Extinctions

Extinctions often happen during simulations. You can control the biomass threshold under which species will be considered as extinct using the extinction_threshold (default is 1e-6):

A = [0 0 0 0 0 0 0 0 0 0; 
     0 1 0 0 0 0 0 0 0 0; 
     0 0 1 0 0 0 0 0 0 0; 
     0 1 1 1 1 1 1 0 0 0; 
     0 1 1 0 0 0 0 0 0 0; 
     1 1 1 1 1 1 1 1 0 0; 
     1 1 1 1 0 0 0 0 0 0; 
     0 0 1 1 1 1 1 0 0 0; 
     0 0 0 1 1 1 1 1 1 1; 
     0 0 0 0 0 0 0 1 0 0]
p = model_parameters(A)
b = rand(size(A, 1))

s = simulate(p, b, start=0, stop=100, extinction_threshold = 1e-12)

The identity of the extinct species are stored in the output as follow:

s[:p][:extinctions] #ordered set of extinct species 
s[:p][:extinctionstime] #each element contains a tuple describing an extinction event as follows:
                        #(time step, species identity)
s[:p][:tmpA] #after each extinction event, the interaction matrix is recalculated 
             #(removing interaction from and to extinct species or rewiring the matrix if rewiring is turned on)
s[:p][:A] #final state of the interaction matrix

It's important to note that for the model to run smoothly, the dimension of the matrix doe not change, so if you want to analyze the structure of the emergent interaction matrix, you need to remove extinct species, as follows:

A_emergent = deepcopy(A)
id_extinct = s[:p][:extinctions] #ordered set of extinct species 
is_extinct = falses(size(A,1))
is_extinct[id_extinct] .= true
A_emergent = A_emergent[is_extinct, is_extinct]

Rewiring (diet switch)

Simulations can be run with rewiring by using the rewiring_method keyword in model_parameters. This allows species to form new links following extinctions according to some set of rules. There are four options for the rewiring_method argument:

• :none - Default setting with no rewiring
• :ADBM - The allometric diet breadth model (ADBM) as described in Petchey et al. (2008). Based on optimal foraging theory.
• :DS - The rewiring mechanism used by Gilljam et al. (2015) based on diet similarity.
• :DO - The rewiring mechanism used by Staniczenko et al. (2010) based on diet overlap.

The simulate function will automatically perform the rewiring depending on which option is chosen. Further arguments can also be supplied to model_parameters to change the parameters of the rewiring models.

Simulations with rewiring are run in the same way as those without, for example using ADBM rewiring:

A = nichemodel(10, 0.3)
p = model_parameters(A,rewire_method = :ADBM)
b = rand(size(A, 1))

s = simulate(p, b, start=0, stop=50, steps=1000)

Rewiring parameters

As for all other parameters, rewiring parameters can be passed to model_parameters. The parameters' default values follow the litterature (see references above). When no alternative value is provided, any value can be passed, as long as it is of the same type as the default value.

For more details on the parameters meaning and value, see the references

Petchey's ADBM model

For this particular model, it is possible to chose how rewiring is triggered: on extinctions (default) or periodically (see example below the table).

Example:

# food web
A = [0 0 0 0 0; 0 1 0 0 0; 0 0 0 0 0; 0 1 1 0 0; 0 1 1 1 1]
# setting the model parameters for ADBM periodic rewiring using biomass to calculate resource densities
p = model_parameters(
                     A #food web
                     , Z = 10.0 #consumer/resource body mass scaling
                     , rewire_method = :ADBM #ADBM reiwring
                     , adbm_trigger = :interval #rewiring triggered periodically ...
                     , adbm_interval = 100 #... with Δt = 100
                     , Nmethod = :biomass #densities = biomass
                     )
# set initial biomass as to have immediate extinction of species 4
Bi = [1.0,1.0,1.0,0.0,1.0]
# perform the simulations
s = simulate(p, Bi)
# you can check the identity of the extinct species :
p[:extinctions]
# extinction times for each species:
p[:extinctionstime]
# and the temporary matrix (A is recorded just before each rewiring event)
p[:tmpA][1] # original matrix, just before extinction of species 4
p[:tmpA][2] # matrix after extinction of species 4, before extinction of species 2
# the new interaction matrix, after the last extinction:
p[:A]

Gilljam's diet similarity model

Staniczenko's diet overlap model

No extra parameters are needed for this rewiring method.

References

• Gilljam, D., Curtsdotter, A., & Ebenman, B. (2015). Adaptive rewiring aggravates the effects of species loss in ecosystems. Nature communications, 6, 8412.

• Petchey, O. L., Beckerman, A. P., Riede, J. O., & Warren, P. H. (2008). Size, foraging, and food web structure. Proceedings of the National Academy of Sciences, 105(11), 4191-4196.

• Staniczenko, P., Lewis, O. T., Jones, N. S., & Reed‐Tsochas, F. (2010). Structural dynamics and robustness of food webs. Ecology letters, 13(7), 891-899.