Package 'BayesFR'

Title: Fitting Functional Responses in 1- and 2-Prey Systems
Description: Easy application of Bayesian inference for functional responses via 'brms'. This package allows to fit various FR models for single- and multi-prey experiments by providing nonlinear prediction functions for 'brms'. It uses dynamical prediction models to correct for prey depletion. The 'brms' framework facilitates statistical modeling and enables users to conveniently incorporate covariates such as temperature gradients, experimental treatment variables, or random effects that account for grouping in experimental units. Default 'brms' functions make it easy to perform model checking, model comparison and hypothesis testing. Potential statistical issues with data from feeding trials, such as overdispersion, can be resolved by effortlessly switching between likelihood functions. This package, together with its tutorials, should provide students and researchers with a comprehensive and integrated statistical framework for easily testing their hypotheses on trophic interactions. References: Rosenbaum and Rall (2018) <doi:10.1111/2041-210X.13039>; Rosenbaum et al. (2024) <doi:10.1111/2041-210X.14372>.
Authors: Benjamin Rosenbaum [aut, cre] (ORCID: <https://orcid.org/0000-0002-2815-0874>)
Maintainer: Benjamin Rosenbaum <[email protected]>
License: GPL (>= 3)
Version: 1.0.1
Built: 2026-06-24 09:30:40 UTC
Source: https://github.com/benjamin-rosenbaum/bayesfr

Help Index

Convert 2-prey data to long format

Description

brms requires univariate response values here. Transforms one row with bivariate (NE1,NE2) to two rows with NE=NE1 and NE=NE2, respectively. Species identity of focal prey is saved in column ID, initial abundance of focal prey in column N0, and initial abundance of non-focal, alternative prey in column N0.alt

Usage

convert_2sp_to_long(df)

Arguments

df

data frame with at least these columns named: N01, N02, NE1, NE2

Value

The transformed data frame

Example dataset for prey mortality

Description

Feeding experiment data from Archer et al. (2019a) were downloaded from Dryad (Archer et al. 2019b). Eaten prey were not replaced during the experiment. Includes data for housefly larvae (Limnophora riparia) and caddisfly larva (Potamophylax cingulatus) feeding on blackfly larvae (Simuliidae). Due to prey background mortality, control experiments without predators were performed, too. Includes a temperature gradient and data from 2 settings (lab/field) and from 2 years.

Usage

data(df_Archer_et_al_2019_JAE)

Format

A data frame with 580 rows and 9 variables:

N0

Number of initial prey

NE

Number of eaten prey

P0

Number of predators (0 or 1)

Time

Duration (h)

Predator

Predator species or control

Prey

Prey species

Temperature

Experimental temperature

Setting

laboratory or field

Year

2013 or 2015

Source

Archer L. C., Sohlström E. H., Gallo B., Jochum M., Woodward G., Kordas R. L., Rall B. C. & O'Gorman E. J. (2019a). Consistent temperature dependence of functional response parameters and their use in predicting population abundance. Journal of Animal Ecology, 88:1670-1683. https://doi.org/10.1111/1365-2656.13060

Archer L. C., Sohlström E. H., Gallo B., Jochum M., Woodward G., Kordas R. L., Rall B. C. & O'Gorman E. J. (2019b). Consistent temperature dependence of functional response parameters and their use in predicting population abundance. Dryad Digital Repository. https://doi.org/10.5061/dryad.tr4v447

Examples

data(df_Archer_et_al_2019_JAE)
head(df_Archer_et_al_2019_JAE)

Example dataset for multi-species FR with 2 prey

Description

Feeding experiment data from Colton (1987) were downloaded from Figshare (Novak & Stouffer 2020). Eaten prey were not replaced during the experiment. Includes data for 10th???instar naiads of a damselfly feeding on a cladoceran (Simocephalus serrulatus, species 1) and a copepod (Diaptomus spatulocrenatus, species 2).

Usage

data(df_Colton_1987_1_ECOLOGY)

Format

A data frame with 108 rows and 6 variables:

N01

Number of initial prey, species 1

N02

Number of initial prey, species 2

NE1

Number of eaten prey, species 1

NE2

Number of eaten prey, species 2

P0

Number of predators (1)

Time

Duration (h)

Details

A single typo (720) was corrected (120).

