Habitat complexity reduces feeding strength of freshwater predators (CRITTER) — Supplemental Statistics Report

Habitat complexity reduces the feeding strength of freshwater predators (CRITTER) Supplemental Statistics Report (v1.0.0) doi: 10.5281/zenodo.17250079 Björn C. Rall∗ . Mireia Aranbarri†Lorea Flores‡Ioar de Guzmán§ Aitor Larrañaga¶Julia Reiss‖ 2025-10-14 ∗ Aquatic Ecology and Evolution, Department of Biology, University of Konstanz, Universitätsstraße 10, 78464 Konstanz, Germany |[email protected] |https://orcid.org/0000-0002-3191-8389 † Laboratory of Stream Ecology, Department of Plant Biology and Ecology, Faculty of Science and Technology, University of the Basque Country, UPV/EHU PO Box 644, 48080 Bilbao, Spain | [email protected] |https://orcid.org/0009-0001-3506-0914 ‡ Laboratory of Stream Ecology, Department of Plant Biology and Ecology, Faculty of Science and Technology, University of the Basque Country, UPV/EHU PO Box 644, 48080 Bilbao, Spain | [email protected] |https://orcid.org/0000-0002-0082-4072 § Laboratory of Stream Ecology, Department of Plant Biology and Ecology, Faculty of Science and Technology, University of the Basque Country, UPV/EHU PO Box 644, 48080 Bilbao, Spain | INRAE, UMR 1224, Ecologie Comportementale et Biologie des Populations de Poissons, Aquapôle, quartier Ibarron, 64310 Saint-Pée sur Nivelle, France | [email protected] | https://orcid.org/0000-0001-8894-8477 ¶ Laboratory of Stream Ecology, Department of Plant Biology and Ecology, Faculty of Science and Technology, University of the Basque Country, UPV/EHU PO Box 644, 48080 Bilbao, Spain | [email protected] |https://orcid.org/0000-0002-0185-9154 ‖ Centre for Pollution Research and Policy, Brunel University of London, Uxbridge, UB8 3PH, UK | [email protected] | https://orcid.org/0000-0002-3740-0046 1 Contents Overview 3 Summary .................................................... 3 License ..................................................... 3 How to cite this report ............................................. 3 Related Works ................................................. 3 Methods 4 Required Packages ............................................... 4 Theoretical Background ............................................ 4 Functional response statistics ......................................... 6 Independence of variables ........................................ 6 Data distribution ............................................. 6 Functional response type selection ................................... 7 Habitat complexity as an independent variable ............................ 7 Data Availability ................................................ 7 Code availability ................................................ 7 Results 8 The functional response type ......................................... 8 Using the phenomenological test .................................... 8 Direct shape estimation using the generalized functional response model .............. 8 Habitat presence, amount, and complexity results .............................. 9 Ischnura elegans results ......................................... 10 Notonecta glauca results ......................................... 13 Funding Information 17 References 18 Appendix 20 Phenomenological Test Rationale ....................................... 20 Full AIC / BIC tables ............................................. 21 Ischnura elegans ............................................. 21 Notonecta glauca ............................................. 22 All summary tables including the original bbmle::summary outputs .................... 23 Ischnura elegans ............................................. 23 Notonecta glauca ............................................. 27 Project Settings ................................................. 31 2 Overview Summary This statistical report is supplementing our peer-reviewed journal article by Aranbarri et al. (2025a) in which we investigate the effect of habitat complexity on the feeding functional response of two freshwater invertebrate predators. Find below information on related works and the location of the data and code. License This report is published under the Creative Commons BY 4.0 license. The underlying code is licensed under the GNU General Public License 3. How to cite this report If you want to cite our report, please use on of the following options: General citation: Rall, B.C., Aranbarri, M., Flores, L., Guzmán, I. de, Larrañaga, A., & Reiss, J.(2025): Habitat complexity reduces the feeding strength of freshwater predators (CRITTER) — Supplemental Statistics Report. Zenodo. https://doi.org/10.5281/zenodo.14898819 Current version: Rall, B.C., Aranbarri, M., Flores, L., de Guzmán, I., Larrañaga, A., and Reiss, J. (2025): Habitat complexity reduces feeding strength of freshwater predators (CRITTER) — Supplemental Statistics Report (v1.0.0). Zenodo. https://doi.org/10.5281/zenodo.17250079 Former versions: Rall, B.C., Aranbarri, M., Flores, L., de Guzmán, I., Larrañaga, A., and Reiss, J. (2025): Habitat complexity reduces feeding strength of freshwater predators (CRITTER) — Supplemental Statistics Report (v0.1.2). Zenodo. https://doi.org/10.5281/zenodo.15348995 Rall, B.C., Aranbarri, M., Flores, L., de Guzmán, I., Larrañaga, A., and Reiss, J. (2025): Complexity reduces feeding strength of freshwater predators (CRITTER) - Supplemental Statistics Report (v0.1.1). Zenodo. https://doi.org/10.5281/zenodo.14907885 Rall, B.C., Aranbarri, M., Flores, L., de Guzmán, I., Larrañaga, A., and Reiss, J. (2025): Complexity reduces feeding strength of freshwater predators (CRITTER) - Supplemental Statistics Report (v0.1.0). Zenodo. https://doi.org/10.5281/zenodo.14898820 Please consider to also cite our main article: Aranbarri, M., Flores, L., de Guzmán, I., Larranaga, A., Rall, B.C., and Reiss, J. (2025): Habitat complexity reduces feeding strength of freshwater predators. Ecology and Evolution, e72258. https://doi.org/10.1002/ece3. 