Compute the Jacobian matrix

Description

Compute the Jacobian matrix

Usage

create.jacob(
  val.mat,
  biomasses,
  efficiencies,
  metabolic.types,
  ef.level = "prey"
)

Arguments

Value

A matrix representing the Jacobian matrix of the dynamical system associated with the fluxes from the fluxing function.

Author(s)

Benoit Gauzens, benoit.gauzens@gmail.com

Examples

# First compute species per unit biomass metabolic rates:
losses = 0.15 * groups.level$bodymasses^(-0.25)


val.mat = fluxing(groups.level$mat, 
                  groups.level$biomasses, 
                  losses, 
                  groups.level$efficiencies, 
                  bioms.pref = TRUE, 
                  ef.level = "prey")
                  
# define metabolic types                   

met.types = rep("animal", nrow(val.mat))
met.types[groups.level$efficiencies == 0.545] = "plant"
met.types[groups.level$efficiencies == 0.158] = "detritus"

create.jacob(val.mat, 
               groups.level$biomasses, 
               groups.level$efficiencies, 
               met.types,
               ef.level = "prey")
               

generate fluxes

Description

Creates a valuated graph adjacency matrix from its binary version.

Usage

fluxing(
  mat,
  biomasses = NULL,
  losses,
  efficiencies,
  bioms.prefs = TRUE,
  bioms.losses = TRUE,
  ef.level = "prey"
)

Arguments

Details

This function computes fluxes in food webs based on an equilibrium hypothesis: for each species, sum of ingoing fluxes (gains from predation) balances the sum of outgoing fluxes. Outgoing fluxes are defined by consumption and the losses argument. Usually losses relate to species metabolic rates and/or natural death rates. For each species i, sum of ingoing fluxes F_i is computed as:

F_{i} = \frac{1}{e_i} (L_i + \sum_j W_{ij}F_j) \quad if \quad \code{ef.level == "pred"}

F_{i} = \frac{L_i + \sum_j W_{ij}F_j}{\sum_j W_{ji}e_j} \quad if \quad \code{ef.level == "pred"}

W set the matrix of preferences estimated from mat, according to bioms.prefs. L is the vector depicting sum of losses (scaled or not by biomasses, accordingly to bioms.losses) and e is the vector of species efficiencies.

• mat: Either a binary or a valuated matrix can be used. A non zero value for mat[i,j] means that species i is consumed by species j. Matrix entries would assess predator preferences on its prey, thus providing a binary matrix assumes no preferences.

• losses: Express species energetic losses not related to consumption. Usually metabolic or death rates. When an array is provided, losses associated to each species correspond to line sums.

• efficiencies: Determines how efficient species are to convert energy (see ef.level for more details). Providing an array will assume values depending on both prey and predator identity.

• bioms.pref: If TRUE, preferences W_{ij} of predator j on prey i are scaled accordingly to species biomass using the following formula:

  W_{i,j} = \frac{mat[i,j] * biomasses[i]}{\sum_k mat[i,k]* biomasses[k]}

  If FALSE, a normalisation on column values is performed.

• bioms.losses: Set to true, function will assume that losses are defined per biomass unit. Thus, total losses will be thereafter multiplied by biomass values for each species.

• ef.level: If "prey" (resp "pred"), the total amount of energy that can be metabolized from a trophic link will be determined by prey (resp predator) identity. "link.specific" assumes that efficiencies are defined for each trophic interaction and implies efficiencies parameter to be a matrix.

Value

Returns an adjacency matrix where entries are the computed energy fluxes between consumer species and their respective resources.

Author(s)

Benoit gauzens, benoit.gauzens@gmail.com

Examples

# first compute species per unit biomass metabolic rates using the metabolic theory:
losses = 0.1 * species.level$bodymasses^(-0.25)

# call of the function:
fluxing(species.level$mat, 
       species.level$biomasses, 
       losses, 
       species.level$efficiencies, 
       bioms.pref = TRUE, 
       ef.level = "prey")

Aggregated version of the Food web of a soil network ecosystem and species general information (species.level).

Description

This dataset contains the matrix describing trophic interactions between trophic groups of a soil food-web (Digel et al. 2014, Oikos) as well as some ecological information on these groups: biomasses, body masses and and species composition.