Source

Colton, T.F. (1987). Extending functional response models to include a second prey type: an experimental test. Ecology, 68: 900-912. https://doi.org/10.2307/1938361

Novak, M., & Stouffer, D. (2020). Data extracted for "Hidden layers of density dependence in consumer feeding rates." Figshare. https://doi.org/10.6084/m9.figshare.12830792

Examples

data(df_Colton_1987_1_ECOLOGY)
head(df_Colton_1987_1_ECOLOGY)

Example dataset for categorical predictors

Description

Feeding experiment data from Cuthbert et al. (2020a) were downloaded from Dryad (Cuthbert et al. 2020b). Eaten prey were not replaced during the experiment. Includes data for two fish species (largemouth bass and bluegill) feeding on tilapia. Both predator and prey were categorized in three size classes, each, with a full factorial treatment.

Usage

data(df_Cuthbert_et_al_2020_ECOL_EVOL)

Format

A data frame with 358 rows and 7 variables:

N0

Number of initial prey

NE

Number of eaten prey

Time

Duration (h)

Predator

Predator species

Prey

Prey species

PredSize

Predator size class

PreySize

Prey size class

Source

Cuthbert R. N., Wassermann R. J., Dalu T., Kaiser H., Weyl O. L. F., Dick J. T. A., Sentis A., McCoy M. W., & Alexander M.E. (2020a). *Influence of intra- and interspecific variation in predator-prey body size ratios on trophic interaction strengths. Ecology and Evolution, 10:5946-5962. https://doi.org/10.1002/ece3.6332

Cuthbert R. N., Wassermann R. J., Dalu T., Kaiser H., Weyl O. L. F., Dick J. T. A., Sentis A., McCoy M. W., & Alexander M.E. (2020b). Influence of intra- and interspecific variation in predator-prey body size ratios on trophic interaction strengths. Dryad Digital Repository. https://doi.org/10.5061/dryad.7m0cfxppt

Examples

data(df_Cuthbert_et_al_2020_ECOL_EVOL)
head(df_Cuthbert_et_al_2020_ECOL_EVOL)

Example dataset for continuous predictors

Description

Feeding experiment data from Davidson et al. (2021) were downloaded from Dryad (Davidson et al. 2020). Eaten prey were not replaced during the experiment. Includes data for two dragonfly nymph species (Pachydiplax and Erythemis) feeding on mosquito larvae. Experiments were performed on a temperature gradient, and predator size was measured, too.

Usage

data(df_Davidson_et_al_2021_FUN_ECOL)

Format

A data frame with 91 rows and 7 variables:

N0

Number of initial prey

NE

Number of eaten prey

Time

Duration (h)

Predator

Predator species

Prey

Prey species

Temperature

Experimental temperature

HeadWidth

Predator size

Source

Davidson A. T., Hamman, E. A., McCoy M. W., and Vonesh J. R. (2021). Asymmetrical effects of temperature on stage-structured predator-prey interactions. Functional Ecology 35: 1041-1054. https://doi.org/10.1111/1365-2435.13777

Davidson A. T., Hamman, E. A., McCoy M. W., and Vonesh J. R. (2020). Asymmetrical effects of temperature on stage-structured predator-prey interactions. Dryad Digital Repository. https://doi.org/10.5061/dryad.j6q573nd4

Examples

data(df_Davidson_et_al_2021_FUN_ECOL)
head(df_Davidson_et_al_2021_FUN_ECOL)

Feeding experiments without prey replacement

Description

Feeding experiment data from Hossie and Murray (2010) downloaded from the FoRAGE database (Uiterwaal et al. 2022). Eaten prey were not replaced during the experiment. Includes data for a dragonfly nymph predator feeding on tadpoles in three leaf litter treatments.

Usage

data(df_Hossie_and_Murray_2010_OECOLOGIA)

Format

A data frame with 91 rows and 6 variables:

N0

Number of initial prey

NE

Number of eaten prey

Time

Duration (h)

Predator

Predator species

Prey

Prey species

ID

Leaf litter treatment

Source

Hossie T. J. and Murray D. S. (2010). You can't run but you can hide: refuge use in frog tadpoles elicits density-dependent predation by dragonfly larvae. Oecologia, 163, 395-404. https://doi.org/10.1007/s00442-010-1568-6

Uiterwaal S. F., Lagerstrom I. T., Lyon S. R., and DeLong, J. P. (2022). FoRAGE Database: A Compilation of Functional Responses for Consumers and Parasitoids. Ecology 103(7): e3706. https://doi.org/10.1002/ecy.3706

FoRAGE database V5 (2024). https://doi.org/10.5063/F1RX99KB

Examples

data(df_Hossie_and_Murray_2010_OECOLOGIA)
head(df_Hossie_and_Murray_2010_OECOLOGIA)

Feeding experiments with prey replacement

Description

Feeding experiment data from Michalko and Pekar (2017) downloaded from the FoRAGE database (Uiterwaal et al. 2022). Eaten prey were replaced during the experiment. Includes three predator-prey combinations with a top predator (Philodromus buchari), a mesopredator (Dictyna spp.) and a pest (C. pyri).

