missing x0!'),
tmax = stop('Missing tmax')){
det.rmpp=function(t, y, parms){
N=y[1] ; P=y[2]
with(as.list(parms),{
dN = (b-d)*N*(1-N/K)-a/(1+w*N)*N*P
dP = c*a/(1+w*N)*N*P-g*P
out=c(dN, dP)
list(out)
})
}
require(odesolve)
time <- seq(0, tmax, by=1)
pop <- c(N=x0[['N']]*0.75, P=x0[['P']]*1.25)
res = as.data.frame(lsoda(pop, time, det.rmpp, parms))
return(ode.res = res)
}

25. The stochastic RMPP in R
> ssa.rmpp <- function(parms = stop('Missing parms!'),
x0 = stop('Missing x0!'),
tmax = stop('Missing tmax')){
command <- paste('../src/ssa_rmpp ',
parms[['b']], parms[['d']],
parms[['K']], parms[['a']],
parms[['w']], parms[['c']],
parms[['g']], x0[['N']], x0[['P']],
tmax, '>../results/out.txt')
system(command)
res <- as.data.frame(read.csv('../results/out.txt',
skip=1, header=TRUE))
return(ssa.res = res)
}

26. Structure of simulation model
rmpp.R
ode.rmpp.R ssa.rmpp.R
ssa_rmpp.c
26

27. Run a simulation in the stable node region
Set the parameters
> alpha <- 0.5
> beta <- 1
> gamma <- 1.2
> alpha; beta; gamma
[1] 0.5
[1] 1
[1] 1.2
Set equilibrium population size
> eq <- 1000
Set the length and the number of simulations
> tmax <- 10000
> n <- 1
27

28. Run a simulation in the stable node region
Run the simulation
> fn.node <- rmpp(alpha, beta, gamma, tmax, eq, n)
Data was saved in
> 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.wolverin
[6] "eq" "fn" "fn.node" "gamma" "i"
[11] "n" "ode.res" "ode.rmpp" "parms" "rmpp"
[16] "ssa.res" "ssa.rmpp" "start.time" "tmax" "x0"
28

29. Plot results
> layout(matrix(c(1,2), ncol=2))
> ymax <- max(c(ode.res$N, ode.res$P, ssa.res$N, ssa.res$P))
> ymin <- min(c(ode.res$N, ode.res$P, ssa.res$N, ssa.res$P))
> 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')
Deterministic Stochastic
1400
1400
1200
1200
Population size
1000
1000
800
800
600
600
400
400
0 50 100 150 200 250 0 50 100 150 200 250
Time Time
Figure: Time series for RMPP model in stable node region (α = 0.5,
β = 1, γ = 1.2).

30. Run a simulation in the limit cycle region
Deterministic Stochastic
3000
3000
2000
2000
Population size
1000
1000
500
500
0
0
0 50 100 150 200 250 0 50 100 150 200 250
Time Time
Figure: Time series for RMPP model in limit cycle region (α = 0.5,
β = 1, γ = 3.5).

31. Run a simulation in the stable focus region
Deterministic Stochastic
1500
1500
Population size
1000
1000
500
500
0 50 100 150 200 250 0 50 100 150 200 250
Time Time
Figure: Time series for RMPP model in stable focus region α = 0.5,
β = 1, γ = 2.5).

32. Detecting periodic fluctuations in simulated time
series
Node
1200
1000
800
600
100 150 200 250 300 350
Focus
Years
1800
Population size
1400
1000
600
100 150 200 250 300 350
Limit cycle
Years
2500
1500
0 500
100 150 200 250 300 350
32

33. Autocorrelation Function (ACF)
Detecting periodic fluctuations
Node Focus Limit cycle
1.0
1.0
1.0
0.8
0.5
0.5
0.6
ACF
0.0
0.4
0.0
−0.5
0.2
−0.5
0.0
0 10 20 30 40 0 5 10 15 20 25 30 0 5 10 15 20 25 30
Lag
√
Evidence of periodicity if ACF(T) > 2/ tm where T is the lag of
the dominant frequency (first maximum), tm is the number of data
points in the time series.
33

