Ecological Archives E086-072-A2

D. B. Stouffer, J. Camacho, R. Guimerà, C. A. Ng, and L. A. Nunes Amaral. 2005. Quantitative patterns in the structure of model and empirical food webs. Ecology 86:1301–1311.

Appendix B. An overview of the models' rules.

The niche model

Consider an ecosystem with $S$ species and $L$ trophic interactions between these species. In the niche model (Williams and Martinez 2000), one first randomly assigns $S$ species to "trophic niches" with niche values $n_i$ mapped uniformly onto the interval [0,1]. For convenience, we will assume that the species are ordered according to their niche value, that is, $n_1 < n_2 < ... < n_S$ .

A species $i$ is characterized by its niche value $n_i$ and by its list of prey. Prey are chosen for all species according to the following rule: A species $i$ preys on all species $j$ with niche value $n_j$ inside a segment of length $a_i$ centered in a position chosen randomly inside the interval $[a_i/2,n_i]$ , with $a_i = x n_i$ and $0 \le x \le 1$ is a random variable with probability density function Eq. (1) (Fig. B1).

 
 
   FIG. B1. Prey selection in the niche model. (A) The hawk preys on all species lying within a segment of niche values having length $a_i$ . (B) and (C) The center of this segment, $c_{a_i}$ , is uniformly drawn from the range $[a_i/2,n_i]$ . Note that this will allow up to half of the potential prey to have niche values higher than the hawk. (D) In this instance, the hawk consumes the prairie dog, the coyote, and the hawk itself.


The nested-hierarchy model

In the nested-hierarchy model (Cattin et al. 2004), the number of prey $k_i$ of a species $i$ is obtained by multiplying the predator's niche value $n_i$ by a value $x_i$ drawn from the interval $[0,1]$ according to Eq. (1).

Prey selection in this model obeys a two-stage, multi-step process (Fig. B2). In stage one, the first prey of species $i$ is selected at random from among species with lower niche values than $i$ . Let $j$ be the first prey of $i$ . If $j$ is a prey of another species, then the next prey of $i$ is chosen from the pool of species eaten by the group of $j$ 's consumers. The group of $j$ 's consumers is defined as all consumers sharing at least one prey, with at least one of these consumers feeding on $j$ . If the required number of links for species $i$ cannot be satisfied by this pool, the remaining prey are chosen randomly from among the species with no predators that have niche values lower than $i$ .

As a last resort, if all possible species with niche values lower than $i$ have been selected and $i$ requires additional prey, selection enters stage two, and prey are chosen from among species with niche values greater than or equal to the niche value of $i$ .

 

\includegraphics[width=0.75\columnwidth]{CattinAnimals}
 
   FIG. B2. Prey selection in the nested-hierarchy model. (A) The wolf randomly selects a prey from the set of all species with lower niche values. The chosen species, the rabbit, is also a prey of the coyote. (B) and (C) The next prey of the wolf is then randomly chosen from among the set of prey of the group of the rabbit's consumers. The group of the rabbit's consumers are the consumers sharing at least one prey, at least one of which must be the rabbit; this group is thus the coyote, the hawk, and now the wolf. (D) The pool of potential prey for the wolf is then the prairie dog, the squirrel, and the mouse. Note that it contains all prey from the group of the rabbit's consumers.

 



The generalized cascade model

In the original cascade model (Cohen and Newman 1985), each species $j$ with $n_j < n_i$ becomes a prey of $i$ with probability $x_0=2CS/(S-1)$ (Figs. B3A and B). We generalize the cascade model in two manners: (i) allowing predation on all species with $n_j \le n_i$ , and (ii) use of a probability function $p(x)$ which is approximately an exponential function of $x$ , where $x$ is drawn independently for each species. Figures B.3C and D describe the prey selection under this formulation. The idea of a predator specific $x$ was originally proposed by Cohen (1990) as an alternative to the original cascade model. Cohen (1990) would consider the niche, nested-hierarchy, and generalized cascade models ``predator dominant'' because $x$ is uniquely defined for each predator. Note that models considered by Cohen et al. (1990) do not satisfy Condition II. Cohen et al. (1990) considers two types of models: (i) "constant-column-sum" models, where all species typically have the same number of prey, and (ii) "increasing-column-sum" models, where a species tends to prey on a greater number of the species with lower niche values as its niche value increases.

 

\includegraphics[width=0.823\columnwidth]{Cascade}
 
   FIG. B3. Prey selection in the cascade model. (A) The coyote can randomly select prey from the set of all species with lower niche values, with equal probability. (B) In this instance, the coyote consumes the squirrel and the mouse, with equal probability $x_{0} = 0.5$ in each case. (C) The generalized cascade model. Prey can be chosen from any species with niche values less than or equal to that of the coyote, allowing for cannibalism. The probability $x_j$ of consuming these species is specific to the predator $j$ . (D) In this instance, the coyote consumes the rabbit and itself with the species-specific probability $x_j$ .


LITERATURE CITED

Cattin, M.-F., L.-F. Bersier, C. Banašek-Richter, R. Baltensperger, and J.-P. Gabriel. 2004. Phylogenetic constraints and adaptation explain food-web structure.
Nature 427:835–839.

Cohen, J. E. 1990. A stochastic theory of community food webs. VI. Heterogeneous alternatives to the cascade model.
Theoretical Population Biology 37:55–90.

Cohen, J. E., F. Briand, and C. M. Newman. 1990. Community food webs: data and theory.
Springer-Verlag, Berlin, Germany.

Cohen, J. E., and C. M. Newman. 1985. A stochastic theory of community food webs. I. Models and aggregated data.
Proceedings of the Royal Society B 224:421–448.

Williams, R. J., and N. D. Martinez. 2000. Simple rules yield complex food webs.
Nature 404:180–183.


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