72258 Related Works •Peer-Reviewed Journal Article (Aranbarri et al., 2025a) •Scientific Preprint Article (Aranbarri et al., 2025b) •Supplemental Statistics Report on Zenodo (Rall et al., 2025b) •Supplemental Statistics Report on GitHub •R-Code on Zenodo (Rall et al., 2025a) 3 •R-Code on GitHub •Data on Zenodo (Flores et al., 2025) •Data on GitHub Methods Required Packages If you want to knit this report for yourself, you need to install: •bbmle (Bolker & R. Development Core Team, 2023) •bookdown (Xie, 2025a) •doParallel (Microsoft & Weston, 2022a) •dplyr (Wickham et al., 2023) •emdbook (Bolker, 2025) •foreach (Microsoft & Weston, 2022b) •frair (Pritchard, 2025) •here (Müller, 2025) •kableExtra (Zhu, 2024) •knitr (Xie, 2025b) •lhs (Carnell, 2024) •MASS (Ripley & Venables, 2025) •odin (FitzJohn, 2025) •png (Urbanek, 2022) •purrr (Wickham & Henry, 2025) •RCurl (CRAN Team & Temple Lang, 2025) •sessioninfo (Csárdi et al., 2025) •supportR (Lyon, 2024) •tinytex (Xie & Posit Software, PBC, 2025) We used R Markdown (Allaire et al., 2025; Xie et al., 2018, 2020) with bookdown (Xie, 2016, 2025a) and TinyTex (Xie & Posit Software, PBC, 2025) to create the pdf report. Please follow the TinyTex installation guidelines on their homepage. You can, of course, use another LaTeX distribution, but we only tested if the R Markdown script works using TinyTex. Find a full list of packages used to compile this report in the section “Project Settings” below. Theoretical Background The feeding functional response describes how a consumer’s feeding rate reacts to its resource density (Holling, 1959a). The most basic shape of the feeding functional response is a hyperbolic curve, also called the type II functional response (Jeschke et al., 2002). The attack rate, a , controls the initial increase in the feeding rate, F , with increasing resource density, N (Holling, 1959a), and the handling time, Th controls the satiation of the curve: F=aN (1+aThN).(1) If the attack rate is a function of resource density, the functional response becomes s-shaped (Holling, 1959b; Kalinkat et al., 2023). In the simplest form of this type III functional response, the attack rate is a linear function of resource density (Juliano, 2001): a=bN, (2) in which bis the attack coefficient. The functional response becomes 4 F=bN2 (1+bThN2).(3) (Real, 1977, 1979) presented an alternative formulation of the functional response inspired by chemical reactions and the Michaelis-Menten kinetics (Barcroft & Hill, 1910). By substituting Th=1 Fmax (4) and a=Fmax Nhalf ,(5) the type II functional response becomes F=FmaxN Nhalf +N.(6) Here, the curve’s satiation is controlled by the maximum feeding rate, Fmax , and the half saturation density, Nhalf controls the steepness of the functional response curve. The type III functional response written in this style is as follows: F=F2 maxN N2 half +N2.(7) The above-described type III functional response model has a linearly increasing attack rate, but the attack rate can also be a more complex function of resource density (Juliano, 2001; Kalinkat et al., 2023). A commonly used version is a power law function of the resource density (Kalinkat et al., 2023; Rall et al., 2008; Williams & Martinez, 2004): a=bq.(8) The Holling-style functional response becomes (Vucic-Pestic et al., 2010): F=bN1+q (1+bThN1+q),(9) and the Real-style functional response becomes (Rall et al., 2008; Williams & Martinez, 2004): F=F1+q maxN N1+q half +N1+q.(10) We will refer to the above-mentioned functional response models as either the Holling-style or the Real-style generalized functional response model. Note that many other functional response models exist, but the ones presented above are most practical and most often used to analyze laboratory functional response experiments (Jeschke et al., 2002; Juliano, 2001; Kalinkat et al., 2023). In our analyses, we will focus on the generalized functional response models and, if statistically appropriate, on the type II or type III functional response models. 5 Functional response statistics Independence of variables Scientific laboratory experiments are designed to manipulate one or more external variables that are typically called the independent or explanatory variables. We measure a response of our experimental system, and this measure is normally called the dependent, or response variable. See, e.g., Zar (2010), chapter 17, or Crawley (2012) chapter 10, for details. In functional response experiments, we manipulate the resource density, N , and measure the number of prey eaten, Neaten , at the end of the experiment. This means that during the experiment, the prey density decreases over time. In contrast, in all the above-mentioned functional response models, the feeding rate, F , depends on a constant prey density, N . To account for the decay of resource density over time, we set up a differential equation, describing the temporal process: dN dt =−F. (11) The change of resource density, dN , over a given time span, dt , is described by the negative of the feeding rate F . F can be any of the functional response models introduced above (or described elsewhere). This differential equation model can not be fitted directly to data, but it needs to be solved beforehand. The solution depends on the type of functional response. For a type II functional response, the solution is (Rogers, 1972; Rosenbaum & Rall, 2018; Royama, 1971): Neaten =Ninital(1 −ea(NeatenTh−P t)).(12) The number of resources eaten, Neaten at the time t (e.g., the end of the experiment) depends on the initial number of resources, Ninitial . All other parameters are as described above. This equation is commonly known as Roger’s Random Predator Equation (RRPE). By substituting Th = 1 Fmax and a = Fmax Nhalf into Roger’s Random Predator Equation, we get the version for the Real-style functional response model: Neaten =Ninital(1 −e Neaten Nhalf −P t Fmax ).(13) Model parameters are as described above. These equations share the issue that Neaten appears on both sides of the equation. Traditionally, this problem is solved using Newton’s method, an iterative numerical procedure (Juliano, 2001; Vonesh & Bolker, 2005), but Bolker (2008) found that the Lambert Wfunction (Corless et al., 1996) allows for an analytical solution: Neaten =Ninital − W(aThNinitiale−a(P t−ThNinitial)) aTh .(14) The Real-style version is: Neaten =Ninital − W(1 Nhalf Ninitiale − Fmax Nhalf (P t− Ninitial Fmax )) 1 Nhalf .