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Modeling food-web dynamics

Modeling food-web dynamics. The time evolution of species’ biomasses in a food web: • Basal species exhibit exponential growth bounded by a carrying capacity • All other species grow according to their feeding rates, feeding preferences, and assimilation efficiencies

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Modeling food-web dynamics

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  1. Modeling food-web dynamics The time evolution of species’ biomasses in a food web: • Basal species exhibit exponential growth bounded by a carrying capacity • All other species grow according to their feeding rates, feeding preferences, and assimilation efficiencies • All species lose energy according to their metabolic rates and the rate at which they are consumed • Functional responses determine how feeding rates vary with the abundance of predator and/or prey species (Based on a 3-species food-chain model proposed by Yodzis and Innes 1992 American Naturalist)

  2. n ( ) å Bi’(t) = Gi(B) – xi Bi (t) + xi yijαij Fij (B) Bi (t) – xj yjiαji Fji (B) Bj (t) / eji j =1 Rate of change = Production rate – Loss of biomass + Gain of biomass – Loss of biomass to in biomass if species i is basal to metabolism from resource spp. consumer spp. Nonlinear bioenergetic ecosystem model The variation of Bi, the biomass of species i, is given by: What factors allow persistence of species in dynamical models of complex food webs? (the “devious strategies”)

  3. G ( B ) i 3 species parameters: : production rate of basal speciesi(Mass/Time) For primary producers, Gi (B) = ri Bi (t) (1 – Bi (t) / K i ), where ri : intrinsic growth rate of species i(1/Time) Ki : carrying capacity of species i(Mass) ______________ xi : mass-specific metabolic rate of species i(Mass/Time * 1/Mass) 4 species interaction parameters: eji : assimilation efficiency of species j consuming species i(fraction of biomass) yij : rate of maximum biomass gain by species i consuming j normalized by metabolic rate of species i (Mass/Time / Mass/Time) αij : relative preference of species ifor species j(fraction of diet) (αij= 0 for producers and sums to 1 for consumers) Fij (B) : non-dimensional functional response (based on parameters q or c) (relative consumption rate of predator species i consuming prey species j as a fraction of the maximum ingestion rate; function of species’ biomass)

  4. Parameterized Functional responses Type II (dominates nonlinear population dynamics modeling; q or c = 0) - ƒ(prey density) - function of predator search and prey handling times Type III (q = 1) - ƒ(prey density) - predation on low-density prey relaxed; successful food searches increases predator’s search effort Predator Interference (c = 1) - ƒ(prey & predator densities) - increase in predators decreases predation due to interference among predators - matches empirical data much better than Type II (Skalski & Gilliam 2001)

  5. q = 1 c = 0 c = 1 B ( t ) Addition of Predator Interference to Type II Functional Response: j = F ( ) B ij n å a + + B ( t ) ( 1 c B ( t )) B ik k ij i 0 ji = k 1

  6. 3-species dynamics & functional response Slight increases in Predator Interference or Type III stabilize dynamics chaotic dynamics  period doubling reversals  stabilization of limit cycles  stable stationary solution Type III Predator Interference biomass local min/max (top predator) local min & max functional response parameters When c or q = 0, the functional response is Type II

  7. biomass min & max 10-species dynamics & functional response Strong Type II FR may stress dynamics by increasing feeding on rarer species while decreasing it on more abundant species. At q = 0 (conventional strong Type II response), only 4 taxa display persistent dynamics. At q > 0.15 (very weak Type III response), all 10 taxa are persistent. At q > 0.3 (weak Type III response), all 10 taxa are steady-state. functional response

  8. Holling Type II / III 0.7 0.6 0.5 0.4 Robustness 0.3 0.2 0.1 0 0 0.05 0.1 0.15 0.2 0.25 0.3 q Stabilization of Dynamics of Ecological Networks (S=30, C=0.15) with Functional Responses Beddington-DeAngelis Predator Interference 0.7 Niche model Niche model 0.6 0.5 Cascade model 0.4 Cascade model  Effects of Structure on Dynamics  0.3 0.2 0.1 Random model Random model 0 0 0.4 0.8 1.2 1.6 2 c  Effects of Dynamics on Structure 

  9. (a) C0 = 0.15 Back to May’s (1973) Stability criteria: i(SC)1/2<1 Linear vs. Hyperbolic De-stabilization of 30-Species Dynamics Due to increases in Diversity and Complexity (b) S0 = 30

  10. Dynamical model Niche model Random deletion of consumers Effectsof Dynamics on Structureq=0.2, c=0robust niche webs have: (A) consumers at lower trophic levels,(B) more basal species,and (C) higher fractions of herbivores B A C

  11. Effects of Omnivore Feeding Preference among Trophic Levels 0.7 q = 0.2 0.6 0.5 0.4 q = 0 Robustness 0.3 0.2 0.1 0 0.1 1 10 Skewness High Trophic-level Prey Low Trophic-level Prey

  12. Factors increasing overall species persistence • Non-type II functional responses • stabilizes chaotic & cyclic dynamics • - more ecologically plausible & empirically supported • Non-random network topology • - especially empirically well-corroborated niche model structure • Decreasing S & C • supporting May’s early analyses • but not fatal to persistence of diverse, complex networks • Consumption weighted to low trophic levels • - eat low on the food chain!

  13. Current & future directions • Non-uniform distributions of functional responses, prey preferences, etc. • Allometric scaling: distribution of metabolic parameters (Body Size!) • Add Nutrients etc. and conduct Invasion & extinction experiments • Data: Coupled human/natural systems (e.g., fisheries) • Ecoinformatics: Webs on the Web (WOW) • Large Diverse Complex Networks need Collaboration • Database: Who eats Whom, Functional Responses, Metabolic Parameters, • Analysis and Visualization

  14. This work was supported by National Science Foundation grants: Scaling of Network Complexity with Diversity in Food Webs  Effects of Biodiversity Loss on Complex Communities:  A Web-Based Combinatorial Approach Webs on the Web: Internet Database, Analysis and Visualization of Ecological Networks Science on the Semantic Web: Prototypes in Bioinformatics Willliams, R. J. and N. D. Martinez .2000.  Simple rules yield complex food webs.  Nature 404:180-183. Willliams, R. J. and N. D. Martinez .2001.  Stabilization of Chaotic and Non-permanent Food-web Dynamics. Santa Fe Inst. Working Paper 01-07-37. Williams, R. J., E. L. Berlow, J. A. Dunne, A-L Barabási. and N. D. Martinez. 2002. Two degrees of Separation in Complex Food Webs. PNAS 99:12917-12922 Dunne, J. A. R. J. Williams and N. D. Martinez. 2002. Food-web structure and network theory: the role of size and connectance. PNAS 99:12917-12922 Brose, U., R.J. Williams, and N.D. Martinez. 2003. The Niche model recovers the negative complexity-stability relationship effect in adaptive food webs.  Science 301:918b-919b Williams, R.J., and N.D. Martinez. Limits to trophic levels and omnivory in complex food webs: theory and data.  In press. American Naturalist.   Dunne, J.A., R.J. Williams, and N.D. Martinez Network structure and robustness of marine food webs In press Marine Ecology Progress Series

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