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Theoretical community ecology P. A. Rossignol F&W-OSU. 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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slide1

Theoretical community ecology

P. A. Rossignol

F&W-OSU

modeling complex systems puccia and levins 1985
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
slide3
What do we want from Nature?
  • Understand
  • Predict
  • Modify
  • What can mathematics provide?
  • Generality
  • Precision
  • Realism
slide4

“…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

community ecology and ecosystem ecology
“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)

slide6

Papers in Ecology 1981

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

basic concepts
Basic concepts
  • Lotka-Volterra model
  • Community matrix
  • Stability
  • Eigenvalues, eigenvectors, isoclines
  • Diversity/stability paradox
  • Predicting the effects of perturbations
  • Turnover
slide8

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

slide9

dx = rx (1- x) - axy

dt K

PREY

How does a simple

system behave?

dy = bxy - dy

dt

PREDATOR

slide10

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’

slide11

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’

slide12

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’

slide13

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’

slide14

t

N*

slide15

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 = 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* =

slide16

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 = 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

slide17

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

slide18

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 = k1•N1-.00013•N1•N1- .00025•N1•N2dtdN2 =.0002•N1•N2- .002•N2•N3dtdN3 =.00025•N2•N3– k3•N3dt

N3

N2

N1

D =

slide20

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)

slide21

It is not however Levins’ original 1968 definition, which was the Jacobian of the per capita equations (following Lotka-Volterra’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

slide22

Two major practical questions:

  • 1) Is the system ‘stable’?
  • If determining the quantities of eigenvalues is not practical,aqualitative evaluation may be possible. We can assess from Hurwitz’s theorem whether or not the system can satisfy conditions for stability
  • 2) How do the variables vary following a press perturbation?
  • Applying Cramer’s 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
1 stability
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’
slide24

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

l=eigenvalues Fn: feedback

what is an eigenvalue
What is an eigenvalue?
  • Technically, eigenvalues are the roots of the characteristic polynomial
  • In population biology, eigenvalues are the solution to Euler’s 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
what happens when the system is not quantifiable
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
slide27

Adolf Hurwitz 1895

F0ln+F1ln-1+…+Fnl0 = 0

det

Not intuitive, but a measure of

imbalance between feedback cycles (overcorrection)

D2 =

>0

Hurwitz determinant(s)

slide28

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, . . . , Δn-1 are all positive, where p0 = +1

Class II models (tend to fail due to overcorrection)

stability diversity paradox
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: Pimm’s proposal to resolve the paradox and to reconcile community ecology theory with ecosystem studies
slide30

+

-

-

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

slide32

The Jacobian or community matrix is useful because the system can be generalized as follows,

A.N* = -k

-A-1.k = N*

(Cramer’s rule)

and we can apply Cramer’s rule for press perturbed equilibria

Gabriel Cramer

1750

slide33

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 =

slide34

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COMMUNITY MATRIX

INVERSE MATRIX

-

-

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-

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-

-

-

.6

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.5

1

0.5

0.1

0.2

0.3

0.7

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)

STONE1990

slide35

a11

a12

0

0

0

a21

a22

0

a24

a25

A=

0

a33

a34

0

0

æ

æ

ö

ö

0

0

a43

a44

a45

ç

ç

÷

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a55

a52

a53

a54

a51

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ø

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5

2

2

1

3

1

2

2

1

3

2

0

4

2

2

2

0

0

2

2

4

0

4

0

4

SYMBOLIC ANALYSIS

OF ADJOINT MATRIX

-

1

-

-

æ

ö

1

.6

0

0

0

0.9

0.4

0.05

0.06

0.2

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0.2

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-A-1=

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-

0

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1

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0

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1.0

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1

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1

0.5

0.1

0.2

0.3

0.7

+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

slide36

a11

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

5

7

2

4

4

2

3

1

3

3

0.1

0.5

0.5

0.3

1

7

1

2

4

4

2

3

1

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

4

0

4

6

4

6

0

4

4

1

0

0.7

0

1

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-

0.9

0.4

0.05

0.06

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-

0.2

0.7

0.09

0.09

0.4

-

-A-1 =

-

0.05

0.01

1.0

0.2

0.07

-

-

0.2

0.06

0.09

0.8

0.4

ç

æ

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0.5

0.1

0.2

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0.7

Dambacher, Li & Rossignol Ecology 2002

predicting changes in life expectancy
Predicting changes in life expectancy
  • Common estimation in system ecology or single population studies, but not in community ecology. No procedure was available
slide38

LIFE EXPECTANCY CHANGE IN PERTURBED COMMUNITIES

Dambacher, Levins & Rossignol

Mathematical Biosciences 2005

overall theoretical developments at osu
Overall theoretical developments at OSU
  • Algorithms -graphical programs (Cleverset & Comp. Sc. – D’Ambrosio) 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 405:340-344Ramsay & Veltman. 2005. Predicting the effects of perturbations on ecological communities. J. Anim. Ecol. 74:905-916

  • 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)
some applications
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)
slide41

Eutrophication in Shallow DanishLakes

THEN    NOW

Mesotrophic State  Eutrophic State

JEPPESEN 1998

slide42

Eutrophic

Shallow Lake

(Jeppesen 1998)

Dambacher, Li & Rossignol. Ecology 2002

slide43

Examples of matching predictions

-Plant eating ducks go down

-Cyprinids go up

adjoint

weighted

predictions

slide44

Specific application:

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

Orme-Zavaleta & Rossignol Trans. R. Soc. Trop. Med. Hyg. 2004

slide45

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(-logeq)-1If BRR >1, then disease is epidemic

Orme-Zavaleta & Rossignol Trans. R. Soc. Trop. Med. Hyg. 2004

loop group
Loop Group
  • Colin Brown (Emeritus, OSU-Env. Eng.)
  • Bruce D’Ambrosio (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)