(15) All parameters are as described above. The generalized functional response cannot be solved analytically and must be solved using numerical simulations (Rosenbaum & Rall, 2018). We follow the methods published by Rosenbaum & Rall (2018), but find more details on the underlying code in our code repository: https://github.com/b-c-r/CRITTERcode (Rall et al., 2025a). Data distribution The mathematical model described above calculates the average number of resources eaten across experimental trials. Each experimentally derived replicate, however, deviates from this overall model mean (see, e.g., Zar (2010), chapter 17, or Crawley (2012) chapter 10, for assumptions of regression statistics). Our dependent data, Neaten , has the following characteristics: integer numbers (0, 1, 2,. . . , n), and the number of prey eaten cannot exceed the number of initially available resource items ( Ninitial ). Data following these characteristics are known as proportion data and follow a binomial probability distribution; see Crawley (2012) chapter 16 for further 6 explanations. We apply this distribution to our data, as it is commonly used for functional response fitting (Bolker, 2008; Rosenbaum & Rall, 2018). Please also read the description of our code in the corresponding README file, either directly in our code repository or download it from Zenodo (Rall et al., 2025a). Functional response type selection We analyzed initally which functional response type describes our data best (Juliano, 2001). We used two methods: (1) the phenomenological test after Juliano, implemented in the frair package (Juliano, 2001; Pritchard et al., 2017); and (2) by fitting the Real-style generalized functional response model and test if q is significantly different from zero (Rosenbaum & Rall, 2018; Vucic-Pestic et al., 2010). If the tests suggest, that at least one functional response treatment is not of type II, we will analyze our data further using the generalized functional response model. If the tests suggest that all treatments are type II, we will analyze our data further using the Roger’s Random Predator Equation models. Habitat complexity as an independent variable The main scientific question of this project is to figure out how habitat complexity affects the interaction strength of two freshwater invertebrate predators (the consumers) on their prey (the resources). We have three different ways to address this question given the data: (1) the absence and presence of habitat; (2) the amount of habitat (either 0, 2, or 3 plastic ring structures); (3) five levels of complexity (0 to 4). The complexity levels originate from different spatial arrangements of the artificial ring structures, where 0 rings are complexity level 0, 2 rings comprise complexity levels 1 and 2, and 3 rings comprise complexity levels 3 and 4. See Flores et al. (2016) and Aranbarri et al. (2025a) for further details. Contrasting the above-described analysis of the functional response type, we use here all data at once, separated by predator species. Each functional response parameter can react to habitat independently of the other parameters. For example, the attack rate can be different for all five complexity levels, whereas the handling time is a function of ring number. For each parameter, we can formulate the following hypotheses: •H0: Habitat has no effect on the functional response parameters. •H1: The sheer presence of habitat changes the functional response parameters. • H 2 : The amount of habitat structure (number of rings, continuous) alters the functional response parameters. • H 3 : The spatial arrangements modify the functional response parameters in addition to the amount of habitat structure. Any functional response parameter independently can fulfill one of the above-stated hypotheses. If we want to test all possible combinations, we need 4×4 models (4 hypotheses per parameter) for each functional response style (Real-style and Holling style), in the case of a type II functional response, resulting in 32 statistical models per predator species. In the case of generalized functional response, we would have to test 64 statistical models for each functional response style, resulting in 128 models per species. Note that some of these models of how habitat affects the functional response are interchangeable, but not all. We will show for which models this is true in the results part below. Data Availability You can either download the data from our data publication on Zenodo (Flores et al., 2025), or from our GitHub repository. Code availability You can either download the code from our code publication on Zenodo (Rall et al., 2025a), or from our GitHub repository. 7 Results The functional response type Using the phenomenological test Table 1: The results of testing for the functional response type by using the phenomenological test (Juliano 2001, Pritchard et al. 2017). We found for all unique treatments (complexity times predator identity) that a type II functional response explains the data best. Predator Complexity Linear (Q) Quadratic (Q) Linear (L) Type Ischnura elegans 0 -(n.s.) +(n.s.) -(*) II Ischnura elegans 1 -(n.s.) +(n.s.) -(*) II Ischnura elegans 2 -(n.s.) +(n.s.) -(***) II Ischnura elegans 3 -(n.s.) +(n.s.) -(**) II Ischnura elegans 4 +(n.s.) -(n.s.) -(**) II Notonecta glauca 0 -(***) +(***) -(***) II Notonecta glauca 1 -(***) +(***) -(***) II Notonecta glauca 2 -(***) +(**) -(***) II Notonecta glauca 3 -(n.s.) -(n.s.) -(***) II Notonecta glauca 4 -(*) -(n.s.) -(***) II We analyzed our data using the so-called phenomenological test by fitting this test to all ten treatments separately, i.e., two predators with five complexity levels each (Juliano, 2001; Pritchard et al., 2017; Pritchard, 2025; Rall et al., 2025a). The only case in which the results suggest a hump-shaped pattern in the proportion data is Ischnura elegans at the complexity level of 4, as the linear term is positive, and the quadratic term is negative (table 1). However, both estimates are not significant. The complexity levels 3 and 4 for Notonecta glauca show a negative quadratic term, which could also indicate a type III functional response, but both results are not significant. All other treatments have both a negative linear and a positive quadratic term, indicating