34. Power spectra
Detecting periodic fluctuations
Node Focus Limit cycle
8
8
8
q
q
q
q
q
q
q
qq
q
qqq
qq
qq
q
q
qq
q
qq q
q
6
6
6
q qq
q
qq q qq
q
qq q
qq q
qq
q
qq
qq
qq qq qq
qq qqq
qqq
qq
q
q qq q q q
q qq q
q
q
qq q q
qq
qqq q qq q q q qq q q
q q q
q qq
q qqqqq
q q qq q qq q
q q
q qq
q
qq
qq q
q q
qq qq q
q q q qq q qq q
q qq q
log10(Power)
qq q q qq q
q qq q
q q q qq q
q q qq q qqq
q q q q q q qq qq qqqq q qqq q q q
qq q q q q q q qqq qqqqq q q
q q q q q
q
q q q qq q qq q qq q q
q q qqq qqq
qq qqq qqqqq q
qq
qqq q qq q q
q q
q qq qq
qqq q
q q
q q q q q qqq q
q q q q qq qq qq qq qq q
qq qqq
q
q qq q q q
q q q q qqqqqqqq qq q q
q q q q q qq qqqq qqqq qqqq
q qqq q q
q qq qq q q qqq qq
qqqq
q qqqq
q qqqq qq
q q q qqqqqq q
q
qqqqqq
q qqq
qq q q q q q qqqqqqq qq
q q q q qqqqqqqq
q qq
q q qq q
q q q q qqqqqqqqqqqqqqqqqq
q q q qqqqq qqqqqqqqqq
q q q qq q q
q q q qq q q qq qq
q q q qq q qq qqq qqq
q q q qqq
qq qqqqqq
qqqq
q qq qq q q q
q q
q qqq qqqqqqqq q q q
q qqqqqqqqqqqqqqq
qq qqq q q qqqqqqqqqqqqqqqqqqq
q qq q qqqqqqq qqqqqqqqqq
qq qqqqqqq qq q q qq q q
qq q qq qqq
q qqqqqqq q
q
q qqqqq
q q
q
qq q q q qqqqqqqq
q qqqqqqqqqqqqqqqq q q q
qqqq qqq qqqqqq
q qq
4
4
4
q q qq
q q qq q q
q q qqqqqqq q
qqqqq q
q q qqqqqqqqq
qqqqqqqqqqqqqqqq q
qqq qqqqqqqqq
q q q qqqqq
qqqqqqqqqqq
qqq qqqqqqqqqqqqqqqqqqq q qq
qq qqqqqqqqqqqq
qqqq qqqqqq q
q q q qq q qq qq q
qq
qq q qqq qqqq
q qq q
q q qqq
q qqqq qq q q qqqqqqqqqqqqqqqqqq
q q q qqqqq
q q q qq qq
q q qq qqqqqqqqqqqqq
qqq qqq qqqqqqqqqqqqqq
qqq qqqqqqqqqqqq
q qqqqqqqqqqqq
q q q qqqqqqqqqqqqqqqq
q q q q q qq q q q
q q qqq qq qqqqq
q
qqqq
qqqqqqq
q qqqqq
q qq
q q
q q q q qqq
qq q
qq qqqqq
q qq
q
q q qqqq qqqqqqqqqqqqqqqqqqq qqqq
q qqqqqqqqqqq
qqqqqqqqqqqqqq
q qq qqqqqqqqqqqqqq
q qqqqqqqqqqqqqq q
q q q q qqqqq
qq q
q qqqqq
q q qqq q q q qq
q qq
qqqq q
q qqqqqqqqqq
qq qqqqq
q qqqqqqq
q q qqq
qq qqqqqqqqqqqqqq
q qqqqqqqqqq
q q qqqqqqqqqqqqqq q q
qqqq
q q qq qqqqqq
q qqqqqqq
qq qq q qq qq
q qq q
qq qq
qqqqq
q q
qqqqq
q
q
q q qqqqqqq qq
qqqqq q
q qq qqqqqqqqqqq
