Consider a biological system described by the nonlinear multiple switch model

| (1.1) |

where the unknowns represent two protein concentrations and represents the strength of unrepressed protein expression. This latter quantity is regarded as a time-constant parameter.

The equilibria of correspond to , or equivalently, , where

| (1.2) |

The following two Hurwitz determinants determine the stability of the hyperbolic equilibria of :

| (1.3) |

| (1.4) |

The semi-algebraic system below encodes the asymptotically stable hyperbolic equilibria:

| (1.5) |

We solve this problem by first computing a real comprehensive triangular decomposition of with respect to the parameter , using the new command __RealComprehensiveTriangularize__:

| (1.6) |

This is a decomposition of the original system into several simpler triangular systems and some additional conditions on the parameter :

| (1.7) |

| (1.8) |

Suppose we are interested in those values of for which the biological system is bistable (that is, there are at least two stable equilibria). This means that we are looking for values of such that has at least 2 positive real solutions. We use the RealComprehensiveTriangularize command again and apply it to our previous result :

| (1.9) |

The condition on that we are looking for is given by the second entry:

| (1.10) |

The locations of the stable equilibria are described by the following triangular system from the first entry:

| (1.11) |

We now illustrate the result by discussing the special case . From the equations above, there are stable equilibria given by:

| (1.12) |

| (1.13) |

We verify that the last inequality is satisfied:

| (1.14) |

| (1.15) |

| (1.16) |

Below, we graphically display trajectories of the dynamical system in the special case. The first plot is a 2D animation, and the second one is a 3D static plot with time as the z-axis. We can see the two stable equilibria well in both plots, but there is also a third, unstable, equilibrium that can be seen best in the 3D plot.

| (1.17) |