in all these cases a declining proportion of eaten prey and therewith type II functional responses (table 1). Direct shape estimation using the generalized functional response model Table 2: The significance of the shape parameter q of all ten separately fitted generalized functional response models. Predator Name Complexity qSignificance Type-o-Response Ischnura elegans 0 0.116 n.s. II Ischnura elegans 1 0.047 n.s. II Ischnura elegans 2 -0.001 n.s. II Ischnura elegans 3 -0.001 n.s. II Ischnura elegans 4 0.193 n.s. II Notonecta glauca 0 -0.011 n.s. II Notonecta glauca 1 -0.006 n.s. II Notonecta glauca 2 0.194 n.s. II Notonecta glauca 3 -0.002 n.s. II Notonecta glauca 4 -0.001 n.s. II We analyzed our data using the generalized functional response model by fitting this model to all ten treatments separately, i.e., two predators with five complexity levels each (Rosenbaum & Rall, 2018; Vucic-Pestic et al., 2010). The shape parameter q is not significantly different from zero (see table 2). This means that in all cases, a simpler type II functional response can be used to describe the data properly. This is in line with the phenomenological test shown above, and we will subsequently use the Roger’s Random Predator Equation. 8 Habitat presence, amount, and complexity results As the type II functional response model is the most appropriate model to describe our data across all treatments (see section “The functional response type” above), we continue our analyses using this functional response model. Based on our hypotheses (see section “Habitat complexity as an independent variable” above), we need to test in total 64 statistical functional response models: 16 models for the Holling-style and 16 models for the Real-style RRPE, resulting in 32 models, and we repeated these 32 models for the two predator species. In the following, we present the description of the 32 models and how habitat affects the functional response parameters: • Model 01h | 01r: Habitat has no effect on Fmax | Th , (H 0 ), and Nhalf | a is also not affected by habitat presence (H0). • Model 02h | 02r: Habitat has no effect on Fmax | Th , (H 0 ), and Nhalf | a is affected by habitat presence (H1). • Model 03h | 03r: Habitat has no effect on Fmax | Th , (H 0 ), and Nhalf | a is affected by the amount of habitat structure (H2). •Model 04h | 04r: Habitat has no effect on Fmax |Th, (H0), and Nhalf |ais affected by complexity (H3). • Model 05h | 05r: Habitat presence affects Fmax | Th , (H 0 ), and Nhalf | a is not affected by habitat presence (H0). • Model 06h | 06r: Habitat presence affects Fmax | Th , (H 0 ), and Nhalf | a is affected by habitat presence (H1). • Model 07h | 07r: Habitat presence affects Fmax | Th , (H 0 ), and Nhalf | a is affected by the amount of habitat structure (H2). •Model 08h | 08r: Habitat presence affects Fmax |Th, (H0), and Nhalf |ais affected by complexity (H3). •Model 09h | 09r: The amount of habitat affects Fmax |Th, (H0), and Nhalf |ais not affected by habitat presence (H0). • Model 10h | 10r: The amount of habitat affects Fmax | Th , (H 0 ), and Nhalf | a is affected by habitat presence (H1). • Model 11h | 11r: The amount of habitat affects Fmax | Th , (H 0 ), and Nhalf | a is affected by the amount of habitat structure (H2). • Model 12h | 12r: The amount of habitat affects Fmax | Th , (H 0 ), and Nhalf | a is affected by complexity (H3). • Model 13h | 13r: Complexity affects Fmax | Th , (H 0 ), and Nhalf | a is not affected by habitat presence (H0). •Model 14h | 14r: Complexity affects Fmax |Th, (H0), and Nhalf |ais affected by habitat presence (H1). • Model 15h | 15r: Complexity affects Fmax | Th , (H 0 ), and Nhalf | a is affected by the amount of habitat structure (H2). •Model 16h | 16r: Complexity affects Fmax |Th, (H0), and Nhalf |ais affected by complexity (H3). The maximum feeding rate, Fmax , and the handling time, Th are always interchangeable as Fmax = 1 Th . However, the attack rate, a , is not only a function of the half saturation density, Nhalf , but also a function of the maximum feeding rate: a = Fmax Nhalf , Nhalf = Fmax a . This circumstance leads to the fact that the interchangeability of these parameters is context-dependent: 1. Attack rate and half saturation density are interchangeable if the maximum feeding rate is a constant across treatments (models 01 to 04). 2. Attack rate and half saturation density are also interchangeable if we fit a model testing for all complexity levels (H3) for the half saturation density (models 04, 08, 12, and 16). 3. Attack rate and half saturation density are also interchangeable if we fit a model testing for the same hypothesis (models 01, 06, 11, and 16). 9 saturation density. The generally decreasing trend of maximum feeding rate with increasing amount of habitat is in line with the models above, only ignoring the up-and-down with complexity levels. We also already saw above (e.g., models 15r and 16r) that the effect of habitat on half saturation is rather weak. Table 14: Results of model 9 (Real style) fitted to the data of Notonecta glauca. Parameter Name Point Estimate Lower CI Upper CI Fmax (intercept) 34.831 31.211 38.894 Fmax (slope) -0.094 -0.110 -0.077 Nhalf 33.937 27.822 41.347 Model 11h and model 11r are interchangeable, and the functional response parameters both depend on the amount of habitat present (tables 15 and 16). Handling time increases, and, of course, maximum feeding rate increases with the same strength. Attack rates decrease, while half saturation densities increase slightly. These results are in line with the above-mentioned models, only missing the up-and-down with complexity levels as in the more complex models. Table 15: Results of model 11 (Holling style) fitted to the data of Notonecta glauca. Parameter Name Point Estimate Lower CI Upper CI Th(intercept) 0.033 0.028 0.038 Th(slope) 0.063 0.031 0.094 a(intercept) 1.265 0.998 1.602 a(slope) -0.144 -0.190 -0.096 Table 16: Results of model 11 (Real style) fitted to the data of Notonecta glauca. Parameter Name Point Estimate Lower CI Upper CI Fmax (intercept) 30.745 26.515 35.727 Fmax (slope) -0.063 -0.095 -0.031 Nhalf (intercept) 24.304 17.192 34.328 Nhalf (slope) 0.081 0.010 0.152 Model 13r (table 17) shows the