# Attractors.basins\_fractions throws "ArgumentError: range must be non-empty"

**URL:** <https://discourse.julialang.org/t/attractors-basins-fractions-throws-argumenterror-range-must-be-non-empty/96531>\
**Category:** Modelling & Simulations\
**Tags:** question, dynamical-systems\
**Created:** [March 23, 2023, 7:57pm UTC](https://discourse.julialang.org/t/attractors-basins-fractions-throws-argumenterror-range-must-be-non-empty/96531 "2023-03-23T19:57:44Z")\
**Posts on this page:** 4\
**Page:** 1

<div class="post-metadata">

**Author:** ![cjoshliu](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/cjoshliu/32/47978_2.png) [@cjoshliu](https://discourse.julialang.org/u/cjoshliu)\
**Post date:** [March 23, 2023, 7:57pm UTC](https://discourse.julialang.org/t/attractors-basins-fractions-throws-argumenterror-range-must-be-non-empty/96531/1 "2023-03-23T19:57:44Z")

</div>

Hi, I’m trying to find basin fractions of an s-species (s-dimensional) generalized Lotka-Volterra model using DynamicalSystems. My model is

d\_t\vec{u}=\vec{\lambda}+\vec{r}\cdot\vec{u}\cdot (1-\hat{\alpha}\vec{u})

where \hat{\alpha}\_{ii}=1 and \hat{\alpha}\_{ij,i\neq j} are distributed c\cdot\text{Gamma}(1/\sigma^2, \sigma^2). My setup, which works fine for some parameters (more on this later), is

```julia
using DifferentialEquations
using Distributions
using DynamicalSystems
using LinearAlgebra
using Random

const SIGMA = 0.5 #sigma
const DISPERSAL = 1.0e-6 #scalar lambda
const GROWTH_RATE = 1.0 #scalar r
const MIN_POP = 1.0e-9 #threshold below which to impose extinction

# function to generate interaction matrix
function interaction_matrix(σ, s, c)
    # initializes s-by-s matrix with entries ~ Gamma
    α = c*rand(Gamma(1/σ^2,σ^2), (s,s))
    α[diagind(α)] .= 1.0 #sets diagonals to 1
    return α
end

# dynamic rule function
function GeneralizedLotkaVolterra!(du, u, p, t)
    λ, r, α = p
    du .= λ+r.*u.*(1 .-α*u)
    return nothing
end

```

When I run `basins_fractions()` with 9 species

```julia
n_species = 9 #number of species s
ij_strength = 1.2 #interaction strength c

SEED = 1234
Random.seed!(SEED)

λ = DISPERSAL*ones(n_species) #vector lambda
r = GROWTH_RATE*ones(n_species) #vector r
α = interaction_matrix(SIGMA, n_species, ij_strength)
p = [λ, r, α]

u0 = rand(n_species) #initial conditions
xg = range(0, 1; length=101) #grid dimension
grid = ntuple(_->xg, n_species)

# checks if species fall below extinction threshold
condition(u, t, integrator) = any(u.<MIN_POP)
# sets species below extinction threshold to 0
function affect!(integrator)
    integrator.u[integrator.u.<MIN_POP] .= 0
end
cb = DiscreteCallback(condition, affect!)
diffeq = (alg=AutoVern7(Rodas4()),
    reltol=1.0e-8,
    abstol=1.0e-8,
    callback=cb)

glv = CoupledODEs(GeneralizedLotkaVolterra!, u0, p; diffeq)
mapper = AttractorsViaRecurrences(glv, grid)
sss, = statespace_sampler(;
    min_bounds=minimum.(grid),
    max_bounds=maximum.(grid)
)

fractions = basins_fractions(mapper, sss; N=10_000)

```

I quickly get a reasonable result

```julia
Mapping initial conditions to attractors: 100%|██████████| Time: 0:00:13
Dict{Int64, Float64} with 3 entries:
  2 => 0.0914
  3 => 0.1504
  1 => 0.7582

```

However, changing only the first line to use 10 species,

```julia
n_species = 10 #number of species s
ij_strength = 1.2 #interaction strength c

SEED = 1234
Random.seed!(SEED)
...
fractions = basins_fractions(mapper, sss; N=10_000)

