systems/rivals.jl

"""
    rivals!(du, u, p, t)
In-place definition of the Rivals system.

See also [`rivals`](@ref).
"""
function rivals!(du, u, p, t)
    x, y = u
    ϵ, α₁, α₂, β₁, β₂ = p

    du[1] = x * (x - α₁) * (1 - x) - β₁ * x * y
    return du[2] = ϵ * (y * (y - α₂) * (1 - y) - β₂ * x * y)
end

"""
    rivals(u, p, t)
Out-of-place definition of the Rivals system.

See also [`rivals!`](@ref).
"""
function rivals(u, p, t)
    x, y = u
    ϵ, α₁, α₂, β₁, β₂ = p

    dx = x * (x - α₁) * (1 - x) - β₁ * x * y
    dy = ϵ * (y * (y - α₂) * (1 - y) - β₂ * x * y)

    return SA[dx, dy]
end

# """
#     rivals_ϵσ(ϵ,σ)
# A shortcut command for returning a CoupledSDEs of the Rivals system in a default setup with multiplicative isotropic noise.

# This setup fixes the parameters α₁ = 0.1, α₂ = 0.3, β₁ = 0.18, β₂ = 0.1 and leaves the value of the time-scale parameter ϵ as a function argument. The prescribed noise process is multiplicative and isotropic: the variables are peturbed by independently drawn identical Gaussian white noise realisations (with standard deviation σ - the other function argument) that are multiplied by the variables' current value.
# """
# function rivals_ϵσ(ϵ, σ) # a convenient two-parameter version of the FitzHugh Nagumo system
#     # defining the CoupledSDEs
#     f(u,p,t) = rivals(u,p,t);
#     α₁ = 0.1; α₂ = 0.3; β₁ = 0.18; β₂ = 0.1; # standard parameters without ϵ (time-scale separation parameter)
#     pf_wo_ϵ = [α₁, α₂, β₁, β₂]; # parameter vector without ϵ
#     dim = 2;
#     g(u,p,t) = multiplicative_idx(u,p,t,[true,true]);
#     pg = nothing;
#     Σ = [1 0; 0 1];
#     process = "WhiteGauss";
#     CoupledSDEs(f, vcat([ϵ], pf_wo_ϵ), dim, σ, g, pg, Σ, process)
# end;