Usage

data(df_Michalko_and_Pekar_2017_AM_NAT)

Format

A data frame with 63 rows and 6 variables:

N0

Number of constant prey

NE

Number of eaten prey

Time

Duration (h)

Predator

Predator species

Prey

Prey species

ID

Predator-prey combination

Source

Michalko R. and Pekar S. (2017). The Behavioral Type of a Top Predator Drives the Short-Term Dynamic of Intraguild Predation. American Naturalist, 189, 242-253. https://doi.org/10.1086/690501

Uiterwaal S. F., Lagerstrom I. T., Lyon S. R., and DeLong, J. P. (2022). FoRAGE Database: A Compilation of Functional Responses for Consumers and Parasitoids. Ecology 103(7): e3706. https://doi.org/10.1002/ecy.3706

FoRAGE database V5 (2024). https://doi.org/10.5063/F1RX99KB

Examples

data(df_Michalko_and_Pekar_2017_AM_NAT)
head(df_Michalko_and_Pekar_2017_AM_NAT)

Example dataset for testing predator interference models

Description

Feeding experiment data from Papanikolaou et al. (2021a) downloaded from Dryad (Papanikolaou et al. 2021b). Eaten prey were not replaced during the experiment. Includes data for two mirid predators (1st and 5th instar nymphs) feeding on Pyralidae eggs. Includes four predator treatments with 1,2,3 or 4 predators, each.

Usage

data(df_Papanikolaou_et_al_2021_ECOL_EVOL)

Format

A data frame with 327 rows and 7 variables:

N0

Number of initial prey

NE

Number of eaten prey

P0

Number of predator individuals

Time

Duration (h)

Predator

Predator species

Prey

Prey species

ID

1st or 5th instar nymphs

Source

Papanikolaou N.E., Dervisoglou S., Fantinou A., Kypraios T., Giakoumaki V., Perdikis D. (2021a). Predator size affects the intensity of mutual interference in a predatory mirid. Ecology and Evolution 2021(11): 1342???1351. https://doi.org/10.1002/ece3.7137

Papanikolaou N.E., Dervisoglou S., Fantinou A., Kypraios T., Giakoumaki V., Perdikis D. (2021b). Data from: Predator size affects the intensity of mutual interference in a predatory mirid. Dryad https://doi.org/10.5061/dryad.2ngf1vhmj

Examples

data(df_Papanikolaou_et_al_2021_ECOL_EVOL)
head(df_Papanikolaou_et_al_2021_ECOL_EVOL)

Example dataset for random effects (predator individual)

Description

Feeding experiment data from Schr??der et al. (2016) were downloaded from Figshare (Kalinkat et al. 2025) under CC BY 4.0. Eaten prey were not replaced during the experiments (2 minutes). Includes data for least killifish (Heterandria formosa, 49 individuals) feeding on nauplii (Artemia salina). Predator individuals were re-used and predator ID was recorded for each trial. Also includes predator size.

Usage

data(df_Schroeder_et_al_2016_OEC)

Format

A data frame with 686 rows and 8 variables:

N0

Number of initial prey

NE

Number of eaten prey

Time

Duration (h): 2 min

ID

Predator individual ID

Size

Predator individual size (mm)

Predator

Predator species

Prey

Prey species

Trial.time

Trials performed in the morning or evening

Source

Schröder A., Kalinkat G. & Arlinghaus R. (2016). Individual variation in functional response parameters is explained by body size but not by behavioural types in a poeciliid fish. Oecologia, 88:1670???1683. https://doi.org/10.1007/s00442-016-3701-7

Kalinkat G., Schröder A. & Arlinghaus R. (2025). Individual variation in functional response parameters is explained by body size but not by behavioural types in a poeciliid fish. Figshare. https://doi.org/10.6084/m9.figshare.24665880

Examples

data(df_Schroeder_et_al_2016_OEC)
head(df_Schroeder_et_al_2016_OEC)

Example dataset for testing type 2 vs. type 3

Description

Feeding experiment data from Sentis et al. (2017) downloaded from the FoRAGE database (Uiterwaal et al. 2022). Eaten prey were not replaced during the experiment. Includes data for three aquatic insect larvae predators feeding on Daphnia prey in two temperature treatments.

