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ECOL/ENGR 8560 Fall 2009 Introduction to Ecological Utility Analysis

ECOL/ENGR 8560 Fall 2009 Introduction to Ecological Utility Analysis Bernard C. Patten 1 and Stuart J. Whipple 1, 2 1 Institute of Ecology, University of Georgia, Athens, Georgia, USA 2 Skidaway Institute of Oceanography, Savannah, Georgia, USA.

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ECOL/ENGR 8560 Fall 2009 Introduction to Ecological Utility Analysis

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  1. ECOL/ENGR 8560 Fall 2009 Introduction to Ecological Utility Analysis Bernard C. Patten1and Stuart J. Whipple1, 2 1Institute of Ecology, University of Georgia, Athens, Georgia, USA 2Skidaway Institute of Oceanography, Savannah, Georgia, USA

  2. Compartmentopen, dissipative, energy–matter storage element in a system j i j i i i Transaction directed energy or matter flow f between two adjacent compartments fij Relation directed proximate or ultimaterelationship r between two adjacent or nonadjacent compartments rij Environsafferent and efferent environments of compartments within a system boundary Utility Utility system–wide measure of value relations transmitted between compartment pairs j i Energy–Matter Stocks and Flows Basic Definitions

  3. rij j i + – Zero–Sum Interactions Proximate fij rij= (+,–) contramensalism (–,+) altruism (0,0) neutralism

  4. Nonzero–Sum Interactions Ultimate (+, –) contramensalism (–, –) competition (–, +) altruism (+, +) mutualism (0, +) commensalism (+, 0) aggradation (–, 0) dissipation (0, –) amensalism (0, 0) neutralism

  5. 90° (-, +) (+, +) 180° 0° (-,-) (+, -) 270° Empirical and Theoretical Linkages in Ecology How have ecologists determined interaction types? Haskell (1947, 1949) . . . social scientist developed coaction scheme: signed interaction pairs (+, –, 0) and coaction (interaction) compass

  6. Empirical and Theoretical Linkages in Ecology (–, –) competition (+, –) predation 90° (-, +) (+, +) (–, +) altruism (+, +) mutualism (+, –) contramensalism (–, –) competition (+, 0) aggradation (0, +) commensalism (–, +) altruism (+, +) mutualism (–, 0) dissipation (0, –) amensalism (+, 0) aggradation (0, +) commensalism 180° 0° (0, 0) neutralism (–, 0) dissipation (0, –) amensalism Hodge and Arthur (1996) presented a revised schemefor ecological interactions and provided a rationalefor using the term ‘contramentalism’ for (+,-) interactions (0, 0) neutralism (-, -) (+, -) How have ecologists dealt with determining interaction types? Burkholder (1952); Odum(1953); Leary (1985);Abrams (1987); Hodge and Arthur (1996) Burkholder (1952) and Odum (1953) brought Haskell’s sign-defined interaction types to ecology Leary (1985) developed Haskell’s interaction scheme and coaction compass into an interaction theory of forest ecology Abrams (1987) discussed issues and difficulties of applying interaction types to ecological components

  7. Empirical and Theoretical Linkages in Ecology How have ecologists dealt with the issue of determining interaction types? Berlow et al. (2004) Methods for determining interaction strengths and types Theoretically- or model-based Elements of the community or Jacobian matrix Elements of the inverse Jacobian matrix Empirically-based: experiments Perturbation effects on population abundances, variability or secondary extinctions Transactions used as a direct measure of a relation Biomass flux

  8. Empirical and Theoretical Linkages in Ecology What are the major issues vexing ecologists in determininginteraction types between organisms? • 1. Defining interactions between organisms • Interaction classification problems • Mechanisms vs. effects Applicability of methods Most methods are valid only for relations (coactions) between living components 3. Indeterminacy in results of perturbation experiments . . .

