---
title: "SuppMat 1: Expanded descriptions of the food web models"
authors:
  - name: Tanya Strydom
    id: ts
    orcid: 0000-0001-6067-1349
abstract: |
  This supplementary material provides expanded methodological descriptions of the food web models summarized in Table 1, detailing their assumptions, parameterization, and link-generation rules.
date: last-modified
bibliography: ../references.bib
---

## Structural models

### Random model

The Erdős–Rényi random graph model [@erdos1959] assigns a fixed number of links ($L$) uniformly at random among a set of $S$ nodes (species). Ecologically, this model assumes that interactions occur independently of species identity, such that no node-level traits influence whether a link is present or absent. As a result, links are randomly distributed throughout the network, producing a food web that is minimally biologically structured while preserving the specified richness ($S$) and connectance ($L/S^{2}$).

### Niche model

The niche model [@williams2000] assumes that trophic interactions are structured by a one-dimensional feeding niche. Each species is assigned a niche range, and all species whose niche values fall within this range are potential prey, allowing for cannibalism. Niche ranges are assigned stochastically, with their sizes constrained in part by the specified connectance ($Co$) of the network. Although several extensions of the niche model have been proposed, increasing model complexity does not appear to improve its ability to reproduce empirically realistic network structure [@williams2008].

Formally, each of the $S$ species is assigned a niche value $n_i$ drawn uniformly from the interval \[0,1\]. Species $i$ consumes all species whose niche values fall within a contiguous range of width $r_i$, with the center of this range ($c_i$) drawn uniformly from the interval $[ \frac{r_i}{2}, n_i ]$. The range width $r_i$ is determined by drawing a value from a beta distribution on \[0,1\] with expected value $2 \times Co$, which is then multiplied by $n_i$ to achieve the desired connectance.

### Allometric diet breadth model

The Allometric Diet Breadth Model (ADBM; @petchey2008) is grounded in optimal foraging theory and predicts trophic interactions based on energetic considerations. In this framework, consumers select prey to maximize their rate of energy intake, such that diet composition is determined by the energetic profitability of potential resources. The model therefore predicts both the number of trophic links and their arrangement based on species-level traits, specifically consumer diet breadth.
The energy intake rate ($K$) of a consumer is defined as:

$$
K = \frac{\sum_{i=1}^{k}\lambda_{ij}E_{i}}{1+\sum_{i=1}^{k}\lambda_{ij}H_{ij}}
$$

where $\lambda_{ij}$ is the encounter rate between consumer $j$ and resource $i$, defined as the product of attack rate ($A_i$) and resource density ($N_i$), $E_i$ is the energy content of the resource, and $H_{ij}$ is the handling time. Handling time depends on the ratio of consumer to resource body mass and is defined as:

$$
H_{ij} = \frac{h}{b - \frac{M_{i}}{M_{j}}} if \frac{M_{i}}{M_{j}} < b
$$

and

$$
H_{ij} = \infty \geq b
$$

All bioenergetic and allometric parameters were fixed to the default values reported in @petchey2008 and were not estimated from data. Attack rates, handling times, and resource densities were parameterised as power-law functions of body mass using empirically derived scaling exponents, while energy content was assumed to scale linearly with prey body mass. Body mass was therefore the sole varying biological input to the model across replicates, with all other parameters held constant.

### Allometric trophic network

The Allometric Trophic Network (ATN; @brose2006) model also uses body mass to infer trophic interactions but places greater emphasis on mechanical constraints associated with consumer–resource size relationships. Interactions are determined by allometric rules based on the ratio of consumer ($M_i$) to resource ($M_j$) body mass and are further constrained to produce networks that resemble realized food webs [@schneider2016; @gauzens2023].

The probability of a trophic interaction ($P_{ij}$) between consumer $i$ and resource $j$ is defined using a Ricker function parameterized by the optimal body-mass ratio ($R_{opt}$) and the shape parameter $\gamma$:

$$
P_{ij} = (L \times \exp(1 - L))^{\gamma}
$$

where

$$
L = \frac{M_{i}}{M_{j} \times R_{opt}}
$$

An optional threshold can be applied to $P_{ij}$, such that interaction probabilities below the threshold are set to zero, further constraining network structure.

### Body size ratio model

The body size ratio model [@rohr2010] infers trophic interactions probabilistically based on the ratio of consumer ($M_i$) to resource ($M_j$) body mass. To represent predator–prey body-mass ratios as a feeding niche, the model modifies this ratio using a quadratic function in log-transformed mass space, producing a unimodal interaction probability distribution.
In its simplified form, the probability of an interaction between consumer $i$ and resource $j$ is defined as:

$$
P_{ij} = \frac{p}{1+p}
$$

where

$$
p = exp[\alpha + \beta log(\frac{M_{i}}{M_{j}}) + \gamma log^{2}(\frac{M_{i}}{M_{j}})]
$$

The original latent-trait formulation introduced by @rohr2010 included an additional interaction term, $v_i \delta f_j$. For simplicity, and following @yeakel2014, this term is omitted here and only the equation above is used. Parameter values are taken directly from @yeakel2014, which were estimated using Serengeti food web data: $\alpha = 1.41$, $\beta = 3.75$, and $\gamma = 1.87$.

## Interaction predictions

### Paleo food web inference model

The Paleo Food Web Inference Model (PFIM; @shaw2024) infers trophic interactions using a rule-based framework applied to discrete trait categories, such as habitat preference and body size. For a given consumer–resource pair, an interaction is deemed feasible only if all trait-based rules across multiple rule classes are satisfied.

The original implementation of PFIM includes an optional downsampling step adapted from @roopnarine2006, which prunes interactions using a power-law distribution defined by the expected link distribution. This approach is conceptually aligned with earlier feasibility-based methods [@roopnarine2017] but does not require explicit assignment of taxa to trophic guilds. We follow the implementation described by @shaw2024, as these approaches are based on the same underlying feasibility framework and are expected to generate highly similar network structures.

# References {.unnumbered}

::: {#refs}
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