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)

Nonlinear bioenergetic ecosystem model What factors allow persistence of species in dynamical models of complex food webs? (the “devious strategies”) The variation of B i, the biomass of species i, is given by:   n j =1 B i ’(t) = G i (B) – x i B i (t) + x i y ij α ij F ij (B) B i (t) – x j y ji α ji F ji (B) B j (t) / e ji 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.

3 species parameters: : production rate of basal species i (Mass/Time) For primary producers, G i (B) = r i B i (t) (1 – B i (t) / K i ), where r i : intrinsic growth rate of species i (1/Time) K i : carrying capacity of species i (Mass) ______________ x i : mass-specific metabolic rate of species i (Mass/Time * 1/Mass) 4 species interaction parameters: e ji : assimilation efficiency of species j consuming species i (fraction of biomass) y ij : rate of maximum biomass gain by species i consuming j normalized by metabolic rate of species i (Mass/Time / Mass/Time) α ij : relative preference of species i for species j (fraction of diet) (α ij = 0 for producers and sums to 1 for consumers) F ij (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) )(BG i

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)

    n k jiiijkik j ij BtBctB tB B F 1 0 )(1()( )( )(  c = 0 c = 1 q = 1 Addition of Predator Interference to Type II Functional Response:

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

biomass min & max 10-species dynamics & functional response 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.

Holling Type II / III Robustness q Stabilization of Dynamics of Ecological Networks (S=30, C=0.15) with Functional Responses  Effects of Dynamics on Structure   Effects of Structure on Dynamics  c Beddington-DeAngelis Predator Interference Niche model Cascade model Random model

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 (a) C 0 = 0.15 (b) S 0 = 30

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

q = 0 q = 0.2 Robustness Skewness Low Trophic-level Prey High Trophic-level Prey Effects of Omnivore Feeding Preference among Trophic Levels

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! Factors increasing overall species persistence

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

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 Simple rules yield complex food webs. Nature 404: Willliams, R. J. and N. D. Martinez Stabilization of Chaotic and Non-permanent Food-web Dynamics. Santa Fe Inst. Working Paper Williams, R. J., E. L. Berlow, J. A. Dunne, A-L Barabási. and N. D. Martinez Two degrees of Separation in Complex Food Webs. PNAS 99: Dunne, J. A. R. J. Williams and N. D. Martinez Food-web structure and network theory: the role of size and connectance. PNAS 99: Brose, U., R.J. Williams, and N.D. Martinez 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