well-known up-and-down pattern of maximum feeding, including a general increase with increasing complexity level. The half saturation density is constant, in line with the rather weak (but significant) positive effect of habitat amount in the models above. Table 17: Results of model 13 (Real style) fitted to the data of Notonecta glauca. Parameter Name Point Estimate Lower CI Upper CI Fmax (C0) 34.784 31.137 39.046 Fmax (C1) 19.350 16.941 22.205 Fmax (C2) 26.070 23.095 29.453 Fmax (C3) 18.371 15.942 21.276 Fmax (C4) 17.969 15.615 20.733 Nhalf 33.928 27.826 41.483 In summary, all models in the range of 2 delta AIC and BIC units suggest that (1) the handling times increase with increasing amount of habitat available, altered by complexity levels. This leads to a decrease in maximum feeding rates, with the mirrored up-and-down effect of habitat complexity. We compared models 15h (the most parsimonious model) with model 11h (the next simple model nested in model 15h) and found that if the amount of habitat is medium (2 rings), the results for complexity levels 2 and 3 differ from the prediction of the simpler 16 model (figure 3). The attack rates decrease with increasing amounts of habitat, and this effect is slightly stronger than the effect on handling time, leading to a slightly increasing half saturation density. An even more complex effect of habitat complexity is given as models 16h and 16r are also within 2 delta AIC units, but as the AIC rather selects models that overfit data, we consider this effect as rather irrelevant. 0.0 1.0 2.0 3.0 15 20 25 30 35 40 Fmax (a) Fmax 0.0 1.0 2.0 3.0 0.03 0.04 0.05 0.06 0.07 Th (b) Th number of rings Figure 3: The functional feeding response parameters Fmax (a) and Th of Notonecta glauca feeding on Asellus aquaticus. We found that model 15h was the most appropriate to describe the data (table 9). This model includes a unique maximum feeding rate for each complexity level and a decreasing attack rate as a function of the amount of habitat provided (table 10). The next simpler model, including the decreasing attack rate with ring count, is model 11h (table 15). We consider the complexity levels as variants of the respective ring levels. Funding Information • Mireia Aranbarri was funded by the Investigo Programm funded by the NextGenerationEU initiative. •Lorea Flores was funded by a grant by the Spanish Ministry of Education and Culture. • Ioar de Guzmán was funded by the Spanish Ministry of Science, Innovation and Universities (TED2021-129966B-C31) and by the University of the Basque Country (POSTUPV24/47). •Julia Reiss was supported by a Royal Society of London Starting Grant. • Björn C. Rall was funded by the German Research Foundation (DFG) via the DynaSym research unit (FOR 5726). 17 References Akaike, H. (1974). A new look at the statistical model identification. IEEE Transactions on Automatic Control, 19(6), 716–723. https://doi.org/10.1109/TAC.1974.1100705 Allaire, J., Xie, Y., Dervieux, C., McPherson, J., Luraschi, J., Ushey, K., Atkins, A., Wickham, H., Cheng, J., Chang, W., & Iannone, R. (2025). rmarkdown: Dynamic documents for R.https://doi.org/10.32614/CRAN. package.rmarkdown Aranbarri, M., Flores, L., de Guzmán, I., Larrañaga, A., Rall, B. C., & Reiss, J. (2025a). Habitat complexity reduces feeding strength of freshwater predators. Ecol Evol, e72258. https://doi.org/10.1002/ece3.72258 Aranbarri, M., Flores, L., de Guzmán, I., Larrañaga, A., Rall, B. C., & Reiss, J. (2025b). Habitat complexity reduces feeding strength of freshwater predators (2025.02.22.639633). bioRxiv. https://doi.org/10.1101/2025. 02.22.639633 Barcroft, J., & Hill, A. V. (1910). The nature of oxyhæmoglobin, with a note on its molecular weight. J. Physiol., 39(6), 411–428. https://doi.org/10.1113/jphysiol.1910.sp001350 Bolker, B. M. (2008). Ecological models and data in R. Princeton University Press. https://math.mcmaster.ca/ ~bolker/emdbook/index.html Bolker, B. M. (2025). emdbook: Support functions and data for "Ecological models and data".https://doi.org/10. 32614/CRAN.package.emdbook Bolker, B. M., & R. Development Core Team. (2023). bbmle: Tools for general maximum likelihood estimation. https://doi.org/10.32614/CRAN.package.bbmle Carnell, R. (2024). lhs: Latin hypercube samples.https://doi.org/10.32614/CRAN.package.lhs Corless, R. M., Gonnet, G. H., Hare, D. E. G., Jeffrey, D. J., & Knuth, D. E. (1996). On the LambertW function. Adv. Comput. Math.,5(1), 329–359. https://doi.org/10.1007/BF02124750 CRAN Team, & Temple Lang, D. (2025). RCurl: General network (HTTP/FTP/...) client interface for R. https://doi.org/10.32614/CRAN.package.RCurl Crawley, M. J. (2012). The R Book (2nd ed.). Wiley. Csárdi, G., Wickham, H., Chang, W., Flight, R., Müller, K., & Hester, J. (2025). sessioninfo: R session information.https://doi.org/10.32614/CRAN.package.sessioninfo FitzJohn, R. (2025). odin: ODE generation and integration.https://doi.org/10.32614/CRAN.package.odin Flores, L., Bailey, R. A., Elosegi, A., Larrañaga, A., & Reiss, J. (2016). Habitat complexity in aquatic microcosms affects processes driven by detritivores. PLOS One,11 (11), e0165065. https://doi.org/10.1371/journal.pone. 0165065 Flores, L., Reiss, J., Larrañaga, A., Rall, B. C., Aranbarri, M., & Guzmán, I. de. (2025). Habitat complexity reduces feeding strength of freshwater predators (CRITTER) — Data. Zenodo. https://doi.org/10.5281/ zenodo.14891980 Holling, C. S. (1959a). Some characteristics of simple types of predation and parasitism. Can. Entomol.,91 (7), 385–398. https://doi.org/10.4039/Ent91385-7 Holling, C. S. (1959b). The components of predation as revealed by a study of small-mammal predation of the european pine sawfly. Can. Entomol.,91(5), 293–320. https://doi.org/10.4039/Ent91293-5 Jeschke, J. M., Kopp, M., & Tollrian, R. (2002). Predator functional responses: Discriminating between handling and digesting prey. Ecol. Monogr.,72(1), 95–112. https://doi.org/10.1890/0012-9615%282002%29072% 5B0095%3APFRDBH%5D2.0.CO%3B2 Juliano, S. A. (2001). Nonlinear curve fitting: Predation and functional response curves. In S. M. Scheiner & J. Gurevitch (Eds.), Design and analysis of ecological experiments (2nd Edition, pp. 178–196). Chapman and Hall. https://doi.org/10.1093/oso/9780195131871.003.0010 