```

I instead get this error message I’m having trouble understanding

```julia
ArgumentError: range must be non-empty

Stacktrace:
  [1] SamplerRangeNDL
    @ /Applications/Julia-1.8.app/Contents/Resources/julia/share/julia/stdlib/v1.8/Random/src/generation.jl:333 [inlined]
  [2] Sampler
    @ /Applications/Julia-1.8.app/Contents/Resources/julia/share/julia/stdlib/v1.8/Random/src/generation.jl:189 [inlined]
  [3] Sampler
    @ /Applications/Julia-1.8.app/Contents/Resources/julia/share/julia/stdlib/v1.8/Random/src/generation.jl:418 [inlined]
  [4] Sampler (repeats 2 times)
    @ /Applications/Julia-1.8.app/Contents/Resources/julia/share/julia/stdlib/v1.8/Random/src/Random.jl:140 [inlined]
  [5] rand!(rng::TaskLocalRNG, A::Vector{CartesianIndex{10}}, X::CartesianIndices{10, NTuple{10, Base.OneTo{Int64}}})
    @ Random /Applications/Julia-1.8.app/Contents/Resources/julia/share/julia/stdlib/v1.8/Random/src/Random.jl:267
  [6] rand
    @ /Applications/Julia-1.8.app/Contents/Resources/julia/share/julia/stdlib/v1.8/Random/src/Random.jl:280 [inlined]
  [7] rand
    @ /Applications/Julia-1.8.app/Contents/Resources/julia/share/julia/stdlib/v1.8/Random/src/Random.jl:281 [inlined]
  [8] rand
    @ /Applications/Julia-1.8.app/Contents/Resources/julia/share/julia/stdlib/v1.8/Random/src/Random.jl:284 [inlined]
  [9] automatic_Δt_basins(ds::CoupledODEs{true, 10, OrdinaryDiffEq.ODEIntegrator{true, CompositeAlgorithm{Tuple{Vern7{Static.False,…}, Rodas4{0,true,…}},…}, Vector{Float64}, Nothing,…}, Vector{Array{Float64}}}, grid::NTuple{10, StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}, Int64}}; N::Int64)
    @ Attractors ~/.julia/packages/Attractors/Y7iTL/src/mapping/attractor_mapping_recurrences.jl:280
 [10] automatic_Δt_basins
    @ ~/.julia/packages/Attractors/Y7iTL/src/mapping/attractor_mapping_recurrences.jl:271 [inlined]
 [11] initialize_basin_info(ds::CoupledODEs{true, 10, OrdinaryDiffEq.ODEIntegrator{true, CompositeAlgorithm{Tuple{Vern7{Static.False,…}, Rodas4{0,true,…}},…}, Vector{Float64}, Nothing,…}, Vector{Array{Float64}}}, grid::NTuple{10, StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}, Int64}}, Δtt::Nothing, sparse::Bool)
    @ Attractors ~/.julia/packages/Attractors/Y7iTL/src/mapping/attractor_mapping_recurrences.jl:219
 [12] AttractorsViaRecurrences(ds::CoupledODEs{true, 10, OrdinaryDiffEq.ODEIntegrator{true, CompositeAlgorithm{Tuple{Vern7{Static.False,…}, Rodas4{0,true,…}},…}, Vector{Float64}, Nothing,…}, Vector{Array{Float64}}}, grid::NTuple{10, StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}, Int64}}; Δt::Nothing, sparse::Bool, force_non_adaptive::Bool, kwargs::Base.Pairs{Symbol, Union{}, Tuple{}, NamedTuple{(), Tuple{}}})
    @ Attractors ~/.julia/packages/Attractors/Y7iTL/src/mapping/attractor_mapping_recurrences.jl:121
 [13] AttractorsViaRecurrences(ds::CoupledODEs{true, 10, OrdinaryDiffEq.ODEIntegrator{true, CompositeAlgorithm{Tuple{Vern7{Static.False,…}, Rodas4{0,true,…}},…}, Vector{Float64}, Nothing,…}, Vector{Array{Float64}}}, grid::NTuple{10, StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}, Int64}})
    @ Attractors ~/.julia/packages/Attractors/Y7iTL/src/mapping/attractor_mapping_recurrences.jl:118
 [14] top-level scope
    @ In[4]:29

```

Surprisingly, 12 species works just fine, and

```julia
n_species = 12 #number of species s
ij_strength = 1.2 #interaction strength c

SEED = 1234
Random.seed!(SEED)
...
fractions = basins_fractions(mapper, sss; N=10_000)

```

gives

```julia
Mapping initial conditions to attractors: 100%|██████████| Time: 0:00:21
Dict{Int64, Float64} with 2 entries:
  2 => 0.0022
  1 => 0.9978

```

Again with 10 species, changing the seed from 1234 to 1243

```julia
n_species = 10 #number of species s
ij_strength = 1.2 #interaction strength c

SEED = 1243
Random.seed!(SEED)
...
fractions = basins_fractions(mapper, sss; N=10_000)

```

throws the same error. Similar errors occur for 11 and 14 dimensions but not 13 or 15.