Usage

data(df_Sentis_et_al_2017_GLOBAL_CHANGE_BIOLOGY)

Format

A data frame with 327 rows and 7 variables:

N0

Number of initial prey

NE

Number of eaten prey

Time

Duration (h)

Predator

Predator species

Prey

Prey species

ID

Predator-Temperature combination

Temperature

Temperature treatment

Source

Sentis A., Gemard C., Jaugeon B., and Boukal D. S. (2017). Predator diversity and environmental change modify the strengths of trophic and nontrophic interactions. Global Change Biology, 23: 2629-2640. https://doi.org/10.1111/gcb.13560

Uiterwaal S. F., Lagerstrom I. T., Lyon S. R., and DeLong, J. P. (2022). FoRAGE Database: A Compilation of Functional Responses for Consumers and Parasitoids. Ecology 103(7): e3706. https://doi.org/10.1002/ecy.3706

FoRAGE database V5 (2024). https://doi.org/10.5063/F1RX99KB

Examples

data(df_Sentis_et_al_2017_GLOBAL_CHANGE_BIOLOGY)
head(df_Sentis_et_al_2017_GLOBAL_CHANGE_BIOLOGY)

Type 2 multi-species FR without replacement

Description

Contains Stan function of the same name as character string. Uses numerical solution of the 2-prey ODE

dN1dt=−a1N11+a1h1N1+a2h2N2P\frac{dN_1}{dt}=-\frac{a_1N_1}{1+a_1h_1N_1+a_2h_2N_2}P

dN2dt=−a2N21+a1h1N1+a2h2N2P\frac{dN_2}{dt}=-\frac{a_2N_2}{1+a_1h_1N_1+a_2h_2N_2}P

to compute number of eaten prey, see Rosenbaum et al. (2024).

Usage

MS_Type2H_dyn_code

Details

Usage in brms formula:

⁠~ MS_Type2H_dyn(N0, N0.alt, ID, P, Time, a1, a2, h1, h2)⁠

⁠N0 ⁠ initial number of focal prey species
⁠N0.alt⁠ initial number of alternative prey species
⁠ID ⁠ ID of focal species (1 or 2)
⁠P ⁠ number of predators
⁠Time ⁠ duration of the experiment
⁠a1 ⁠ attack rate for prey species 1
⁠a2 ⁠ attack rate for prey species 2
⁠h1 ⁠ handling time for prey species 1
⁠h2 ⁠ handling time for prey species 2

Requires the data to be in a specific "long" format. Use the function ⁠convert_2sp_to_long()⁠ to transform from a dataframe with columns N01, N02, NE1, NE2

Value

Number of eaten prey

References

Rosenbaum, B., Li, J., Hirt, M. R., Ryser, R., & Brose, U. (2024). Towards understanding interactions in a complex world: Design and analysis of multi-species functional response experiments. Methods in Ecology and Evolution, 15, 1704-1719. https://doi.org/10.1111/2041-210X.14372

Type 3 multi-species FR without replacement

Description

Contains Stan function of the same name as character string. Uses numerical solution of the 2-prey ODE

dN1dt=−b1N11+q1+b1h1N11+q+b2h2N21+qP\frac{dN_1}{dt}=-\frac{b_1N_1^{1+q}}{1+b_1h_1N_1^{1+q}+b_2h_2N_2^{1+q}}P

dN2dt=−b2N21+q1+b1h1N11+q+b2h2N21+qP\frac{dN_2}{dt}=-\frac{b_2N_2^{1+q}}{1+b_1h_1N_1^{1+q}+b_2h_2N_2^{1+q}}P

to compute number of eaten prey, see Rosenbaum et al. (2024).

Usage

MS_Type3H_dyn_code

Details

Usage in brms formula:

⁠~ MS_Type3H_dyn(N0, N0.alt, ID, P, Time, b1, b2, h1, h2, q)⁠

⁠N0 ⁠ initial number of focal prey species
⁠N0.alt⁠ initial number of alternative prey species
⁠ID ⁠ ID of focal species (1 or 2)
⁠P ⁠ number of predators
⁠Time ⁠ duration of the experiment
⁠b1 ⁠ attack coefficient for prey species 1
⁠b2 ⁠ attack coefficient for prey species 2
⁠h1 ⁠ handling time for prey species 1
⁠h2 ⁠ handling time for prey species 2
⁠q ⁠ attack rate exponent

