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

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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 • Complete description of complex systems beyond our time frame or funding • Quantification not always necessary or valuable**What do we want from Nature?**• Understand • Predict • Modify • What can mathematics provide? • Generality • Precision • Realism**“…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”**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)**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**• Lotka-Volterra model • Community matrix • Stability • Eigenvalues, eigenvectors, isoclines • Diversity/stability paradox • Predicting the effects of perturbations • Turnover**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**dx = rx (1- x) - axy**dt K PREY How does a simple system behave? dy = bxy - dy dt PREDATOR**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’**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’**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’**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’**t**N***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* =**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**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**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 =**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 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**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**• 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’**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?**• 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?**• 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**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)**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**• 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**+**- - 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 -A-1.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 -A-1.k = N* (-A)-1 =**æ**ö ? ? ? ? + ç ÷ ? ? ? ? + ç ÷ ç ÷ + ? ? ? + ç ÷ + ? ? ? + ç ÷ è ø + ? ? ? + COMMUNITY MATRIX INVERSE MATRIX - - æ ö 1 .6 0 0 0 0.9 0.4 0.05 0.06 0.2 ç ÷ - - - .6 1 0 .1 .6 0.2 0.7 0.09 0.09 0.4 - ç ÷ ç ÷ - - 0 0 1 .2 0 0.05 0.01 1.0 0.2 0.07 ç ÷ - - - - 0 0 .2 1 .5 0.2 0.06 0.09 0.8 0.4 ç ÷ è ø - - - - - .6 .6 .2 .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**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 ÷ ÷ a55 a52 a53 a54 a51 ç ç ÷ ÷ ç ç ÷ ÷ ç ç ÷ ÷ ç ç ÷ ÷ è è ø ø 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 ç ÷ - - - .6 1 0 .1 .6 0.2 0.7 0.09 0.09 0.4 - ç ÷ -A-1= ç ÷ - - = - 0 0 1 .2 0 0.05 0.01 1.0 0.2 0.07 ç ÷ - - - - 0 0 .2 1 .5 0.2 0.06 0.09 0.8 0.4 ç ÷ è ø - - - - - .6 .6 .2 .5 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**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 è æ - 0.9 0.4 0.05 0.06 0.2 ç - 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 è - - 0.5 0.1 0.2 0.3 0.7 Dambacher, Li & Rossignol Ecology 2002**Predicting changes in life expectancy**• Common estimation in system ecology or single population studies, but not in community ecology. No procedure was available**LIFE EXPECTANCY CHANGE IN PERTURBED COMMUNITIES**Dambacher, Levins & Rossignol Mathematical Biosciences 2005**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**• 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)**Eutrophication in Shallow DanishLakes**THEN NOW Mesotrophic State Eutrophic State JEPPESEN 1998**Eutrophic**Shallow Lake (Jeppesen 1998) Dambacher, Li & Rossignol. Ecology 2002**Examples of matching predictions**-Plant eating ducks go down -Cyprinids go up adjoint weighted predictions**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**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**• 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)

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