How might I fix this? Thanks!

---

<div class="post-metadata">

**Author:** ![cjoshliu](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/cjoshliu/32/47978_2.png) [@cjoshliu](https://discourse.julialang.org/u/cjoshliu)\
**Post date:** [March 23, 2023, 11:44pm UTC](https://discourse.julialang.org/t/attractors-basins-fractions-throws-argumenterror-range-must-be-non-empty/96531/2 "2023-03-23T23:44:07Z")

</div>

I’ve found another way to trigger this error. When I use the same setup as above with 17 dimensions and dimension 1e-2 grid cells

```julia
n_species = 17 #number of species s
ij_strength = 1.2 #interaction strength c

SEED = 1234
Random.seed!(SEED)

λ = DISPERSAL*ones(n_species) #vector lambda
r = GROWTH_RATE*ones(n_species) #vector r
α = interaction_matrix(SIGMA, n_species, ij_strength)
p = [λ, r, α]

u0 = rand(n_species) #initial conditions
xg = range(0, 1; length=101) #grid dimension
grid = ntuple(_->xg, n_species)

# checks if species fall below extinction threshold
condition(u, t, integrator) = any(u.<MIN_POP)
# sets species below extinction threshold to 0
function affect!(integrator)
    integrator.u[integrator.u.<MIN_POP] .= 0
end
cb = DiscreteCallback(condition, affect!)
diffeq = (alg=AutoVern7(Rodas4()),
    reltol=1.0e-8,
    abstol=1.0e-8,
    callback=cb)

glv = CoupledODEs(GeneralizedLotkaVolterra!, u0, p; diffeq)
mapper = AttractorsViaRecurrences(glv, grid)
sss, = statespace_sampler(;
    min_bounds=minimum.(grid),
    max_bounds=maximum.(grid)
)

fractions = basins_fractions(mapper, sss; N=10_000)

```

I get a reasonable result:

```julia
Mapping initial conditions to attractors: 100%|██████████| Time: 0:00:35
Dict{Int64, Float64} with 3 entries:
  2 => 0.201
  3 => 0.0146
  1 => 0.7844

```

However, when I use 1e-3 grid cells,

```julia
n_species = 17 #number of species s
ij_strength = 1.2 #interaction strength c
SEED = 1234
Random.seed!(SEED)
...
xg = range(0, 1; length=1001) #grid dimension
grid = ntuple(_->xg, n_species)
...
fractions = basins_fractions(mapper, sss; N=10_000)

```

I get the error:

```julia
ArgumentError: range must be non-empty

Stacktrace:
  [1] SamplerRangeNDL
    @ /Applications/Julia-1.8.app/Contents/Resources/julia/share/julia/stdlib/v1.8/Random/src/generation.jl:333 [inlined]
  [2] Sampler
    @ /Applications/Julia-1.8.app/Contents/Resources/julia/share/julia/stdlib/v1.8/Random/src/generation.jl:189 [inlined]
  [3] Sampler
    @ /Applications/Julia-1.8.app/Contents/Resources/julia/share/julia/stdlib/v1.8/Random/src/generation.jl:418 [inlined]
  [4] Sampler (repeats 2 times)
    @ /Applications/Julia-1.8.app/Contents/Resources/julia/share/julia/stdlib/v1.8/Random/src/Random.jl:140 [inlined]
  [5] rand!(rng::TaskLocalRNG, A::Vector{CartesianIndex{17}}, X::CartesianIndices{17, NTuple{17, Base.OneTo{Int64}}})
    @ Random /Applications/Julia-1.8.app/Contents/Resources/julia/share/julia/stdlib/v1.8/Random/src/Random.jl:267
  [6] rand
    @ /Applications/Julia-1.8.app/Contents/Resources/julia/share/julia/stdlib/v1.8/Random/src/Random.jl:280 [inlined]
  [7] rand
    @ /Applications/Julia-1.8.app/Contents/Resources/julia/share/julia/stdlib/v1.8/Random/src/Random.jl:281 [inlined]
  [8] rand
    @ /Applications/Julia-1.8.app/Contents/Resources/julia/share/julia/stdlib/v1.8/Random/src/Random.jl:284 [inlined]
  [9] automatic_Δt_basins(ds::CoupledODEs{true, 17, OrdinaryDiffEq.ODEIntegrator{true, CompositeAlgorithm{Tuple{Vern7{Static.False,…}, Rodas4{0,true,…}},…}, Vector{Float64}, Nothing,…}, Vector{Array{Float64}}}, grid::NTuple{17, StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}, Int64}}; N::Int64)
    @ Attractors ~/.julia/packages/Attractors/Y7iTL/src/mapping/attractor_mapping_recurrences.jl:280
 [10] automatic_Δt_basins
    @ ~/.julia/packages/Attractors/Y7iTL/src/mapping/attractor_mapping_recurrences.jl:271 [inlined]
 [11] initialize_basin_info(ds::CoupledODEs{true, 17, OrdinaryDiffEq.ODEIntegrator{true, CompositeAlgorithm{Tuple{Vern7{Static.False,…}, Rodas4{0,true,…}},…}, Vector{Float64}, Nothing,…}, Vector{Array{Float64}}}, grid::NTuple{17, StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}, Int64}}, Δtt::Nothing, sparse::Bool)
    @ Attractors ~/.julia/packages/Attractors/Y7iTL/src/mapping/attractor_mapping_recurrences.jl:219
 [12] AttractorsViaRecurrences(ds::CoupledODEs{true, 17, OrdinaryDiffEq.ODEIntegrator{true, CompositeAlgorithm{Tuple{Vern7{Static.False,…}, Rodas4{0,true,…}},…}, Vector{Float64}, Nothing,…}, Vector{Array{Float64}}}, grid::NTuple{17, StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}, Int64}}; Δt::Nothing, sparse::Bool, force_non_adaptive::Bool, kwargs::Base.Pairs{Symbol, Union{}, Tuple{}, NamedTuple{(), Tuple{}}})
    @ Attractors ~/.julia/packages/Attractors/Y7iTL/src/mapping/attractor_mapping_recurrences.jl:121
 [13] AttractorsViaRecurrences(ds::CoupledODEs{true, 17, OrdinaryDiffEq.ODEIntegrator{true, CompositeAlgorithm{Tuple{Vern7{Static.False,…}, Rodas4{0,true,…}},…}, Vector{Float64}, Nothing,…}, Vector{Array{Float64}}}, grid::NTuple{17, StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}, Int64}})
    @ Attractors ~/.julia/packages/Attractors/Y7iTL/src/mapping/attractor_mapping_recurrences.jl:118
 [14] top-level scope
    @ In[39]:29