  9. Empirical and Theoretical Linkages in Ecology 3. Indeterminancy in results of perturbation experiments Yodzis (1988) used the inverse of a set of ‘plausible community matrices’ to generate outcomes for all possible ‘press’ perturbation experiments • Directional Indeterminacy: • If the interaction strengths were varied within an order of magnitude, the result is directionally determined if the sign is the same in at least 95% of the cases • How are long-term press perturbation experiments affected? • In a sample of 16 empirical food webs • 27% of self effects • 52% of effects of predators on prey • 54% of effects of prey on predators • 58% of effects between competitors • were found to be directionally undetermined Environ utility theory accounts for these strange results

  10. Empirical and Theoretical Linkages in Ecology Yodzis (1988, p. 511) summarized reasons for the indeterminacy found in ecological interactions as follows: “The net effect … [of an interaction] is the sum of the direct effect plus all the individual indirect effects. It is this net effect that we observe in press perturbations ... indeterminacy (and the ‘reverse determinacy’) … can come about only if the direct component is ‘swamped’ by a stronger indirect component. Because of this prevalence of indirect effects, even our most elementary expectations for press experiments (say, that adding predators will result in fewer prey) are not to be taken for granted.” Environ utility theory demonstrates this 'swamping' effect

  11. Turnbull,WA NWR + + Rapid River, ID + 0 rij http://tolweb.org/tree?group=Greya_politella + 0 - - http://www.ucsc.edu/currents/01-02/06-24/coevolution.html http://www.dereila.ca/dereilaimages/Wildflowers5.html Empirical and Theoretical Linkages in Ecology Context dependency, indirect effects, and variability in interaction strengths and types: Abrams (1987); Thompson (1988); Abrams (1995); Thompson & Fernandez (2006) Relations between Lithophragma parviflorum (flowering herb) and Greya politella (moth) differ consistently among communities and within communities at different times (Thompson & Fernandez, 2006)

  12. Empirical and Theoretical Linkages in Ecology Does utility analysis provide possible solutions to the above four issues in the determination of ecological interaction types? Use of transactions to determine relations Quantitative and qualitative results: Magnitude of relations  utility measure Interaction types  sign pair measure Relations involving nonliving as well as living compartments can be determined Whole-system basis addresses issues of indirect effects and context dependency Context dependency in utility analysis involves the adequacy of model representation

  13. Empirical and Theoretical Linkages in Ecology Theoretical example: How interaction types change is reflected in environ utility analysis Holt and Polis (1997)Lotka-Volterra models of intraguild predation generate alternate stable equilibria, which arise, in particular, when the intraguild predator does not gain substantially from consumption of intraguild prey Utility analysis Intraguild predation community modules produce interaction types between the intraguil predator and the intraguild prey that change from contramensalism (+,-) to mutualism (+,+) when intraguild predator consumption of intraguild prey is reduced These results show a consistency that indicates they may be revealing the same fundamental characteristics of intraguild predation

  14. bacteria S. acutus commensalism Light P rij + 0 P DOM http://protist.i.hosei.ac.jp/PDB/Images/Chlorophyta/Scenedesmus/acutus/costulatus_5.html http://www.com.univ-mrs.fr/IRD/atollpol/commatoll/ukbactpl.htm Empirical and Theoretical Linkages in Ecology Empirical example: how interaction type changes are reflected in utility analysis Gurung et al. (1999) Determination of the relationship between Scenedesmus acutus and heterotrophic bacteria by variation in boundary inputs  light and phosphorus (P) competition - - Now . . . our utility-based results High light, moderate P level Low light, all P levels Utility analysis results provide examples of changes ininteraction type caused by changes in the strength ofboundary flows – “exogenous parametric determination”

  15. ABCD's The of Environ AnalysisMathematical Methods Different methods describe different properties ofenvirons They employ matrix state transition equations forobject and subject descriptions They employ matrix power series for analysis Structural analysis … APathways: identifies, counts, and classifies pathways in networks Functional analyses … BThroughflow:maps boundary inputszjand outputsyk into interior throughflowsTi CStorage:maps boundary inputszjand outputsykinto interior stocks xi Value analysis … DUtility:measures direct and indirect values (benefits and costs) conferred to objects and subjects by their participation in networks There are four basic matrices derived fromFandFTin: ’:dx/dt= F•1+ z= –FT•1– y