Requires the data to be in a specific "long" format. Use the function ⁠convert_2sp_to_long()⁠ to transform from a dataframe with columns N01, N02, NE1, NE2

Value

Number of eaten prey

References

Rosenbaum, B., Li, J., Hirt, M. R., Ryser, R., & Brose, U. (2024). Towards understanding interactions in a complex world: Design and analysis of multi-species functional response experiments. Methods in Ecology and Evolution, 15, 1704-1719. https://doi.org/10.1111/2041-210X.14372

Yodzis FR without replacement

Description

Contains Stan function of the same name as character string. Uses numerical solution of the 2-prey ODE

dN1dt=−w1a1N11+r(w1N1r+(1−w1)N2r)+w1a1h1N11+r+(1−w1)a2h2N21+rP\frac{dN_1}{dt}=-\frac{w_1a_1N_1^{1+r}}{(w_1N_1^r+(1-w_1)N_2^r)+w_1a_1h_1N_1^{1+r}+(1-w_1)a_2h_2N_2^{1+r}}P

dN2dt=−(1−w1)a2N21+r(w1N1r+(1−w1)N2r)+w1a1h1N11+r+(1−w1)a2h2N21+rP\frac{dN_2}{dt}=-\frac{(1-w_1)a_2N_2^{1+r}}{(w_1N_1^r+(1-w_1)N_2^r)+w_1a_1h_1N_1^{1+r}+(1-w_1)a_2h_2N_2^{1+r}}P

to compute number of eaten prey, see Rosenbaum et al. (2024).

Usage

MS_TypeY_dyn_code

Details

Usage in brms formula:

⁠~ MS_TypeY_dyn(N0, N0.alt, ID, P, Time, a1, a2, h1, h2, w1, r)⁠

⁠N0 ⁠ initial number of focal prey species
⁠N0.alt⁠ initial number of alternative prey species
⁠ID ⁠ ID of focal species (1 or 2)
⁠P ⁠ number of predators
⁠Time ⁠ duration of the experiment
⁠a1 ⁠ attack rate for prey species 1
⁠a2 ⁠ attack rate for prey species 2
⁠h1 ⁠ handling time for prey species 1
⁠h2 ⁠ handling time for prey species 2
⁠w1 ⁠ preference weight for species 1 in multi-prey
⁠r ⁠ additional exponent in multi-prey only

Requires the data to be in a specific "long" format. Use the function ⁠convert_2sp_to_long()⁠ to transform from a dataframe with columns N01, N02, NE1, NE2

Value

Number of eaten prey

References

Rosenbaum, B., Li, J., Hirt, M. R., Ryser, R., & Brose, U. (2024). Towards understanding interactions in a complex world: Design and analysis of multi-species functional response experiments. Methods in Ecology and Evolution, 15, 1704-1719. https://doi.org/10.1111/2041-210X.14372

Generalized switching FR without replacement

Description

Contains Stan function of the same name as character string. Uses numerical solution of the 2-prey ODE

dN1dt=−w1b1N11+q+r(w1N1r+(1−w1)N2r)+w1b1h1N11+q+r+(1−w1)b2h2N21+q+rP\frac{dN_1}{dt}=-\frac{w_1b_1N_1^{1+q+r}}{(w_1N_1^r+(1-w_1)N_2^r)+w_1b_1h_1N_1^{1+q+r}+(1-w_1)b_2h_2N_2^{1+q+r}}P

dN2dt=−(1−w1)b2N21+q+r(w1N1r+(1−w1)N2r)+w1b1h1N11+q+r+(1−w1)b2h2N21+q+rP\frac{dN_2}{dt}=-\frac{(1-w_1)b_2N_2^{1+q+r}}{(w_1N_1^r+(1-w_1)N_2^r)+w_1b_1h_1N_1^{1+q+r}+(1-w_1)b_2h_2N_2^{1+q+r}}P

to compute number of eaten prey, see Rosenbaum et al. (2024).

