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We briefly describe the two most widely used structural food web models—the Gen- eralized Cascade Model and the Generalized Niche Model—before introducing the starting point of modelling food webs with trophic structure via the Preferential Preying Model.

Cascade Model and Generalized Cascade Model The original Cascade Model [61] assigns a a random number ni to each species i from the uniform distribution

on[0,1]. Then for all pairs(i, j),iis set to be the consumer ofj with some constant probabilitypifni > nj and with probability zero otherwise. If the number of species

is set to beN and the expected number of links toL, the constant probability takes the form

p= 2L

N(N₋1). (3.2)

The Cascade Model thus assumes a “pecking order” of species and a rule under which no species can consume prey higher than itself in the order. On the other hand, the probability of a species preying on a lower-ranked species is constant regardless of how far down the hierarchy the potential prey exists. This was the first attempt to show that networks similar to real food webs could be generated via simple rules and imposing a hierarchy of species.

The model was later modified so that the number of prey would be instead drawn from a Beta distribution as in the Niche Model [250] and the new model was called the Generalized Cascade Model [227]. This amendment improved the model’s predictions with respect to the distribution of prey and predators (i.e. the empirically observed degree distributions).

Niche Model and Generalized Niche Model As in the Cascade Model, in the Niche Model [250] each speciesi is awarded a uniform random number in[0,1)

called the niche value, but the model refines the Cascade Model by constraining a speciesidiet to a subset of speciesjwith lower niche value. Specifically, speciesiis constrained to consume speciesj such thatci−ri/2≤nj < ci+ri/2, i.e. all species

within an interval of size ri centred at ci and no others. The range parameter is

defined as ri =xini wherexi is drawn from the Beta distribution with parameters (1, β). For a fixed number of species N and an expected number of links L, the parameterβ takes the form

β= N(N −1)

2L −1. (3.3)

The centres of the intervalsci are drawn from the uniform distribution on[ri/2, ni].

The Niche Model is an improvement on the Cascade Model by allowing for cannibalism as well as looping—propensity to consume prey above a species trophic niche. The motivation behind the Niche Model is the so called intervality hypoth- esis—that there is an intrinsic ordering of species such that the prey of any given predator are contiguous [63]. Recent work has shown that food webs are biased to- wards intervality [228], although they are generally not perfectly interval, especially if species body size is taken as a proxy for niche value [57]. Nevertheless, the Niche Model has been very successful as it outperforms the Cascade Model in predicting many structural features of food webs [250].

The Niche Model was later modified to account for the fact that real food webs are not maximally interval by introducing an additionalcontiguityparametercwhich determines the proportion of prey to be allocated according to the original Niche Model prescription and the rest according to the Generalized Cascade Model [228]. This modified model is known as the Generalized Niche Model and is implemented as the original Niche Model, but with reduced ranges ri = cxini. Then for each

species a number of extra preykcascade

i = (1−c)xiniN is drawn randomly from all

the available species with niche values lower thanni as in the Generalized Cascade

Model. Thus, the Generalized Niche Model interpolates between the Niche Model

(c= 1)and the Generalized Cascade Model (c= 0).

Preferential Preying Model (PPM) The Preferential Preying Model (PPM) was introduced by Johnson et al. [131] as the first food web model to explicitly model trophic coherence in food webs. It is a growing network model, inspired by the Barabási-Albert model of preferential attachment [20], but in the context of a growing ecosystem through immigration of new species. The model starts with B

nodes (basal species) with trophic levels s = 1 and no links. New nodes are then added sequentially until the total number of nodes reaches N. Every new node i

is first awarded a random prey node j from all the existing nodes in the network and assigned a temporary trophic levelˆsi= 1 +sj. Following this, anotherκi prey

nodeslare chosen with a probability that decays with the tentative trophic distance

ˆ xil =sl−sˆi, specifically: Pil∝exp −|xˆil−1| T , (3.4)

whereT is a “temperature” parameter that tunes the trophic coherence: for T = 0, perfectly coherent networks are generated (q = 0), whileq increases monotonically forT >0.

The number of preyκi in the original PPM is obtained similarly to the Niche

Model method [227] by setting κi = xini, where ni is the number of nodes in the

network when nodeiarrives andxi is a Beta random variable with parameters(1, β)

such that

β = N

2_−B2

2L −1, (3.5)

where L is the expected number of links in the resulting network. This way the Generalized Cascade Model is recovered in theT _{→ ∞} limit while the degree distri- butions are as in the Generalized Niche Model.

Even though the PPM captures many empirically observed patterns in food webs, it suffers from a major drawback, namely it can only generate acyclic networks. While this is a property that is shared by many food webs, it is far from universal [56] and, in particular, makes the PPM unsuitable for studying mutual predation or food web motifs. To remedy this, we now describe an extension of the PPM that can generate cycles.