  16. ABCD's The of Environ AnalysisMathematical Methods direct (m = 1) boundary (m = 0) indirect (m > 1) A: I+ (A) + (A2)+ (A3)+ … + (Am) + … ∞ AT: I+ (AT) + (AT)2 + (AT)3 + … + (AT)m+ … ∞ B: I+ (I+B) + (I+B)2 + (I+B)3 + … + (I+B)m+ … =–B–1 T =–B–1 z B': I + (I+B') + (I+B')2 + (I+B')3 + … + (I+B)m +…] = –B'–1 T =–B'–1 y C: I+ (–+C) + (–+C)2 + (–+C)3 + … + (–+C)m+ … =–C–1 x =–C–1 z C': I + (–+C') + (–+C')2 + (–+C')3 + … + (–+C)m+ …] =–C'–1 x =–C'–1 y DT = B'– B Dx = C'– C ’: dx/dt = F1 + z = –FT1 – y = 0 (steady state) Am, (AT)m = # of pathways of lengthsm Pathways Throughflow Storage – = diagonal turnover rate matrix UT = (I –DT)–1T = UTT Utility Ux =(I –Dx)–1T = Uxx

  17. These interaction types are exemplified in a recent food-web study in Banff National Park, Canada Hebblewhite, M., White, C. A., Nietvelt, C. B., McKenzie, J. A., Hurd, T. E., Fryxell, J. M., Bayley, S. E., and Paquet, P. C. 2005 Human activity mediates a trophic cascade caused by wolves Ecology 86(8): 2135-2144 Images:http://www.google.com

  18. Banff Trophic Cascade Wolf + Riparian P a s s e r i n e s – + Elk – Beaver + + + – + – + Aspen Willow basal compartments Images:http://www.google.com

  19. + – – Grizzly + + Wolf + + + Riparian P a s s e r i n e s + – – – – Elk – – – Deer Beaver + + + + + + – – – – Aspen – Willow – – – – – + + + + – + + + + + Microbes + + + + now, spaghetti ! spaghetti ? Images:http://www.google.com

  20. + – – + + + + + + – – + + – – – – – + + + + + + + – – – – + In the scrambled interactive networks of established ecosystem compartments and processes, what are the determinants of ultimate interaction types between arbitrary compartment pairs? – – – – – – + + + + – + + + + + Images:http://www.google.com Question … We investigate a modular approach . . . How do we unscramble the web?

  21. Questions … How do we unscramble complex webs to determine ultimate interaction types in ecosystems? Is web structure (as in food-web theory) enough, or must linkages be quantified? Utility analysis of “community modules” suggests two main answers … Banff National Park, Canada In some cases, food web topology is sufficient to specify the interaction types. The values of the internal or boundary flows have no influence. We call this: “structural determination” In other cases, topology AND the values of the internal or boundary flows determine interaction types. We call this: “parametric determination” Images:http://www.google.com

  22. 1.1 Canonical Form: Feeding Link + – For adjacent compartment pairs the relation is always + for the recipient and – for the donor, (+,–) = contramensalism Images:http://www.google.com Community Modules: Acyclic Reference: Holt, R. D. 1997. Community modules. Chapter 17 in Gonge, A. C. and Brown, V. K. (eds.), Multitrophic Interactions in Terrestrial Systems. Blackwell Science, Ltd., Oxford, U. K., pp. 333-350. 448 pp. Case 1. Structurally Determined Interaction type determined strictly by the graph topology Rule

  23. Community Modules: Acyclic Case 1. Structurally Determined + + 1.2 Sequential Chains (any length) + + + Rules – + + + – + 3 3 2 4 2 2 + For odd transfers > 1 between compartment pairs the relation is always ultimate contramensalism (+,–) + – + – + – + – + + Images:http://www.google.com Adjacent predations produce ultimate contramensalism (+,–) For even transfers ≥ 2 between compartment pairs the relation is always ultimate mutualism (+,+)

  24. Community Modules: Acyclic Case 1. Structurally Determined 1.3 Divergent (Exploitative) Competition (extends to other relations) Rules + + + + – + Adjacent predations produce ultimate contramensalism (+,–) – – + + – – For odd transfers ≥ 1 between compartment pairs the ultimate relation is always competition (–,–) + + + – – – 2 1 For even transfers ≥ 2 between compartment pairs the ultimate relation is always mutualism (+,+) + + Other relations are also structurally determined Images:http://www.google.com