Kalinkat, G., Rall, B. C., Uiterwaal, S., & Uszko, W. (2023). Empirical evidence of type III functional responses and why it remains rare. Front. Ecol. Evol.,11, 1033818. https://doi.org/10.3389/fevo.2023.1033818 Lyon, N. J. (2024). supportR: Support functions for wrangling and visualization.https://doi.org/10.32614/ CRAN.package.supportR Microsoft, & Weston, S. (2022a). doParallel: Foreach parallel adaptor for the ’parallel’ package.https: //doi.org/10.32614/CRAN.package.doParallel Microsoft, & Weston, S. (2022b). foreach: Provides foreach looping construct.https://doi.org/10.32614/CRAN. package.foreach Müller, K. (2025). here: A simpler way to find your files.https://doi.org/10.32614/CRAN.package.here Pritchard, D. W. (2025). frair: Tools for functional response analysis.https://doi.org/10.32614/CRAN.package. frair Pritchard, D. W., Paterson, R. A., Bovy, H. C., & Barrios-O’Neill, D. (2017). frair: An R package for fitting and comparing consumer functional responses. Methods Ecol. Evol.,8(11), 1528–1534. https: //doi.org/10.1111/2041-210X.12784 Rall, B. C., Aranbarri, M., Flores, L., de Guzmán, I., Larrañaga, A., & Reiss, J. (2025a). Habitat complexity 18 reduces feeding strength of freshwater predators (CRITTER) — Code. Zenodo. https://doi.org/10.5281/ zenodo.14894598 Rall, B. C., Aranbarri, M., Flores, L., de Guzmán, I., Larrañaga, A., & Reiss, J. (2025b). Habitat complexity reduces feeding strength of freshwater predators (CRITTER) — Supplemental Statistics Report. Zenodo. https://doi.org/10.5281/zenodo.14898819 Rall, B. C., Guill, C., & Brose, U. (2008). Food-web connectance and predator interference dampen the paradox of enrichment. Oikos,117(2), 202–213. https://doi.org/10.1111/j.2007.0030-1299.15491.x Real, L. A. (1977). The kinetics of functional response. Am. Nat.,111(978), 289–300. https://doi.org/10.1086/ 283161 Real, L. A. (1979). Ecological determinants of functional response. Ecology,60 (3), 481–485. https://doi.org/10. 2307/1936067 Ripley, B., & Venables, B. (2025). MASS: Support functions and datasets for Venables and Ripley’s MASS. https://doi.org/10.32614/CRAN.package.MASS Rogers, D. (1972). Random search and insect population models. J. Anim. Ecol.,41(2), 369–383. https: //doi.org/10.2307/3474 Rosenbaum, B., & Rall, B. C. (2018). Fitting functional responses: Direct parameter estimation by simulating differential equations. Methods Ecol. Evol.,9(10), 2076–2090. https://doi.org/10.1111/2041-210X.13039 Royama, T. (1971). A comparative study of models for predation and parasitism. Res. Popul. Ecol.,13(1), 1–91. https://doi.org/10.1007/BF02511547 Schwarz, G. (1978). Estimating the dimension of a model. The Annals of Statistics,6(2), 461–464. https: //doi.org/10.1214/aos/1176344136 Urbanek, S. (2022). png: Read and write PNG images.https://doi.org/10.32614/CRAN.package.png Vonesh, J. R., & Bolker, B. M. (2005). Compensatory larval responses shift trade-offs associated with predatorinduced hatching plasticity. Ecology,86(6), 1580–1591. https://doi.org/10.1890/04-0535 Vucic-Pestic, O., Rall, B. C., Kalinkat, G., & Brose, U. (2010). Allometric functional response model: Body masses constrain interaction strengths. J. Anim. Ecol.,79(1), 249–256. https://doi.org/10.1111/j.13652656.2009.01622.x Wickham, H., François, R., Henry, L., Müller, K., & Vaughan, D. (2023). dplyr: A grammar of data manipulation. https://doi.org/10.32614/CRAN.package.dplyr Wickham, H., & Henry, L. (2025). purrr: Functional programming tools.https://doi.org/10.32614/CRAN. package.purrr Williams, R. J., & Martinez, N. D. (2004). Stabilization of chaotic and non-permanent food-web dynamics. Eur. Phys. J. B.,38(2), 297–303. https://doi.org/10.1140/epjb/e2004-00122-1 Xie, Y. (2016). bookdown: Authoring books and technical documents with R Markdown. Chapman and Hall/CRC. https://bookdown.org/yihui/bookdown Xie, Y. (2025a). bookdown: Authoring books and technical documents with R Markdown.https://doi.org/10. 32614/CRAN.package.bookdown Xie, Y. (2025b). knitr: A general-purpose package for dynamic report generation in R.https://doi.org/10.32614/ CRAN.package.knitr Xie, Y., Allaire, J. J., & Grolemund, G. (2018). R Markdown: The definitive guide. Chapman and Hall/CRC. https://bookdown.org/yihui/rmarkdown Xie, Y., Dervieux, C., & Riederer, E. (2020). R Markdown cookbook. Chapman and Hall/CRC. https: //bookdown.org/yihui/rmarkdown-cookbook Xie, Y., & Posit Software, PBC. (2025). tinytex: Helper functions to install and maintain TeX Live, and compile LaTeX documents.https://doi.org/10.32614/CRAN.package.tinytex Zar, J. H. (2010). Biostatistical Analyses (5th ed.). Prentice Hall. https://elibrary.pearson.de/book/99.150005/ 9781292037110 Zhu, H. (2024). kableExtra: Construct complex table with ’kable’ and pipe syntax.https://doi.org/10.32614/ CRAN.package.kableExtra 19 Appendix Phenomenological Test Rationale Instead of fitting the number eaten prey items as depended variable as a function of the initial prey density, the ratio of prey eaten and initial prey items is used as depended variable (Juliano, 2001; Pritchard et al., 2017; Pritchard, 2025). In the case of a type II functional response, the proportion of eaten prey is constantly declining with increasing prey density, whereas in the case of a type III functional response, the proportion of prey eaten increases at low prey densities and declines after reaching a maximum proportion rate. The main trend of any functional response, if looking at the proportion data, is negative. This negative trend is caused by the fact, that initial prey density linearly increases, whereas the eaten prey items are leveling off. Please see Juliano (2001) for a full description. Pritchard et al. (2017) implemented this test using generalized linear models with a binomial distribution, a standard method to analyze proportion data based on yes/no data (i.e., whether a prey item is eaten or not eaten), but see Crawley (2012), chapter 16, for an introduction on this topic. We use here the customized function phen_type_test() which is a wrapper around the frair_test() function from the frair package (Pritchard, 2025). phen_type_table() creates a high-quality PDF report table based on the output of phen_type_test(). Both functions are located in the project’s sub-folder functions. 