Theoretical community ecology P. A. Rossignol F&WOSU. 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
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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
Resource mgt models
Mechanistic/Stats models
Levins Am. Sci. 1965
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 (predatorprey, competition, press perturbation, natural selection etc)
Community
ecology
(living)
Ecosystem
ecology
(environment)
Papers in Ecology 1981


1990
1990
(1,253 papers)
(1,253 papers)
Number of Papers
Number of Papers
Number of Species Considered
Number of Species Considered
Who says they do ‘Community Ecology’ and who actually does it?
Kareiva Ecology 1994
Per Capita Change in PREY:N1 = births  a12N2
Per Capita Change in PREDATOR: N2 = +a21N1  deaths
Alfred Lotka
1925
Vito Volterra
1926
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’
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’
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’
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’
N*
Let us assume that we observe a threespecies predatorprey trophic chain: N1, N2, N3where N1 exhibits intraspecific 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 LotkaVoltera equations
dN1 = k1•N1 –a11•N1•N1 – a12•N1•N2 dtdN2 = a21 •N1•N2 – a23•N2•N3 dtdN3 = a32•N2•N3 – k3•N3 dt
N3
N2
N1
N* =
Over a determined period of time, densitydependent 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 = k1•N1 –a11•N1•N1 – a12•N1•N2 dtdN2 = a21•N1•N2 – a23•N2•N3 dtdN3 = a32•N2•N3 – k3•N3 dt
due to interaction with N1 N2 N3
N1
Change in N2 N3
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
These matrices are simply another way of representingthe LotkaVolterra equations, where each element of the interaction matrix corresponds to a parameter in the LV equations
dN1 = k1•N1.00013•N1•N1 .00025•N1•N2dtdN2 =.0002•N1•N2 .002•N2•N3dtdN3 =.00025•N2•N3– k3•N3dt
N3
N2
N1
D =
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)
It is not however Levins’ original 1968 definition, which was the Jacobian of the per capita equations (following LotkaVolterra’s 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
Two major practical questions:
The General Problem of
the Stability of Motion
Characteristic Equation
n + F1 n1 + F2 n2 …+ Fn= 0
Roots () with Negative Real Parts
N3
N2
N1
Qualitatively,
the characteristic eq. =
det
l=eigenvalues Fn: feedback
F0ln+F1ln1+…+Fnl0 = 0
det
Not intuitive, but a measure of
imbalance between feedback cycles (overcorrection)
D2 =
>0
Hurwitz determinant(s)
Hurwitz’s (1895) Principal Theorem
Proposed “Hurwitz Criteria” and discovery of two behaviorsDambacher, 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 Δ2, Δ3, Δ4, . . . , Δn1 are all positive, where p0 = +1
Class II models (tend to fail due to overcorrection)


Stability Criteria
F3 =  a11a23a32 +a31a23a12 a33a12a21
F2 =  a23a32 a11a33 a12a21
F1 =  a33 a11
F0 = 1
Ambiguity: if a31 is too strong, system is unstable
i)
ii)
F1 F3
1 F2
F1F2 + F3 >0
>0
The Jacobian or community matrix is useful because the system can be generalized as follows,
A.N* = k
A1.k = N*
(Cramer’s rule)
and we can apply Cramer’s rule for press perturbed equilibria
Gabriel Cramer
1750
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
A1.k = N*
(A)1 =
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COMMUNITY MATRIX
INVERSE MATRIX


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But qualitative predictions were generally ambiguous and often did not match quantitative predictions
4) P: phytoplankton
5) N: nutrients
Qualitative analysis(ambiguous predictions)
STONE1990
a12
0
0
0
a21
a22
0
a24
a25
A=
0
a33
a34
0
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a43
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a45
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SYMBOLIC ANALYSIS
OF ADJOINT MATRIX

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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 a44+a51 a22 a43 a34
a12
0
0
0
a21
a22
0
a24
a25
A=
0
a33
a34
0
0
0.7
0.5
0.5
0.3
1
0
0
a43
a44
a45
a55
a52
a53
a54
a51
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3
3
÷
=
1
0
0.50
0.5
1
2
2
0
2
8
4
4
2
2
2
WEIGHTED
PREDICTIONS MATRIX
2
2
0
2
6
0
4
2
2
2
1
0
0
0.5
1
ADJOINT
MATRIX
TOTAL
FEEDBACK MATRIX
4
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Dambacher, Li & Rossignol Ecology 2002
LIFE EXPECTANCY CHANGE IN PERTURBED COMMUNITIES
Dambacher, Levins & Rossignol
Mathematical Biosciences 2005
Recent tests and validation of qualitative analysis by outside researchers:Hulot et al. 2000. Functional diversity governs ecosystem response to nutrient enrichment. Nature 405:340344Ramsay & Veltman. 2005. Predicting the effects of perturbations on ecological communities. J. Anim. Ecol. 74:905916
Eutrophication in Shallow DanishLakes
THEN NOW
Mesotrophic State Eutrophic State
JEPPESEN 1998
Examples of matching predictions
Plant eating ducks go down
Cyprinids go up
adjoint
weighted
predictions
Lyme disease prediction model
(system described by Ostfeld et al. 1996)
Tools for system analysis:
Powerplay program allows drawing and quantification
(D’Ambrosio & students, Comp. Sc. OSU)
Maple program (Dambacher, Li and Rossignol 2002) evaluates stability criteria and generates predictions
OrmeZavaleta & Rossignol Trans. R. Soc. Trop. Med. Hyg. 2004
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(logeq)1If BRR >1, then disease is epidemic
OrmeZavaleta & Rossignol Trans. R. Soc. Trop. Med. Hyg. 2004