  25. Community Modules: Acyclic Type 1. Structurally Determined 1.4 Convergent (Apparent) Competition (extends to other relations) Rules + + Adjacent predations (sdij, sdji) = (+,–) produce ultimate contramensalism (suij, suji) = (+,–) + + + – – – For odd transfers ≥ 1 between compartment pairs the relation is always ultimate competition: (suij, suji) = (–,–), denoted by – – + + + – – – – – 1 2 + + + + + For even transfers ≥ 2 between compartment pairs the relation is always ultimate mutualism: (suij, suji) = (+,+), denoted by + Other relations are also structurally determined Images:http://www.google.com

  26. Community Modules: Acyclic Case 1. Structurally Determined 1.5 Divergent/Convergent Competition (scale symmetric; extends to other relations) Rules Adjacent predations again produce ultimate contramensalism (+,–) + – + – – Adjacent (1-step) transactions (divergent or convergent) yield ultimate competition (–,–) + + + – – 1 2 – Two-step transactions yield ultimate mutualism (+,+) + Images:http://www.google.com Note that the parallel pathways are of equal lengths

  27. Community Modules: Acyclic Case 1. Structurally Determined (probably) 1.6 Convergent/Divergent Competition (scale-symmetric) Rules We have not yet investigated this case Structural determination ends here

  28. ? 2.1 Divergent/Convergent (scale-asymmetric: omnivory, mixotrophy, intraguild predation, etc.) Rules ? ? ? – – + + Mechanism ? – – Unequal pathway lengths in parallel from originating to terminating compartments produce conflicting interaction types resolved by the quantitative flows ? Images:http://www.google.com Community Modules: Acyclic Case 2. Parametrically Determined—Endogenous Interaction type determined by internal flow values Divergent competition (–,–) at a fixed trophic level within a feeding guild … … becomes, on introduction of cross-level feeding, structurally indeterminate interaction types (?, ?)

  29. z1 =100 z1 =100 40 40 1 1 - 30 + 55 15 r31 = ? 15 2 30 r31 = ? 2 5 + + 15 40 45 45 3 3 Case 1 Case 2 contramensalism mutualism 3 + ↔ - 1 sU = + + + – + + – – + sU = + + + – + + + – + 3 + ↔ + 1 1 2 3 1 2 3 1 1 2 2 3 3 Intraguild Predation  Parametric Determination

  30. 3.1 Food Cycles, Biogeochemical Cycling, Microbial Loops, etc. Rules ? – ? + + ? Presence and strength of inputs to compartments in cycles determines signs for each compartment pair Mechanism ? – Dissipation constrains feedback cycles from altering established relations; inputs to compartments in cycles relax this constraint and allow exogenous parametric determination of internal relational types ? – ? + Images:http://www.google.com Community Modules: Cyclic References: Lindeman, R. L. 1942.The trophic-dynamic aspect of ecology. Ecology 28: 399-418 Redfield, A. C. 1958. The biological control of chemical factors in the environment. American Scientist 46: 205-221. Pomeroy, L. R. 1974. The ocean’s food web: a changing paradigm. BioScience 24: 499-504. Case 3. Parametrically Determined—Exogenous Interaction type determined by existence and values of boundary flows

  31. z1 =100 z1 =100 1 1 70 60 29 19 20 20 z2 =10 2 2 1 39 39 1 + - r31 = ? + r31 = ? - 3 3 39 39 mutualism competition Case 1 Case 2 3 + ↔ + 2 3 – ↔ – 2 1 2 3 1 2 3 sU = sU = 1 1 + – + + + + – + + + + + – + – – – + 2 2 3 3 Divergent Series with Cycles  Parametric Determination

  32. Link sequences of indefinite length . . . . . . Divergent sequences of indefinite length without cross-level coupling . . . . . . . . . Convergent sequences of indefinite length without cross-level coupling . . . . . . . . . . . . . . . Summary In community modules, interaction types between compartment pairs are determined as follows: Case 1 Acyclic graphs with no cross-level coupling are structurally determined. These include: A single path or multiple pathways of equal length interconnecting focal compartment pairs confers structural determination

  33. . . . Divergent link sequences of indefinite length with cross-level convergent coupling ( ) . . . . . . . . . . . . . . . Convergent link sequences of indefinite length with cross-level divergent coupling ( ) . . . . . . . . . . . . . . . Summary In community modules, interaction types between compartment pairs are determined as follows: Case 2 Acyclic graphs with cross-level coupling (by omnivory, mixotrophy, intraguild predation, etc.) are parametrically determined. These include: Unequal pathway lengths interconnecting focal compartment pairs confers parametric determination

