Theoretical community ecology

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Theoretical community ecology P. A.
Rossignol FW-OSU
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Modeling complex systems(Puccia and Levins 1985)

• Nature cannot be made uniform
• Conflicting interests and goals
• Some important variables will never be
  quantifiable or measurable
• Complete description of complex systems beyond
  our time frame or funding
• Quantification not always necessary or valuable

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• What do we want from Nature?
• Understand
• Predict
• Modify
• What can mathematics provide?
• Generality
• Precision
• Realism

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our truth is the intersection of independent
lies.
Theor. Comm. Ecol.
PRECISION
GENERALITY
REALISM
Richard Levins
PRECISION
PRECISION
GENERALITY
REALISM
GENERALITY
REALISM
Mechanistic/Stats models
Resource mgt models
Levins Am. Sci. 1965
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Community Ecology and Ecosystem Ecology
Ecosystem ecology Study of flows between
compartments. Emphasis on physical/chemical
aspects (hydrology, carbon, energy, functional
ecology etc) Community ecology Study of
Darwinian interactions between species
(predator-prey, competition, press perturbation,
natural selection etc)
Community ecology (living)
Ecosystem ecology (environment)
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Who says they do Community Ecology and who
actually does it?
Kareiva Ecology 1994
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Basic concepts

• Lotka-Volterra model
• Community matrix
• Stability
• Eigenvalues, eigenvectors, isoclines
• Diversity/stability paradox
• Predicting the effects of perturbations
• Turnover

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LOTKA-VOLTERRA EQUATIONS
Per Capita Change in PREY???N1 births - a12N2
Per Capita Change in PREDATOR ??N2 a21N1 -
deaths

• LAWS OF
• MASS ACTION
• Matter Constant
• Energy Flows and Degrades

Alfred Lotka 1925
Vito Volterra 1926
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dx rx (1- x) - axy dt K
PREY
How does a simple system behave?
dy bxy - dy dt
PREDATOR
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dx rx (1- x) - axy dt K
Phase plane
PREY
dy 0 dt
dy bxy - dy dt
PREDATOR
r a
Prey isocline
Equilibrium
dx 0 dt
Predator isocline
Equilibrium
Prey, x
K
b d
Predator, y
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dx rx (1- x) - axy dt K
PREY
Qualitatively stable
dy 0 dt
dy bxy - dy dt
PREDATOR
r a
Prey isocline
Stable equilibrium
dx 0 dt
Predator isocline
Prey, x
K
b d
Predator, y
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dx rx (1- x) - axy dt K
PREY
Quantitative behavior
dv2 0 dt
dy 0 dt
l(max)
dy bxy - dy dt
PREDATOR
eigenvector
r a
Prey isocline
l(min)
dv1 0 dt
dx 0 dt
Predator isocline
Prey, x
K
b d
Predator, y
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dx rx (1- x) - axy dt K
PREY
dv2 0 dt
dy 0 dt
dy bxy - dy dt
l(max)
PREDATOR
eigenvector
r a
Trajectory Return time µ 1/Re(l(min))
Prey isocline
l(min)
dv1 0 dt
dx 0 dt
Predator isocline
Prey, x
K
b d
Predator, y
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t
N
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What is the community matrix? Let us assume that
we observe a three-species predator-prey trophic
chain N1, N2, N3where N1 exhibits
intra-specific competition andN2 and N3 are
totally dependent on prey N1 and N2,
respectively, and with stable equilibrium
levels of N1 800 N2 100 N3 80
Corresponding to the Lotka-Voltera equations
dN1 k1N1 a11N1N1 a12N1N2 dtdN2
a21 N1N2 a23N2N3 dtdN3 a32N2N3
k3N3 dt
N3 N2 N1
N
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Over a determined period of time,
density-dependent changes observedfor the
variables are such that 80 of N1 die due to
interaction with other N1 20 of N1 die due to
predation by N2 16 of N2 are born from preying on
N1 16 of N2 die to predation by N3 2 of N3 are
born from preying on N2 Tabulate these numbers
as follows (creating a matrix)
dN1 k1N1 a11N1N1 a12N1N2 dtdN2
a21 N1N2 a23N2N3 dtdN3 a32N2N3
k3N3 dt
due to interaction with N1 N2 N3
N1 Change in N2
N3
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The values (e.g. -80) are for the whole
population. We would like a general
representation of the system, independent of
density. Given equilibrium values, N1 800 N2
100 N3 80
D