Usage

MS_TypeZ_dyn_code

Details

Usage in brms formula:

⁠~ MS_TypeZ_dyn(N0, N0.alt, ID, P, Time, b1, b2, h1, h2, w1, q, r)⁠

⁠N0 ⁠ initial number of focal prey species
⁠N0.alt⁠ initial number of alternative prey species
⁠ID ⁠ ID of focal species (1 or 2)
⁠P ⁠ number of predators
⁠Time ⁠ duration of the experiment
⁠b1 ⁠ attack coefficient for prey species 1
⁠b2 ⁠ attack coefficient for prey species 2
⁠h1 ⁠ handling time for prey species 1
⁠h2 ⁠ handling time for prey species 2
⁠w1 ⁠ preference weight for species 1 in multi-prey
⁠q ⁠ attack rate exponent
⁠r ⁠ additional exponent in multi-prey only

Requires the data to be in a specific "long" format. Use the function ⁠convert_2sp_to_long()⁠ to transform from a dataframe with columns N01, N02, NE1, NE2

Value

Number of eaten prey

References

Rosenbaum, B., Li, J., Hirt, M. R., Ryser, R., & Brose, U. (2024). Towards understanding interactions in a complex world: Design and analysis of multi-species functional response experiments. Methods in Ecology and Evolution, 15, 1704-1719. https://doi.org/10.1111/2041-210X.14372

Type 1 FR with prey depletion

Description

Contains Stan function of the same name as character string. Uses analytical solution (exponential function) of the ODE

dNdt=−aNP\frac{dN}{dt}=-aNP

to compute number of eaten prey.

Usage

Type1_dyn_code

Details

Usage in brms formula:

⁠~ Type1_dyn(N, P, Time, a)⁠

⁠N ⁠ initial number of prey
⁠P ⁠ number of predators
⁠Time ⁠ duration of the experiment
⁠a ⁠ attack rate

Value

Number of eaten prey

Type 1 FR with prey replacement

Description

Contains Stan function of the same name as character string. Number of eaten prey:

NE=aNPTN_E=aNPT

Usage

Type1_fix_code

Details

Usage in brms formula:

⁠~ Type1_fix(N, P, Time, a)⁠

⁠N ⁠ initial number of prey
⁠P ⁠ number of predators
⁠Time ⁠ duration of the experiment
⁠a ⁠ attack rate

Value

Number of eaten prey

Functional response models with predator interference

Description

Contains Stan function of the same name as character string. Uses analytical solution of the Beddington-DeAngelis model

dNdt=−aN1+c(P−1)+ahNP\frac{dN}{dt}=-\frac{aN}{1+c(P-1)+ahN}P

to compute number of eaten prey. Rogers random predator equation with LambertW function is used with modified attack rates. Predator interference affects attack rates only.

Usage

Type2BD_dyn_code

Details

Usage in brms formula:

⁠~ Type2BD_dyn(N, P, Time, a, h, c)⁠

⁠N ⁠ initial number of prey
⁠P ⁠ number of predators
⁠Time ⁠ duration of the experiment
⁠a ⁠ attack rate for P=1
⁠h ⁠ handling time
⁠c ⁠ predator interference coefficient

Value

Number of eaten prey

Functional response models with predator interference

Description

Contains Stan function of the same name as character string. Uses analytical solution of the Crowley-Martin model

dNdt=−aN(1+ahN)(1+c(P−1))P\frac{dN}{dt}=-\frac{aN}{(1+ahN)(1+c(P-1))}P

to compute number of eaten prey. Rogers random predator equation with LambertW function is used with modified attack rates and handling times. Predator interference affects attack rates and handling times, both.

Usage

Type2CM_dyn_code

Details

Usage in brms formula:

⁠~ Type2CM_dyn(N, P, Time, a, h, c)⁠

⁠N ⁠ initial number of prey
⁠P ⁠ number of predators
⁠Time ⁠ duration of the experiment
⁠a ⁠ attack rate for P=1
⁠h ⁠ handling time
⁠c ⁠ predator interference coefficient

Value

Number of eaten prey

Type 2 FR (Holling) with prey depletion

Description

Contains Stan function of the same name as character string. Uses analytical solution (Rogers random predator equation with LambertW function) of the ODE

dNdt=−aN1+ahNP\frac{dN}{dt}=-\frac{aN}{1+ahN}P

to compute number of eaten prey.