20 Full AIC / BIC tables Ischnura elegans Table 18: AIC / BIC table of all 32 possible models for Ischnura elegans. Model (AIC) df dAIC Model (BIC) df dBIC Model 5r 3 0.000 Model 5r 3 0.000 Model 7r 4 0.265 Model 5h 3 1.583 Model 6h 4 0.952 Model 7r 4 2.731 Model 6r 4 0.952 Model 6h 4 3.418 Model 5h 3 1.583 Model 6r 4 3.418 Model 13r 6 2.107 Model 7h 4 4.718 Model 7h 4 2.252 Model 2h 3 7.001 Model 8h 7 2.446 Model 2r 3 7.001 Model 8r 7 2.446 Model 9r 3 7.189 Model 14r 7 2.928 Model 10h 4 7.768 Model 15r 7 3.495 Model 13r 6 9.505 Model 14h 7 3.834 Model 10r 4 10.128 Model 13h 6 4.026 Model 9h 3 11.069 Model 15h 7 4.039 Model 13h 6 11.424 Model 12h 7 5.271 Model 11h 4 11.468 Model 12r 7 5.271 Model 11r 4 11.468 Model 10h 4 5.303 Model 3h 3 11.887 Model 16h 10 5.945 Model 3r 3 11.887 Model 16r 10 5.945 Model 8h 7 12.309 Model 2h 3 7.001 Model 8r 7 12.309 Model 2r 3 7.001 Model 14r 7 12.791 Model 9r 3 7.189 Model 15r 7 13.358 Model 10r 4 7.663 Model 14h 7 13.697 Model 4h 6 8.317 Model 15h 7 13.902 Model 4r 6 8.317 Model 12h 7 15.135 Model 11h 4 9.002 Model 12r 7 15.135 Model 11r 4 9.002 Model 4h 6 15.715 Model 9h 3 11.069 Model 4r 6 15.715 Model 3h 3 11.887 Model 1h 2 19.128 Model 3r 3 11.887 Model 1r 2 19.128 Model 1h 2 21.594 Model 16h 10 23.206 Model 1r 2 21.594 Model 16r 10 23.206 21 Notonecta glauca Table 19: AIC / BIC table of all 32 possible models for Notonecta glauca. Model (AIC) df dAIC Model (BIC) df dBIC Model 15h 7 0.000 Model 15h 7 0.000 Model 16h 10 1.274 Model 9r 3 1.004 Model 16r 10 1.274 Model 11h 4 1.485 Model 15r 7 1.990 Model 11r 4 1.485 Model 14r 7 4.189 Model 13r 6 1.507 Model 13r 6 4.854 Model 15r 7 1.990 Model 12h 7 8.352 Model 7h 4 2.516 Model 12r 7 8.352 Model 14r 7 4.189 Model 8h 7 8.459 Model 10r 4 4.478 Model 8r 7 8.459 Model 7r 4 5.947 Model 14h 7 9.938 Model 12h 7 8.352 Model 11h 4 11.526 Model 12r 7 8.352 Model 11r 4 11.526 Model 8h 7 8.459 Model 7h 4 12.557 Model 8r 7 8.459 Model 9r 3 14.392 Model 14h 7 9.938 Model 10r 4 14.519 Model 3h 3 10.853 Model 7r 4 15.988 Model 3r 3 10.853 Model 4h 6 20.141 Model 16h 10 11.316 Model 4r 6 20.141 Model 16r 10 11.316 Model 10h 4 21.456 Model 10h 4 11.415 Model 3h 3 24.241 Model 5r 3 16.378 Model 3r 3 24.241 Model 4h 6 16.794 Model 6h 4 28.922 Model 4r 6 16.794 Model 6r 4 28.922 Model 6h 4 18.880 Model 5r 3 29.767 Model 6r 4 18.880 Model 13h 6 33.151 Model 13h 6 29.804 Model 2h 3 43.253 Model 2h 3 29.864 Model 2r 3 43.253 Model 2r 3 29.864 Model 9h 3 44.833 Model 9h 3 31.445 Model 5h 3 53.840 Model 5h 3 40.451 Model 1h 2 125.439 Model 1h 2 108.704 Model 1r 2 125.439 Model 1r 2 108.704 22 All summary tables including the original bbmle::summary outputs The tables below are build using the function bbmle::summary from the bbmle package (Bolker & R. Development Core Team, 2023). This means that all values except the slopes are displayed on log10 scale in the column Orig. Est. (original estimate). The column Estimate. displays the values on normal scale. The significance values are all tested against “0”, in the case of the log10 -transformed parameters they are tested against a value of 1 on normal scale. As example, the maximum feeding rate value is tested if it differs significantly from 1 itemsprey day eaten, not if Fmax differs between treatments. Therefore, the significance for the log10 -transformed parameters are not relevant. Only the significance for the slopes is of interest as it tells if the model finds a significant effect of the amount of habitat in the experimental trial (0, 2, or 3 ring structures). Ischnura elegans Table 20: All 32 summary tables from Ischnura elegans model fits. Name Orig. Est. SE z p Estimate CI low CI up Model 1h t_h_log10 -1.204 0.065 -18.648 0.0000 0.063 0.047 0.083 a_log10 -0.510 0.054 -9.505 0.0000 0.309 0.243 0.394 Model 1r f_max_log10 1.204 0.065 18.648 0.0000 15.996 11.938 21.486 n_half_log10 1.714 0.107 16.036 0.0000 51.718 32.054 84.229 Model 2h t_h_log10 -1.277 0.077 -16.592 0.0000 0.053 0.038 0.075 a_hab0_log10 -0.257 0.085 -3.038 0.0024 0.553 0.377 0.813 a_hab1_log10 -0.643 0.061 -10.541 0.0000 0.228 0.173 0.300 Model 2r f_max_log10 1.277 0.077 16.592 0.0000 18.906 13.439 26.754 n_half_hab0_log10 1.534 0.134 11.424 0.0000 34.166 18.511 61.663 n_half_hab1_log10 1.920 0.126 15.274 0.0000 83.088 47.009 145.743 Model 3h t_h_log10 -1.254 0.073 -17.191 0.0000 0.056 0.040 0.077 a_intercept_log10 -0.307 0.082 -3.746 0.0002 0.493 0.338 0.711 a_slope -0.121 0.034 -3.509 0.0005 -0.121 -0.187 -0.054 Model 3r f_max_log10 1.254 0.073 17.191 0.0000 17.957 12.959 24.865 n_half_intercept_log10 1.562 0.129 12.102 0.0000 36.451 20.396 64.587 n_half_slope 0.121 0.034 3.509 0.0005 0.121 0.054 0.188 Model 4h t_h_log10 -1.277 0.075 -16.930 0.0000 0.053 0.038 0.074 a_0_log10 -0.257 0.084 -3.047 0.0023 0.553 0.378 0.804 a_1_log10 -0.813 0.099 -8.248 0.0000 0.154 0.099 0.242 a_2_log10 -0.554 0.104 -5.319 0.0000 0.279 0.176 0.445 a_3_log10 -0.612 0.094 -6.534 0.0000 0.244 0.161 0.370 a_4_log10 -0.589 0.105 -5.636 0.0000 0.257 0.161 0.413 Model 4r f_max_log10 1.277 0.075 16.930 0.0000 18.907 13.475 26.373 n_half_0_log10 1.534 0.132 11.576 0.0000 34.168 18.725 62.361 n_half_1_log10 2.089 0.142 14.724 0.0000 122.880 64.842 232.165 n_half_2_log10 1.831 0.152 12.023 0.0000 67.712 34.019 132.893 n_half_3_log10 1.889 0.140 13.493 0.0000 77.424 40.938 143.808 n_half_4_log10 1.866 0.154 12.150 0.0000 73.468 37.366 143.832 Model 5h t_h_hab0_log10 -1.792 0.237 -7.545 0.0000 0.016 0.005 0.047 t_h_hab1_log10 -1.084 0.064 -16.928 0.0000 0.082 0.062 0.110 a_log10 -0.499 0.054 -9.273 0.0000 0.317 0.248 0.405 (continued on next page...) 23 Table 20: All 32 summary tables from Ischnura elegans model fits. (continued) Name Orig. Est. SE z p Estimate CI low CI up Model 5r f_max_hab0_log10 1.448 0.083 17.543 0.0000 28.081 19.385 40.615 f_max_hab1_log10 1.164 0.069 16.839 0.0000 14.578 10.687 19.836 n_half_log10 1.756 0.111 15.797 0.0000 57.000 34.351 93.273 Model 6h t_h_hab0_log10 -1.578 0.176 -8.959 0.0000 0.026 0.012 0.059 t_h_hab1_log10 -1.126 0.074 -15.163 0.0000 0.075 0.054 0.104 a_hab0_log10 -0.378 0.092 -4.091 0.0000 0.419 0.276 0.637 a_hab1_log10 -0.561 0.065 -8.607 0.0000 0.275 0.205 0.369 Model 6r f_max_hab0_log10 1.578 0.176 8.959 