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

## 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?

## 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