Supplemental materials for “Effects of Network and Dynamical Model Structure on Species Persistence in Large Model Food Webs” by Richard J. Williams
Appendix A
1.Structural Models
The original niche model(Williams & Martinez 2000)orders all S species according to a uniformly random ‘niche value’ (ni) assigned to each species. This value randomly places the species somewhere along a “niche dimension” from 0 to 1 (0 ≤ ni≤ 1). A consumer eats all species whose niche values fall within arange (ri) whose center(ci) is a uniformly random number between ri/2 and min(ni, 1-ri/2). This constraint on ci ensures that cini, that ri fits entirely on the niche dimension, and that consumers’ diets are strongly biased towards resource species with niche values < ni. The niche range ri=xniand 0 ≤ x≤ 1 is a random variable with beta-distributed probabilitydensity function with. This causes species with higher ni to tend to eat more species and ensures that the average of all species’ ri equals C further causing the model to create food webs with connectance close to, if not equal to, the C put into the model.
The generalized cascade model(Stouffer et al. 2005)also accepts S and C as independent input parameters. The model maintains the hierarchical ordering constraint of the original cascade model(Cohen et al. 1990) between consumer species i and resource species j with the exception that species can be cannibalistic i.e., nj ≤ ni . As in the niche model, all S species are ordered according to a uniformly random ‘niche value’ (ni) assigned to each species. This value randomly places the species somewhere from 0 to 1 (0 ≤ ni≤ 1). Again as in the niche model, a valueri=xniis chosen for each species, where 0 ≤ x≤ 1 is a random variable with beta-distributed probabilitydensity function with. Unlike the niche model, the resource species of each consumer are not required to be adjacent on a niche dimension. Instead, each species has a probability (p) of feeding on any species with lower or equal niche value pi = ri/ni.
The cascade model(Cohen et al. 1990; Williams & Martinez 2000)orders the S species and each species has a probability (p) of feeding on any species lower in the hierarchyp = 2CS/S-1.
The random model(Erdos & Renyi 1959) assigns every link with probability p = C.
For all models, when networks are constructed with disconnected species or with trophically identical species (species with identical predators and prey), the network is rejected and replaced by running the model again. All models were also constrained to produce energetically feasible food webs by requiring that every species have at least one path connecting it to a basal species.
2. Bioenergetic Model
A population’s rate of change of biomass over time is given by
(A1a)
for primary producers; and
(A1b)
for consumer species.
In these equations, Bi is the biomass of population i, r’i is the maximum mass-specific growth rate and is the normalized growth rate of primary producer population i and is potentially a function of the biomasses of all of the populations in the system, Ti is the mass-specific respiration rate, Jij is the maximum ingestion rate of prey item j by consumer i andis the normalized multi-species functional response, also potentially a function of the biomasses of all species in the system, eij is an assimilation efficiency equal to the fraction of the biomass of species j lost due to consumption by species i that is actually metabolized, and feij is ingestion efficiency equal to the fraction of biomass lost from resource j that is actually ingested by consumer i. Assimilation efficiency is separated from ingestion efficiency because the former can theoretically be allometrically scaled while the latter is less systematic and contingent on the natural history of consumption.
As in earlier work, the biological rates of production, metabolism and maximum consumption are scaled with the species’ body mass (Brown et al. 2004; Enquist et al. 1999; Yodzis & Innes 1992). The mass-specific respiration rate, Ti, is given by
.(A2)
The mass-specific maximum assimilation rate, , is given by
.(A3)
The mass-specific maximum growth rate of a producer species is given by
.(A4)
The constants aT, ar, and aJ, all with units of (mass0.25time-1) have been determined from empirical data. These constants vary between metabolic groups of organisms including plants, invertebrates, and ectotherm and endotherm vertebrates but remain the same for species within the same metabolic group despite dramatic variation in mean body mass (Brown et al. 2004; Ernest et al. 2003; Yodzis & Innes 1992). These values are not universal for the whole system when the system is made up of species with different metabolic types. The constants fJij and fri are fractional quantities whose value may be specified for each specific population or feeding interaction in a particular ecological context.
The model’s time scale is normalized to the growth rate of a chosen primary producer k by introducing a new non-dimensional time variable :
. (A5)
This means that a unit of time is defined as the inverse of the growth rate of primary producer k. Once species k is chosen and time is scaled accordingly, several constants are defined as follows:
,(A6)
,(A7)
.(A8)
The first parameter is the relative mass-specific growth rate of producer species i normalized with the growth rate of the chosen producer species k. Similarly, xi is the mass specific metabolic rate of species i relative to the chosen time scale of the system. Finally, the non-dimensional constant yij is the maximum ingestion rate (biomass per unit time) of prey species j by predator species i relative to the metabolic rate of species i (biomass per unit time). Further details and discussion can be found in (Williams et al. 2007).
3. Sample Model Time Series
Figure A.1 shows a sample time series from this model for aS = 30, C = 0.15 niche model web with strong generalists, weak type III (q = 0.2) functional response and competitive primary producers. Biomass is plotted on a log transformed axis so as to clearly show the biomass decline and eventual extinction of several species. The time series plot allows evaluation typical biomasses, the timescale of extinctions and the potential effects of changes in various model choices, such as the duration of the time series and the extinction threshold. This system has chaotic dynamics; looking across the range of simulations, dynamics range from steady state to periodic fluctuations to chaos. The chosen time series shows some of the more complex dynamics and longer time scale of extinctions.
Figure Legend
Figure A.1 Time series fora model witha S = 30, C = 0.15 niche model webwith strong generalists, weak type III (q = 0.2) functional response and competitive primary producers.
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