  34. • • • environment Cycling enables any environmental input to a cycle to feed back to focal compartments ( ) and determine ultimate interaction types Summary In community modules, interaction types between compartment pairs are determined as follows: Case 3 Cyclic graphs are parametrically determined In cyclic subgraphs the upsteam–downstream distinction does not apply In this case the parametric determination shifts from endogenous to exogenous

  35. The environ utility methodology has been applied to a marsh food-web model for Okefenokee Swamp Patten, B. C. 1991 Network ecology: indirect determination of the life–environment relationship in ecosystems. In Higashi, M. & Burns, T. P. (eds.), Theoretical Ecosystem Ecology: The Network Perspective. London, Cambridge University Press. pp. 238-351. Images:http://www.google.com

  36. Okefenokee Marsh Food-Web Model Specifications 24 compartments 7 sectors: Organic Matter Microinvertebrates Nutrients Macroinvertebrates Decomposers Vertebrates Primary Producers 21% connectivity 116 links/552 possible (without loops) 44,025,553 simple paths max length 21 links mean length 15.52 links 3,953,202 simple cycles max length 20 links mean length 14.81 links Images:http://www.google.com

  37. Okefenokee Marsh Food-Web Model Proximate to UltimateInteraction-Type Transitions 300 proximate and 300 ultimate pairwise interactions Images:http://www.google.com

  38. CH-7 Network Mutualism Number of + 95 317 Nature not red in tooth and claw Number of – 95 64 CH-8 Network Synergism Nature much more green than mean Okefenokee Marsh Food-Web Model Cardinal Hypotheses 7 and 8 Ultimate signs Interaction signs Proximate signs +/– ratio 1.00 4.95 Utility summary Proximate utiles Ultimate utiles +4914 +15721 Sum of + utilities Sum of – utilities –4914 –3789 |Benefit(+)/Cost(–)| ratio 1.00 4.15 Images:http://www.google.com

  39. Conclusions Ecologists sometimes speak loosely of fixed (implying structurally determined) interactions—competition, commensalism, mutualism, etc. Relations in nature, however, are never fixed because … Cross-linkage truncates structural determination and establishes endogenous parametric determination Cycling adds exogenous parametric determination As compartments, flows and linkage patterns are always changing in time, relations between organisms in ecosystems are fluid and changing also Now . . . toward implicate order This is the significance of parametric determination—and it is universal!

  40. Theoretical Development 1.1 2 3 40 20 20 40 1 40 z1 =100 F = FT= Orientation and flow are the same Orientation and flow are opposite

  41. Theoretical Development 1.2 Donor-normalized flow intensity Recipient-normalized flow intensity B’ = B = D = B’ - B Definition of D matrix D =

  42. Theoretical Development 1.3 Power series of D matrix: D2 = D*D * D2 =

  43. Theoretical Development 1.4 = = 0 Zero terms: 0 = 0 = = = = 0 = = D2 = d11(2) = d12(2) = d13(2) = d21(2) = d22(2) = d23(2) = d31(2) = d32(2) = d33(2) =

  44. Theoretical Development 1.5 D2 = D2 =

  45. Theoretical Development 1.6 - - 2 2 3 3 1 1 d32(2)= - 0.2 d23(2) = d32(2) = - 0.4 d23(2) = This produces the competition (-,-) relationship for (2,3) and (3,2)

  46. Theoretical Development 1.7 - d11(2) = 2 3 1 - 0.6 d11(2) =

  47. Theoretical Development 1.8 - d22(2) = 2 3 1 - 0.4 d22(2) =

  48. Theoretical Development 1.9 - d33(2) = 2 3 1 - 0.2 d33(2) = The same reasoning applies to all relational orders

  49. Theoretical Development 2.0 A AT= A* A* or AU AT A = Now . . . inescapable holism Directed forward Directed backward Bi-directed adjacency matrix used to generate utility sequences

  50. Network Enfolding Implications for Observations, Experiments, and Models of Ecosystems Stuart J. Whipple Bernard C. Patten

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