Usage

Type2H_dyn_code

Details

Usage in brms formula:

⁠~ Type2H_dyn(N, P, Time, a, h)⁠

⁠N ⁠ initial number of prey
⁠P ⁠ number of predators
⁠Time ⁠ duration of the experiment
⁠a ⁠ attack rate
⁠h ⁠ handling time

Value

Number of eaten prey

Type 2 FR (Holling) with prey replacement

Description

Contains Stan function of the same name as character string. Number of eaten prey:

NE=aN1+ahNPTN_E=\frac{aN}{1+ahN}PT

Usage

Type2H_fix_code

Details

Usage in brms formula:

⁠~ Type2H_fix(N, P, Time, a, h)⁠

⁠N ⁠ initial number of prey
⁠P ⁠ number of predators
⁠Time ⁠ duration of the experiment
⁠a ⁠ attack rate
⁠h ⁠ handling time

Value

Number of eaten prey

Functional response models with predator interference

Description

Contains Stan function of the same name as character string. Uses analytical solution of the Hassell-Varley model

dNdt=−aNPc+ahNP\frac{dN}{dt}=-\frac{aN}{P^c+ahN}P

to compute number of eaten prey. Rogers random predator equation with LambertW function is used with modified attack rates. Predator interference affects attack rates only.

Usage

Type2HV_dyn_code

Details

Usage in brms formula:

⁠~ Type2HV_dyn(N, P, Time, a, h, c)⁠

⁠N ⁠ initial number of prey
⁠P ⁠ number of predators
⁠Time ⁠ duration of the experiment
⁠a ⁠ attack rate for P=1
⁠h ⁠ handling time
⁠c ⁠ predator interference coefficient

Value

Number of eaten prey

Generalized type 3 FR (Holling) with prey depletion

Description

Contains Stan function of the same name as character string. Uses numerical solution of the ODE

dNdt=−bN1+q1+bhN1+qP\frac{dN}{dt}=-\frac{bN^{1+q}}{1+bhN^{1+q}}P

to compute number of eaten prey.

Usage

Type3GenH_dyn_code

Details

Usage in brms formula:

⁠~ Type3GenH_dyn(N, P, Time, b, h, q)⁠

⁠N ⁠ initial number of prey
⁠P ⁠ number of predators
⁠Time ⁠ duration of the experiment
⁠b ⁠ attack coefficient
⁠q ⁠ attack exponent
⁠h ⁠ handling time

Value

Number of eaten prey

Generalized type 3 FR (Holling) with prey replacement

Description

Contains Stan function of the same name as character string. Number of eaten prey:

NE=bN1+q1+bhN1+qPTN_E=\frac{bN^{1+q}}{1+bhN^{1+q}}PT

Usage

Type3GenH_fix_code

Details

Usage in brms formula:

⁠~ Type3GenH_fix(N, P, Time, b, h, q)⁠

⁠N ⁠ initial number of prey
⁠P ⁠ number of predators
⁠Time ⁠ duration of the experiment
⁠b ⁠ attack coefficient
⁠q ⁠ attack exponent
⁠h ⁠ handling time

Value

Number of eaten prey

Functional response models with prey mortality

Description

Contains Stan function of the same name as character string. Uses numerical solution of the generalized type 3 ODE with an additional mortality term

dNdt=−bN1+q1+bhN1+qP−mN\frac{dN}{dt}=-\frac{bN^{1+q}}{1+bhN^{1+q}}P-mN

to compute number of eaten / dead prey. It can compute predictions for feeding trials (observations with ⁠P>0⁠) and also control treatments (⁠P=0⁠), for which the ODE reduces to

dNdt=−mN\frac{dN}{dt}=-mN

The exponent ⁠q⁠ can be fixed for fitting type 2 responses (⁠q=0⁠) or type 3 responses (⁠q=1⁠), which both do not have an analytical solution with additional and prey mortality have to be predicted using the ODE.

Usage

Type3GenH_mort_dyn_code

Details

Usage in brms formula:

⁠~ Type3GenH_dyn(N, P, Time, b, h, q, m)⁠

⁠N ⁠ initial number of prey
⁠P ⁠ number of predators
⁠Time ⁠ duration of the experiment
⁠b ⁠ attack coefficient
⁠q ⁠ attack exponent
⁠h ⁠ handling time
⁠m ⁠ mortality rate

Type 2 functional response:

⁠~ Type3GenH_dyn(N, P, Time, a, h, 0, m)⁠

Type 3 functional response:

⁠~ Type3GenH_dyn(N, P, Time, b, h, 1, m)⁠

Value

Number of eaten prey

Type 3 FR (Holling) with prey depletion

Description

Contains Stan function of the same name as character string. Uses analytical solution (quadratic equation) of the ODE

dNdt=−bN21+bhN2P\frac{dN}{dt}=-\frac{bN^2}{1+bhN^2}P

to compute number of eaten prey.

