A simple non-gradient simulation function for testing

This is a toy stochastic non-gradient system which can have multistability in some conditions. Model specification:

Usage

sim_fun_nongrad(
  initial = list(x1 = 0, x2 = 0, a = 1),
  parameter = list(b = 1, k = 1, S = 0.5, n = 4, lambda = 0.01, sigmasq1 = 8, sigmasq2 =
    8, sigmasq3 = 2),
  constrain_a = TRUE,
  amin = -0.3,
  amax = 1.8,
  length = 1e+05,
  stepsize = 0.01,
  seed = NULL,
  progress = TRUE
)

Arguments

initial, parameter: Two sets of parameters. initial contains the initial value of x1, x2, and a; parameter contains b,k,S,n,lambda, which control the model dynamics, and sigmasq1,sigmasq2,sigmasq3, which are the squares of $\sigma_1,\sigma_2,\sigma_3$ and controls the amplitude of noise.
constrain_a: Should the value of a be constrained? (TRUE by default).
amin, amax: If constrain_a, the minimum and maximum values of a.
length: The length of simulation.
stepsize: The step size used in the Euler method.
seed: The initial seed that will be passed to set.seed() function.
progress: Show progress bar of the simulation?

Value

A matrix of simulation results.

Details

$$\frac {dx_ {1}}{dt} = \frac {ax_ {1}^ {n}}{S^ {n}+x_ {1}^ {n}} + \frac {bS^ {n}}{S^ {n}+x_ {2}^ {n}} - kx_ {1}+ \sigma_1 dW_1/dt$$ $$\frac {dx_ {2}}{dt} = \frac {ax_ {2}^ {n}}{S^ {n}+x_ {2}^ {n}} + \frac {bS^ {n}}{S^ {n}+x_ {1}^ {n}} - kx_ {2}+ \sigma_2 dW_2/dt$$ $$\frac {da}{dt} = -\lambda a+ \sigma_3 dW_3/dt$$

References

Wang, J., Zhang, K., Xu, L., & Wang, E. (2011). Quantifying the Waddington landscape and biological paths for development and differentiation. Proceedings of the National Academy of Sciences, 108(20), 8257-8262. doi:10.1073/pnas.1017017108