Editing Bistability (section)

==Mathematical modelling==

In the mathematical language of [[Dynamical systems theory|dynamic systems analysis]], one of the simplest bistable systems is{{citation needed|date=December 2022}}

:<math>
\frac{dy}{dt} = y (1-y^2).
</math>

This system describes a ball rolling down a curve with shape <math>\frac{y^4}{4} - \frac{y^2}{2}</math>, and has three equilibrium points: <math> y = 1 </math>, <math> y = 0 </math>, and <math> y = -1</math>. The middle point <math>y=0</math> is [[Marginal stability|marginally stable]] (<math> y = 0 </math> is stable but <math> y \approx 0 </math> will not converge to <math> y = 0 </math>), while the other two points are stable. The direction of change of <math>y(t)</math> over time depends on the initial condition <math>y(0)</math>.  If the initial condition is positive (<math>y(0)>0</math>), then the solution <math>y(t)</math> approaches 1 over time, but if the initial condition is negative (<math>y(0)< 0</math>), then <math>y(t)</math> approaches −1 over time. Thus, the dynamics are "bistable". The final state of the system can be either <math> y = 1 </math> or <math> y = -1 </math>, depending on the initial conditions.<ref name="Chong">{{cite journal | author = Ket Hing Chong | author2 = Sandhya Samarasinghe | author3 = Don Kulasiri | author4 = Jie Zheng | name-list-style = amp | year = 2015| title = Computational techniques in mathematical modelling of biological switches | journal = Modsim2015 | pages =   578–584 }} For detailed techniques of mathematical modelling of bistability, see the tutorial by Chong et al. (2015) http://www.mssanz.org.au/modsim2015/C2/chong.pdf The tutorial provides a simple example illustration of bistability using a synthetic toggle switch proposed in {{cite journal |last1=Collins |first1=James J.  |author-link1=James Collins (bioengineer)|last2=Gardner |first2=Timothy S. |last3=Cantor |first3=Charles R. |title=Construction of a genetic toggle switch in Escherichia coli |journal=Nature |volume=403 |issue=6767 |pages=339–42 |year=2000 |pmid=10659857 |doi=10.1038/35002131 |bibcode=2000Natur.403..339G |s2cid=345059 }}. The tutorial also uses the dynamical system software XPPAUT http://www.math.pitt.edu/~bard/xpp/xpp.html to show practically how to see bistability captured by a saddle-node bifurcation diagram and the hysteresis behaviours when the bifurcation parameter is increased or decreased slowly over the tipping points and a protein gets turned 'On' or turned 'Off'.</ref>

The appearance of a bistable region can be understood for the model system
<math>
\frac{dy}{dt} = y (r-y^2)
</math>
which undergoes a supercritical [[pitchfork bifurcation]] with [[Bifurcation theory|bifurcation parameter]] <math> r </math>.