q q q qq qqqqqqqqqqqqqqq
q qq qqqqqqqqqqq
q qqqqqqq
q qq q q qqq qqqqqqqqqqqqqq
q qqqqqqqqqq
q q qqqqqqqqqqqqqq
qqqqqqq q
q qq
q q q qq
q qqqqqqqqqqqqqq
qqqq qqqqqqqq qq q
qq qq
qq
q qq
qqqq q q q qqq
q qq
qqq qq
q
q
qq
q qqqqqqqqqqqqqqqqqqqqqqqqqqq
q qqqqqqqq
qqqqq
q qqqqqqqqqq
qqqqqqqqq
qqqq
q qqqqqqq
q qq qq
q q qqqq
q qqq
q qq q qqq qqq
q qqq
q qqqq
q q qqqqq
qq qqqqqqqqqqqqqqq
qq q qqqqqqqqqqq
q q qqqqqqqqqq
qq qq qqqqqqqqqqqqqqq
qqqq
qqqqqqqq
qqqqqqqq q q q qq q qq
q q q qqqqq qqqqqqqqqqq
q q qqqqq
q qqqqqqqqqqq
q qq qqqqqq q qq
q q qqqqq
q q qqqq
q qqqqqqqqqq
q qqq qqqqqqqqq q
q
qqq
q q q q q qq
qq qq qq
qq qq
qq
q qq qqqq
q qq
q qq
qqq
q qq qqqqqqqqqqqq
q qqqqqqqqq
q qq qqqq
q qq q qqqqqqqqqqqqq
q q q qq qqqqqqqqq qqq
qqq
q q q q q qq
q qq
q qq q
q qq q qq
q
q q q q qqqqqqqqqqqqqq qqqqqq
qqqqqq
qqqqq
qqqqqqq
q qqqqqqqqqq
qqqqqqqqq
qqqqq
q qqqqqqqqqq
qqqqqq
qq qqqqqqq
q
qq qqq
qqq qq qq q q
qqqq
qqq
q q q qqqqqqqqqq
q q qq qqqqqq
q q q qqqqq
q qqqq
qqqqqq
qq
qqqqq
qq q q qqqqqqqq q q
q
q q qqqq
qqq
q qqqq
qqqqqq qqqqqqqq
q qqqqqqq
q q qqqqqqq
q q q qqqqqq
qqq
q q
q qqq
q qqq
q q qqq qq qq
qq
qq q qq
q qq q
qq qqqqq
q qqqqqq
qqq
q qqqq
qq qqqq
qq qqqqqq q q qq qq
q
q qq qq qqqq
qq qq qqqqqqqq
q q qq q qqq
q qqqqq
q qq q q qqqqqqq
q
qq
qq
qq q
q
q
q q qqq q
q
q q
qqqq
q q qqqqqqq
qqq
q q qqq
q q qqq
q q q q qqqqqq
q q qqqqqq qq q qq q
2
2
2
q qqqqq
q q qqq qqqq
q qqqqq
qq
qq q
q q q q qq qqqq
q qqq
qq q q
qq qq q qqq q q
q qqq
q q qqq q q
q qqqq qq
q qq qq
q qq
qq q qq q
qq q qqq qqq
qq
q q q
q q qq
q qq q q
q qq
q q
0
0
0
q
−4 −3 −2 −1 −3.0 −2.5 −2.0 −1.5 −1.0 −0.5 −3.0 −2.5 −2.0 −1.5 −1.0 −0.5
log10(Cycle frequency)
Evidence of periodicity: presence of dominant frequency peak
34

35. Marginal distribution
Distinguishing between quasi-cycles and noisy limit cycles
Node Focus Limit cycle
150
3000
200
2500
150
100
2000
Frequency
1500
100
1000
50
50
500
0
0
0
600 800 1000 1200 600 1000 1400 1800 0 500 1500 2500
Prey population size
Evidence of limit cycle: bimodal distribution
35