D is the interaction matrix
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These matrices are simply another way of
representingthe Lotka-Volterra equations, where
each element of the interaction matrix
corresponds to a parameter in the L-V
equations dN1 k1N1 -.00013N1N1 -
.00025N1N2 dtdN2 .0002 N1N2 - .002N2N3
dtdN3 .00025N2N3 k3N3 dt
N3 N2 N1
D
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The Jacobian matrix is
J
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that at equilibrium simplifies to
J
N3 N2 N1
and that can be expressed numerically as
The above, J, is the most widely accepted
definition of the community matrix and was
proposed by May (1973)
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It is not however Levins original 1968
definition, which was the Jacobian of the per
capita equations (following Lotka-Volterras
formulation), which in this case would be the
same as D (above), the matirx of interaction
coefficients He later represented the community
matrix, A, in terms of signs only,
A
N3 N2 N1
or sometimes symbolically,
which corresponds to a signed digraph
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• Two major practical questions
• 1) Is the system stable?
• If determining the quantities of eigenvalues is
  not practical, a qualitative evaluation may be
  possible. We can assess from Hurwitzs theorem
  whether or not the system can satisfy conditions
  for stability
• 2) How do the variables vary following a press
  perturbation?
• Applying Cramers rule, we can assess direction
  of change to equilibrium levels
• Press perturbation permanent leads to new
  equilibria
• Pulse perturbation one time leads to return to
  original equilibria

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1) STABILITY

• 120 definitions in ecology, 70 distinct (Grimm
  Wissell 1997 Oecologia)
• Mathematically, ability to return to
  equilibrium following a local disturbance
  (Logofet 1993 reviews a number mathematical
  definitions)
• Generally reducible to the Routh-Hurwitz
  criteria

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Aleksandr Lyapunov 1892 The General Problem
of the Stability of Motion
Characteristic Equation ?n F1 ?n-1 F2 ?n-2
Fn 0 Roots (?) with Negative Real Parts
N3 N2 N1
Qualitatively, the characteristic eq.
det
leigenvalues Fn feedback
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What is an eigenvalue?

• Technically, eigenvalues are the roots of the
  characteristic polynomial
• In population biology, eigenvalues are the
  solution to Eulers equation (a specific
  characteristic polynomial)
• The best known eigenvalue is population growth.
  For population stability, one eigenvalue must
  have a positive real only solution. Stability
  occurs when all age stages reach a constant ratio
  (i.e. age pyramid is constant even though
  population may be growing or declining)
• In community ecology, all eigenvalues must have
  negative real parts for stability
• In community ecology, a common stability
  criterion is return time, the inverse of the
  largest (closest to zero) real part
• Note coefficients of the characteristic
  polynomial are the feedback cycles of the system

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What happens when the system is not quantifiable?

• The standard ecological approach to stability is
  to evaluate the Routh-Hurwitz Criteria, which
  are redundant
• All coefficients (feedback levels) of
  characteristic polynomial are the same sign
  (negative in ecology) necessary but not
  sufficient
• Hurwitz determinants are positive necessary and
  sufficient

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Adolf Hurwitz 1895
F0lnF1ln-1Fnl0 0 det
Not intuitive, but a measure of imbalance
between feedback cycles (overcorrection)
D2
gt0
Hurwitz determinant(s)
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Hurwitzs (1895) Principal Theorem

• Proposed Hurwitz Criteria and discovery of two
  behaviors Dambacher, Luh, Li Rossignol. Am.
  Nat. (2003)
• (i) Polynomial coefficients F0, F1, F2, . . . ,
  Fn are all of the same sign
• Class I models (tend to fail due to lack of
  negative feedback)
• (ii) Hurwitz determinants ?2, ?3, ?4, . . . ,
  ?n-1 are all positive, where p0 1
• Class II models (tend to fail due to
  overcorrection)

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Stability-Diversity Paradox

• We observe great complexity and diversity
  (Elton), supported by ecosystem persistence and
  stability (MacArthur)
• Based on mathematics of evolutionary theory,
  however, we are led to conclude that stability
  decreases with increasing diversity (May,
  Levins), hence a paradox between stability and
  divesity (Goodman 1975). The paradox was stated
  most famously by Hutchinson (1961) as the
  paradox of the plankton
• Eltonian perspective Natural history suggests
  that diversity is stabilizing (Elton 1927, 1958).
  Most ecologists are Eltonian at heart
  (Schoener)
• Food Web Theory Pimms proposal to resolve the
  paradox and to reconcile community ecology theory
  with ecosystem studies

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Stability Criteria
F3 - a11 a23 a32 a31 a23 a12 - a33 a12 a21 F2
- a23 a32 - a11 a33 - a12 a21 F1
- a33 - a11 F0
-1
Ambiguity if a31 is too strong, system is
unstable
i)
ii)
F1 F3 -1 F2
F1F2 F3 gt 0
gt 0
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2) PREDICTIONS
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The Jacobian or community matrix is useful
because the system can be generalized as follows,
A.N -k -A-1.k N (Cramers rule)
and we can apply Cramers rule for press
perturbed equilibria
Gabriel Cramer 1750
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Economists (Quirk, Rupert, Maybee, Hale, Lady
etc, based on Samuelson) demonstrated that one
can reformulate the system in terms of
qualitative values and eventually derive
qualitative predictions
A.N -k
N3 N2 N1
A
Press perturbation Read direction of change down
a column
and the inverse will indicate the qualitative
direction of change -A-1.k N
(-A)-1
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But qualitative predictions were generally
ambiguous and often did not match quantitative
predictions