```

Nevertheless, when I use the even finer 1e-4 grid cells,

```julia
n_species = 17 #number of species s
ij_strength = 1.2 #interaction strength c
SEED = 1234
Random.seed!(SEED)
...
xg = range(0, 1; length=10001) #grid dimension
grid = ntuple(_->xg, n_species)
...
fractions = basins_fractions(mapper, sss; N=10_000)

```

I get a reasonable result quantitatively different from but qualitatively similar to the one I obtained with 1e-2 grid cells:

```julia
Dict{Int64, Float64} with 3 entries:
  2 => 0.111
  3 => 0.0134
  1 => 0.8756

```

🥲 please help, thanks!

---

<div class="post-metadata">

**Author:** ![cjoshliu](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/cjoshliu/32/47978_2.png) [@cjoshliu](https://discourse.julialang.org/u/cjoshliu)\
**Post date:** [March 24, 2023, 12:55am UTC](https://discourse.julialang.org/t/attractors-basins-fractions-throws-argumenterror-range-must-be-non-empty/96531/3 "2023-03-24T00:55:42Z")

</div>

Using `mapper = AttractorsViaRecurrences(glv, grid; Δt=0.1)` (adding the kwarg `Δt=0.1`) seems to fix it. Running `automatic_Δt_basins(glv, grid)` throws the same error. I would still like help getting some intuition on this. Thanks!

---

<div class="post-metadata">

**Author:** ![KalelR](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/kalelr/32/43305_2.png) [@KalelR](https://discourse.julialang.org/u/KalelR)\
**Post date:** [March 24, 2023, 10:06am UTC](https://discourse.julialang.org/t/attractors-basins-fractions-throws-argumenterror-range-must-be-non-empty/96531/4 "2023-03-24T10:06:02Z")

</div>

Hey!

From the stacktrace you can see that the error occurs initially in the function `automatic_Δt_basins`. As the name suggests, the function finds a nice value for `Δt` for the integration. So from this we can understand why your solution works: by defining yourself the value of `Δt`, the function is entirely skipped and the error avoided. This is completely fine, and a nice solution.

But to understand the error, I’ve looked at the function code and found that the error is quite strange. It boils down to just:

```julia
a = CartesianIndices((101, 101, 101, 101, 101, 101, 101, 101, 101, 101))
N = 5000
rand(a, N)

```

which gives `ArgumentError: range must be non-empty`.

I have no clue why this occurs, but it is unrelated to the Attractors.jl or related packages. @Datseris do you have any clues?

Anyways, Josh, if you can just give the `Δt` value you should be good to go!