Usage

Type3H_dyn_code

Details

Usage in brms formula:

⁠~ Type3H_dyn(N, P, Time, b, h)⁠

⁠N ⁠ initial number of prey
⁠P ⁠ number of predators
⁠Time ⁠ duration of the experiment
⁠b ⁠ attack coefficient
⁠h ⁠ handling time

Value

Number of eaten prey

Type 3 FR (Holling) with prey replacement

Description

Contains Stan function of the same name as character string. Number of eaten prey:

NE=bN21+bhN2PTN_E=\frac{bN^2}{1+bhN^2}PT

Usage

Type3H_fix_code

Details

Usage in brms formula:

⁠~ Type3H_fix(N, P, Time, b, h)⁠

⁠N ⁠ initial number of prey
⁠P ⁠ number of predators
⁠Time ⁠ duration of the experiment
⁠b ⁠ attack coefficient
⁠h ⁠ handling time

Value

Number of eaten prey

Functional response models with predator interference

Description

Contains Stan function of the same name as character string. Uses numerical solution of the (generalized) Beddington-DeAngelis model

dNdt=−bN1+q1+c(P−1)+bhN1+qP\frac{dN}{dt}=-\frac{bN^{1+q}}{1+c(P-1)+bhN^{1+q}}P

to compute number of eaten prey. Predator interference affects attack rates only.

Usage

TypeGenBD_dyn_code

Details

The exponent ⁠q⁠ can be fixed for fitting type 2 responses (⁠q=0⁠) or type 3 responses (⁠q=1⁠).

Usage in brms formula:

⁠~ TypeGenBD_dyn(N, P, Time, b, h, q, c)⁠

⁠N ⁠ initial number of prey
⁠P ⁠ number of predators
⁠Time ⁠ duration of the experiment
⁠b ⁠ attack coefficient for P=1
⁠q ⁠ attack exponent
⁠h ⁠ handling time
⁠c ⁠ predator interference coefficient

Type 2 functional response:

⁠~ TypeGenBD_dyn(N, P, Time, a, h, 0, c)⁠

Type 3 functional response:

⁠~ TypeGenBD_dyn(N, P, Time, b, h, 1, c)⁠

Value

Number of eaten prey

Functional response models with predator interference

Description

Contains Stan function of the same name as character string. Uses numerical solution of the (generalized) Crowley-Martin model

dNdt=−bN1+q(1+bhN1+q)(1+c(P−1))P\frac{dN}{dt}=-\frac{bN^{1+q}}{(1+bhN^{1+q})(1+c(P-1))}P

to compute number of eaten prey. Predator interference affects attack rates and handling times, both.

Usage

TypeGenCM_dyn_code

Details

The exponent ⁠q⁠ can be fixed for fitting type 2 responses (⁠q=0⁠) or type 3 responses (⁠q=1⁠).

Usage in brms formula:

⁠~ TypeGenCM_dyn(N, P, Time, b, h, q, c)⁠

⁠N ⁠ initial number of prey
⁠P ⁠ number of predators
⁠Time ⁠ duration of the experiment
⁠b ⁠ attack coefficient for P=1
⁠q ⁠ attack exponent
⁠h ⁠ handling time
⁠c ⁠ predator interference coefficient

Type 2 functional response:

⁠~ TypeGenCM_dyn(N, P, Time, a, h, 0, c)⁠

Type 3 functional response:

⁠~ TypeGenCM_dyn(N, P, Time, b, h, 1, c)⁠

Value

Number of eaten prey

Functional response models with predator interference

Description

Contains Stan function of the same name as character string. Uses numerical solution of the (generalized) Hassell-Varley model

dNdt=−bN1+qPc+bhN1+qP\frac{dN}{dt}=-\frac{bN^{1+q}}{P^c+bhN^{1+q}}P

to compute number of eaten prey. Predator interference affects attack rates only.

Usage

TypeGenHV_dyn_code

Details

The exponent ⁠q⁠ can be fixed for fitting type 2 responses (⁠q=0⁠) or type 3 responses (⁠q=1⁠).

Usage in brms formula:

⁠~ TypeGenHV_dyn(N, P, Time, b, h, q, c)⁠

⁠N ⁠ initial number of prey
⁠P ⁠ number of predators
⁠Time ⁠ duration of the experiment
⁠b ⁠ attack coefficient for P=1
⁠q ⁠ attack exponent
⁠h ⁠ handling time
⁠c ⁠ predator interference coefficient

Type 2 functional response:

⁠~ TypeGenHV_dyn(N, P, Time, a, h, 0, c)⁠

Type 3 functional response:

⁠~ TypeGenHV_dyn(N, P, Time, b, h, 1, c)⁠

Value

Number of eaten prey