• R protozoa
• B bacteria
• Z zooplankton

4) P phytoplankton 5) N nutrients
Qualitative analysis(ambiguous predictions)
STONE 1990
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SYMBOLIC ANALYSIS OF ADJOINT MATRIX
1
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a22 a33 a44 a55 a22 a33 a45 a54 a22 a43 a34
a55 a52 a33 a24 a45 a52 a33 a25 a44 a52 a43
a25 a34 a22 a53 a34 a45 a21 a53 a34 a45 a51
a33 a24 a45 a51 a33 a25 a44 a51 a43 a25 a34
a21 a33 a44 a55 a21 a33 a45 a54 a21 a43 a34
a55 a21 a52 a34 a45 a51 a22 a34 a45 a21 a52
a33 a45 a51 a22 a33 a45 a21 a52 a33 a44 a21
a52 a43 a34 a51 a22 a33 a44a51 a22 a43 a34
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Dambacher, Li Rossignol Ecology 2002
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Predicting changes in life expectancy

• Common estimation in system ecology or single
  population studies, but not in community ecology.
  No procedure was available

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LIFE EXPECTANCY CHANGE IN PERTURBED COMMUNITIES
Dambacher, Levins Rossignol Mathematical
Biosciences 2005
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Overall theoretical developments at OSU

• Algorithms -graphical programs (Cleverset
  Comp. Sc. DAmbrosio) with Maple program
  (Dambacher et al) -website available in 2006
  (Hans Luh P. Rossignol)
• Predicting ambiguous responses
  -weighted-feedback metrics (Ecology 2002)
• Recent tests and validation of qualitative
  analysis by outside researchers Hulot et al.
  2000. Functional diversity governs ecosystem
  response to nutrient enrichment. Nature
  405340-344 Ramsay Veltman. 2005. Predicting
  the effects of perturbations on ecological
  communities. J. Anim. Ecol. 74905-916
• Hurwitz theorem on stability -resolve redundancy
  and classify system responses (Am. Nat. 2003)
• Life expectancy -develop algorithm for
  predicting changes (Math. Biosc. 2005)
• Effect of press extends only three links away
  (Dambacher Rossignol SIGSAM 2001, Berlow et al
  2004)

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Some Applications

• Analyze systems in literature Danish shallow
  lakes (Jeppesen 1998) Old field systems
  (Schmitz 1997) Plankton system (Stone 1990)
  Freshwater pelagic (McQueen et al 1989)
  Mosquito ecology (Wilson et al 1990)
• Novel analyses Salmon toxicology (Can. J. Fish.
  Aq. Sc. 2004) Lyme disease ecology (Tr. Roy.
  Soc. Trop. Med. Hyg. 2004) West Nile virus
  ecology (Risk Analysis in press)

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Eutrophication in Shallow Danish
Lakes THEN ? ? ? ? ? ?
? ? ? NOW
Mesotrophic State ? Eutrophic State
JEPPESEN 1998
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Eutrophic Shallow Lake (Jeppesen 1998)
Dambacher, Li Rossignol. Ecology 2002
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Examples of matching predictions -Plant eating
ducks go down -Cyprinids go up
adjoint
weighted predictions
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Specific application Lyme disease prediction
model (system described by Ostfeld et al. 1996)
Tools for system analysis Powerplay program
allows drawing and quantification (DAmbrosio
students, Comp. Sc. OSU)

Maple program (Dambacher, Li and Rossignol 2002)
evaluates stability criteria and generates
predictions
Orme-Zavaleta Rossignol Trans. R. Soc. Trop.
Med. Hyg. 2004
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Specific application Lyme disease risk
assessment model Changes in abundance Changes
in life expectancy
Analysis predicts changes in vectorial capacity
following El Nino events
Basic reproduction rate mba2pnqr(-logep)-1(-loge
q)-1 If BRR gt1, then disease is epidemic
Orme-Zavaleta Rossignol Trans. R. Soc. Trop.
Med. Hyg. 2004
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Loop Group

• Colin Brown (Emeritus, OSU-Env. Eng.)
• Bruce DAmbrosio (OSU-Comp. Sc./Cleverset)
• Pete Eldridge (EPA)
• Selina Heppell (OSU-FW)
• Geoff Hosack (OSU-FW)
• Jane Jorgensen (www.cleverset.com)
• Hiram Li (USGS/OSU-FW)
• Michael Liu (OSU-FW)
• Hans Luh (OSU-Forestry)
• Matt Mahrt (OSU-FW)
• Peter McEvoy (OSU-Botany)
• Lea Murphy (OSU-Math)
• Jennifer Orme-Zavaleta (EPA)
• Grant Thompson (NOAA/OSU-FW)

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