Format

a list of 5 elements:

mat

the network adjacency matrix

biomasses

groups total biomasses (g)

bodymasses

group mean bodymasses of species (g)

efficiencies

group species mean assimilation efficiencies

species.tgs

groups' species composition

sensitivity analysis

Description

Assesses how sensitive the results from argument function are to variability of input parameter through coefficient of variation.

Usage

sensitivity(fun.name, param.name, var, n, full.output = FALSE, ...)

Arguments

Details

At each replicate, a coefficient of variation is computed (relative to results obtained form fun.name without random variation). if full.output is FALSE (default) a list of two objects of the same type as the one produced by fun.name is returned, first element contains the mean coefficient of variation in comparison to non randomised inputs among all the replicates, second element contains the standard deviation of these coefficients of variation If full.output is TRUE, a list of size n with of objects containing the coefficients of variation is returned.

Argument for ... should be passed with their names.

Value

a list of two elements of the same type as param.name: first element contains the mean coefficient of variation in comparison to non randomised inputs among all the replicates, second element contains the standard deviation of these coefficient of variation

Examples

# first compute species per unit biomass metabolic rates using the metabolic theory:
losses = 0.1 * species.level$bodymasses^(-0.25)


res = sensitivity(fluxing, "mat", 0.1, 5, full.output = TRUE, 
                 mat = species.level$mat, 
                 biomasses = species.level$biomasses, 
                 losses = losses, 
                 efficiencies = species.level$efficiencies)
res = sensitivity(fluxing, "efficiencies", 0.01, 50, 
                 mat = species.level$mat, 
                 biomasses = species.level$biomasses, 
                 losses = losses, 
                 efficiencies = species.level$efficiencies)

# growth rates of basal species
growth.rates = rep(NA, dim(species.level$mat)[1])
growth.rates[colSums(species.level$mat) == 0] = 0.5

val.mat = fluxing(species.level$mat, species.level$biomasses, losses, species.level$efficiencies)

Food web of a soil network ecosystem and species general information.

Description

This dataset correspond to the food web of a microcosm assembled from the Chesapeake Bay estuary (Lefcheck and Duffy 2010, Ecology)

Format

a list of 4 elements:

mat

the network adjacency matrix

met.rate

metabolic rates of species (J.h-1)

biomasses

species biomasses (g)

efficiencies

species assimilation efficiencies

names

species names

Food web of a soil network ecosystem and species general information.

Description

This dataset contains the matrix describing trophic interactions from a deutsch soil food-web (Digel et al. 2014, Oikos) as well as some ecological information on species: biomasses, body masses and and species names.

Format

a list of 5 elements:

mat

the network adjacency matrix

biomasses

species biomasses (g)

bodymasses

species bodymasses (g)

efficiencies

species assimilation efficiencies

names

species names

Estimates network stability

Description

Computes resilience of the system through Jacobian matrix eigenvalues.

Usage

stability.value(
  val.mat,
  biomasses,
  efficiencies,
  metabolic.types,
  ef.level = "prey",
  full.output = FALSE
)

Arguments

Details

• efficiencies: Determines how efficient species are to convert energy (see ef.level for more details). Providing an array will assume values depending on both prey and predator identity.

• ef.level: If "prey" (resp "pred"), the total amount of energy that can be metabolized from a trophic link will be determined by prey (resp pred) identity. "link.specific" assumes that efficiencies are defined for each trophic interaction and implies efficiencies parameter to be a matrix

• full.output: If TRUE, function result is a list of eigenvalues and eigenvectors of the Jacobian matrix.

Value

Maximum eigenvalue of the Jacobian matrix of a Lotka Voltera like system of equations. If full.output, Jacobian eigenvalues and eigenvectors are returned.

Author(s)

Benoit Gauzens, benoit.gauzens@gmail.com

Examples

losses = 0.15 * groups.level$bodymasses^(-0.25)
# define metbolic types:
met.types = rep('animal', length(losses))
met.types[groups.level$efficiencies == 0.545] = 'plant'

val.mat = fluxing(groups.level$mat, 
                  groups.level$biomasses, 
                  losses, 
                  groups.level$efficiencies, 
                  bioms.pref = TRUE, 
                  ef.level = "pred")
                  
stability.value(val.mat, 
                groups.level$biomasses, 
                groups.level$efficiencies,
                metabolic.types = met.types,
                ef.level = "pred")