0.0000 37.807 17.087 83.325 f_max_hab1_log10 1.126 0.074 15.163 0.0000 13.361 9.510 18.842 n_half_hab0_log10 1.955 0.250 7.837 0.0000 90.245 28.903 270.351 n_half_hab1_log10 1.687 0.126 13.389 0.0000 48.647 27.588 85.918 Model 7h t_h_hab0_log10 -1.639 0.198 -8.274 0.0000 0.023 0.009 0.056 t_h_hab1_log10 -1.107 0.070 -15.876 0.0000 0.078 0.057 0.107 a_intercept_log10 -0.420 0.087 -4.817 0.0000 0.380 0.256 0.560 a_slope -0.047 0.040 -1.157 0.2471 -0.047 -0.126 0.033 Model 7r f_max_hab0_log10 1.563 0.133 11.770 0.0000 36.596 20.103 67.390 f_max_hab1_log10 1.132 0.071 15.910 0.0000 13.563 9.870 18.770 n_half_intercept_log10 1.935 0.185 10.474 0.0000 86.008 37.716 200.551 n_half_slope -0.094 0.074 -1.272 0.2035 -0.094 -0.241 0.051 Model 8h t_h_hab0_log10 -1.578 0.176 -8.959 0.0000 0.026 0.012 0.059 t_h_hab1_log10 -1.133 0.074 -15.380 0.0000 0.074 0.053 0.103 a_0_log10 -0.378 0.092 -4.091 0.0000 0.419 0.273 0.637 a_1_log10 -0.748 0.108 -6.904 0.0000 0.179 0.110 0.292 a_2_log10 -0.468 0.111 -4.201 0.0000 0.341 0.205 0.562 a_3_log10 -0.541 0.104 -5.219 0.0000 0.288 0.182 0.460 a_4_log10 -0.504 0.110 -4.585 0.0000 0.313 0.192 0.511 Model 8r f_max_hab0_log10 1.578 0.176 8.959 0.0000 37.807 16.923 84.661 f_max_hab1_log10 1.133 0.074 15.380 0.0000 13.569 9.694 18.962 n_half_0_log10 1.955 0.250 7.837 0.0000 90.243 28.569 278.225 n_half_1_log10 1.880 0.150 12.570 0.0000 75.896 39.054 151.398 n_half_2_log10 1.600 0.156 10.243 0.0000 39.837 19.612 80.945 n_half_3_log10 1.674 0.148 11.291 0.0000 47.174 24.132 92.077 n_half_4_log10 1.637 0.153 10.687 0.0000 43.351 21.851 85.612 Model 9h t_h_intercept_log10 -1.528 0.132 -11.571 0.0000 0.030 0.016 0.055 t_h_slope 0.162 0.050 3.261 0.0011 0.162 0.065 0.259 a_log10 -0.495 0.053 -9.382 0.0000 0.320 0.251 0.406 Model 9r f_max_intercept_log10 1.402 0.082 17.130 0.0000 25.244 17.486 36.804 f_max_slope -0.091 0.022 -4.179 0.0000 -0.091 -0.133 -0.048 n_half_log10 1.750 0.110 15.840 0.0000 56.226 34.670 93.451 Model 10h t_h_intercept_log10 -1.428 0.120 -11.919 0.0000 0.037 0.022 0.064 t_h_slope 0.096 0.051 1.887 0.0592 0.096 -0.004 0.195 a_hab0_log10 -0.320 0.087 -3.692 0.0002 0.479 0.325 0.711 (continued on next page...) 24 Table 20: All 32 summary tables from Ischnura elegans model fits. (continued) Name Orig. Est. SE z p Estimate CI low CI up a_hab1_log10 -0.598 0.064 -9.422 0.0000 0.252 0.187 0.338 Model 10r f_max_intercept_log10 1.339 0.094 14.301 0.0000 21.827 14.223 33.230 f_max_slope -0.048 0.041 -1.160 0.2461 -0.048 -0.129 0.035 n_half_hab0_log10 1.622 0.153 10.633 0.0000 41.882 20.931 82.863 n_half_hab1_log10 1.834 0.136 13.524 0.0000 68.270 37.019 124.623 Model 11h t_h_intercept_log10 -1.440 0.124 -11.641 0.0000 0.036 0.021 0.063 t_h_slope 0.111 0.052 2.135 0.0327 0.111 0.009 0.212 a_intercept_log10 -0.372 0.084 -4.453 0.0000 0.425 0.291 0.619 a_slope -0.077 0.039 -2.001 0.0454 -0.077 -0.156 -0.002 Model 11r f_max_intercept_log10 1.440 0.124 11.640 0.0000 27.542 15.637 48.333 f_max_slope -0.111 0.052 -2.136 0.0327 -0.111 -0.210 -0.009 n_half_intercept_log10 1.812 0.184 9.852 0.0000 64.841 28.173 149.807 n_half_slope -0.034 0.079 -0.430 0.6669 -0.034 -0.188 0.123 Model 12h t_h_intercept_log10 -1.476 0.134 -11.024 0.0000 0.033 0.018 0.061 t_h_slope 0.117 0.055 2.132 0.0330 0.117 0.007 0.224 a_0_log10 -0.339 0.088 -3.863 0.0001 0.458 0.309 0.681 a_1_log10 -0.798 0.100 -7.961 0.0000 0.159 0.101 0.249 a_2_log10 -0.533 0.105 -5.093 0.0000 0.293 0.183 0.466 a_3_log10 -0.536 0.107 -5.010 0.0000 0.291 0.179 0.469 a_4_log10 -0.500 0.113 -4.429 0.0000 0.316 0.190 0.523 Model 12r f_max_intercept_log10 1.476 0.134 11.024 0.0000 29.898 16.646 54.936 f_max_slope -0.117 0.055 -2.132 0.0330 -0.117 -0.225 -0.011 n_half_0_log10 1.815 0.200 9.092 0.0000 65.239 27.270 161.085 n_half_1_log10 2.039 0.138 14.746 0.0000 109.348 58.517 204.919 n_half_2_log10 1.774 0.147 12.083 0.0000 59.458 30.684 116.514 n_half_3_log10 1.660 0.164 10.137 0.0000 45.741 21.537 95.276 n_half_4_log10 1.624 0.168 9.686 0.0000 42.029 19.249 88.181 Model 13h t_h_0_log10 -1.765 0.224 -7.883 0.0000 0.017 0.006 0.048 t_h_1_log10 -0.944 0.111 -8.513 0.0000 0.114 0.069 0.187 t_h_2_log10 -1.044 0.117 -8.904 0.0000 0.090 0.054 0.152 t_h_3_log10 -1.056 0.126 -8.379 0.0000 0.088 0.049 0.154 t_h_4_log10 -1.203 0.097 -12.358 0.0000 0.063 0.040 0.097 a_log10 -0.488 0.055 -8.902 0.0000 0.325 0.255 0.415 Model 13r f_max_0_log10 1.442 0.082 17.509 0.0000 27.663 19.146 39.952 f_max_1_log10 1.056 0.090 11.759 0.0000 11.369 7.598 17.055 f_max_2_log10 1.185 0.088 13.388 0.0000 15.298 10.277 22.818 f_max_3_log10 1.169 0.091 12.886 0.0000 14.750 9.781 21.994 f_max_4_log10 1.213 0.077 15.704 0.0000 16.325 11.502 23.021 n_half_log10 1.745 0.111 15.673 0.0000 55.625 33.461 92.193 Model 14h t_h_0_log10 -1.578 0.176 -8.959 0.0000 0.026 0.012 0.058 t_h_1_log10 -0.981 0.119 -8.209 0.0000 0.104 0.062 0.180 t_h_2_log10 -1.079 0.127 -8.508 0.0000 0.083 0.047 0.150 t_h_3_log10 -1.107 0.142 -7.810 0.0000 0.078 0.041 0.147 t_h_4_log10 -1.240 0.109 -11.393 0.0000 0.058 0.035 0.093 a_hab0_log10 -0.378 0.092 -4.091 0.0000 0.419 0.277 0.636 (continued on next page...) 25 Table 22: All loaded R packages that we used in this report, including base packages and dependencies. (continued) Package Name Version Is Package Attached? Is a Base Package? MASS 7.3-65 FALSE FALSE Matrix 1.7-4 FALSE FALSE mvtnorm 1.3-3 FALSE FALSE numDeriv 2016.8-1.1 FALSE FALSE pillar 1.11.1 FALSE FALSE pkgconfig 2.0.3 FALSE FALSE pkgload 1.4.1 FALSE FALSE plyr 1.8.9 FALSE FALSE png 0.1-8 FALSE FALSE purrr 1.1.0 FALSE FALSE R6 2.6.1 FALSE FALSE rappdirs 0.3.3 FALSE FALSE RColorBrewer 1.1-3 FALSE FALSE Rcpp 1.1.0 FALSE FALSE RcppParallel 5.1.11-1 FALSE FALSE RCurl 1.98-1.17 FALSE FALSE rlang 1.1.6 FALSE FALSE rmarkdown 2.30 FALSE FALSE rprojroot 2.1.1 FALSE FALSE rstudioapi 0.17.1 FALSE FALSE scales 1.4.0 FALSE FALSE sessioninfo 1.2.3 FALSE FALSE stringi 1.8.7 FALSE FALSE stringr 1.5.2 FALSE FALSE supportR 1.5.0 FALSE FALSE svglite 2.2.1 FALSE FALSE systemfonts 1.3.1 FALSE FALSE textshaping 1.0.4 FALSE FALSE tibble 3.3.0 FALSE FALSE tidyselect 1.2.1 FALSE FALSE tinytex 0.57 FALSE FALSE vctrs 0.6.5 FALSE FALSE viridisLite 0.4.2 FALSE FALSE withr 3.0.2 FALSE FALSE xfun 0.53 FALSE FALSE xml2 1.4.0 FALSE FALSE yaml 2.3.10 FALSE FALSE 32