Stochastic Arithmetic as a Tool to Study the Stability of Biological Models

Abstract

The theoretical study of the stability of the numerical solution of a differential system may be complicated or even not feasible when the system is large and nonlinear. Here it is shown that such a study can be experimentally done by using stochastic arithmetic and its discrete approach known as the CESTAC method.
The CESTAC method has been first proposed since more than forty years by M. La Porte and J. