Ecological Utility Analysis: On the Emergence of Positive Interactions between Organisms in Ecosystems Bernard C. Patten 1 and Stuart J. Whipple 1,2 1 Odum School of Ecology, University of Georgia, Athens, Georgia, USA 2 Skidaway Institute of Oceanography, Savannah, Georgia, USA
A Brief History 1981 The Legović connection We had fun trying nonassociative: : 4 of 27 cases connection (,,,,,, etc.) We did show: A minimal generating set for ecological interactions was (0,) ) and (,0)( It was unlikely that adjacent and ultimate interaction types were e the same Positive nonadjacent relations emerged freely in our examples Ca. 1990 The Ulanowicz/Puccia connection (D)( Patten, B. C. 1991. Network ecology: indirect determination of the lifeenvironment relationship in ecosystems. In Higashi, M. & Burns, T. P. (eds.), Theoretical Ecosystem Ecology: The Network Perspective. London, Cambridge University Press. pp. 238-351. 19982007 Subsequent refining papers by Brian Fath, Stuart Whipple, myself
EnergyMatter Stock and Flow Models Basic Definitions Compartment open, dissipative, energymatter storage element in a system i Transaction directed energy or matter flow f between two adjacent compartments j f ij i Relation directed proximate or ultimate relationship r between two adjacent or nonadjacent compartments j r ij i Environs afferent and efferent environments of compartments within a system boundary i Utility systemwide measure of value relations transmitted between compartment pairs j i
The of Environ Analysis Mathematical Methods Different methods describe different properties of environs They employ matrix state transition equations for compartment descriptions They employ inverse matrices and matrix power series for analysis Structural analysis A Pathways: identifies, counts, and classifies pathways in networks Functional analyses B C Throughflow: maps boundary inputs z j and outputs y k into interior throughflows T i Storage: maps boundary inputs z j and outputs y k into interior stocks x i There are four basic matrices derived from F and F T in: φ : dx/dt = F1 z = F T 1 y Value analysis D Utility: measures direct, D = (d ij ), and direct plus indirect, U = (ID) 1, values (benefits and costs) conferred to compartments by their participation in networks; ordered sign pairs (sd ij,sd ji ) and (su ij,su the interactions sd ij sd ji su ij su ji ji ) give
Three ZeroSum Interactions Proximate j f out f ij ij f in ij r ij i f ij out f ij in = 0 r ij = (,) contramensalism (e. g., predation) (,) altruism (0, 0) neutralism
Nine NonzeroSum Interactions Ultimate (, ) contramensalism (, ) altruism (, ) competition (, ) mutualism (, 0) aggradation (, 0) dissipation (0, ) commensalism (0, ) amensalism (0, 0) neutralism
Questions How do we unscramble complex webs to determine ultimate interaction types in ecosystems? Is unweighted web structure enough, or must linkages be quantified? Utility analysis of community modules suggests two main answers In some cases, web topology is sufficient to specify the interaction types. The values of the internal or boundary flows have no influence. Banff National Park, Canada Reference: 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 We call this: structural determination In other cases, topology AND the values of the internal or boundary flows determine the interaction types. We call this: parametric determination
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 1.1 Canonical Form: Feeding Link Rule For adjacent compartment pairs the relation is always for the recipient and for the donor, (sd ij, sd ji ) = (su ij ) = (,) = contramensalism
Community Modules: Acyclic Case 1. Structurally Determined 3 2 1.2 Sequential Chains (any length) Rules 2 4 Adjacent predations, (sd ij, sd ji ) = (,) produce ultimate contramensalisms, (su ij ) = (,) For odd transfers > 1 between compartment pairs the relation is always ultimate contramensalism, (su ij ) = (,) 3 2 For even transfers 2 between compartment pairs the relation is always ultimate mutualism, (su ij ) =(,)
Community Modules: Acyclic Case 1. Structurally Determined 2 1 1.3 Divergent (Exploitative) Competition (extends to other relations) Rules Adjacent predations produce ultimate contramensalisms (su ij ) = (,) For odd transfers 1 between compartment pairs the ultimate relation is always competition, (su ij ) = (,) For even transfers 2 between compartment pairs the ultimate relation is always mutualism, (su ij )=(,) Other relations are also structurally determined
Community Modules: Acyclic Type 1. Structurally Determined 1.4 Convergent (Apparent) Competition (extends to other relations) Rules 1 2 Adjacent predations (sd ij, sd ji ) = (,) produce ultimate contramensalism (su ij ) = (,) For odd transfers 1 between compartment pairs the relation is always ultimate competition: (su ij ) = (,) For even transfers 2 between compartment pairs the relation is always ultimate mutualism: (su ij ) = (,) Other relations are also structurally determined Structural determination ends here
Community Modules: Acyclic Case 2. Parametrically Determined Endogenous Interaction type determined by internal flow values 2.1 Omnivory, Mixotrophy, Intraguild Predation, etc.??? Rules Divergent competition, (su ij ) = (,), at a fixed trophic level within a feeding guild??? becomes, on introduction of cross-level feeding, structurally indeterminate interaction types (su ij ) = (?,?) Mechanism Cross-linkage occurs when a link causes convergence in a divergent network, or divergence in a convergent network. The resultant graph is a lattice element
Community Modules: Acyclic Case 2. Parametrically Determined Endogenous 2.1 Divergent/Convergent Competition 2 1 This structure is also internally parametrically determined it has the lattice structure of the previous example (Fath, B. D., 2007. Network mutualism: positive community-level relations in ecosystems. Ecol. Mod., in press)
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?????? 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
Summary In community modules, interaction types between compartment pairs appear to be determined as follows: Case 1 Case 2 Acyclic graphs with no cross-level coupling are structurally determined Acyclic graphs with cross-level coupling are internally parametrically determined Case 3 Cyclic graphs are externally parametrically determined These are hypotheses pending more exhaustive exploration of cases, or where possible, statement and proof of theorems
Proximate vs. ultimate interaction types in a marsh food-web model for Okefenokee Swamp Patten, B. C. 1991 Network ecology: indirect determination of the lifeenvironment relationship in ecosystems. In Higashi, M. & Burns, T. P. (eds.), Theoretical Ecosystem Ecology: The Network Perspective. London, Cambridge University Press. pp. 238-351.
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
Okefenokee Marsh Food-Web Model Proximate to Ultimate Interaction-Type Transitions 300 proximate and 300 ultimate pairwise interactions Utility values near zero Predation Neutralism Altruism Ultimate set equal to zero (,) (0,0) (,) (nonzero-sum) Contramensalism (,) 0 0 0 0 Neutralism (0,0) 0 0 0 0 Altruism (,) 0 0 0 0 Dissipation (,0) 0 0 0 0 Competition (,) 0 0 0 0 Amensalism (0,) 3 51 10 64 Commensalism (0,) 0 23 6 49 Aggradation (,0) 45 58 3 106 Mutualism (,) 3 73 5 81 Proximate (zero-sum) 51 205 44 300
Okefenokee Marsh Food-Web Model Summary of Results Interaction signs Proximate signs Ultimate signs Number of 95 317 Number of 95 64 / ratio 1.00 4.95 Utility summary Proximate utiles Ultimate utiles Sum of utilities Sum of utilities 4914 4914 15721 3789 Benefit()/Cost() ratio 1.00 4.15
Okefenokee Marsh Food-Web Model Summary of Results Interaction signs Proximate signs Ultimate signs Number of 95 317 Number of 95 64 / ratio 1.00 4.95 Utility summary Proximate utiles Ultimate utiles Sum of utilities Sum of utilities 4914 4914 15721 3789 Benefit()/Cost() ratio 1.00 4.15
Conclusions Ecologists often speak of fixed (implying structurally determined) ultimate interactions especially competition and mutualism Nonadjacent relations in nature, however, are apparently not fixed because Cross-linkage truncates structural determination and establishes endogenous parametric determination Cycling adds exogenous parametric determination As cross-linkage and cycling are ubiquitous properties of ecological networks, and compartments, flows and linkage patterns are always changing in time, relations between organisms in ecosystems are fluid and changing also In nature's complex networks, parametric determination is universal!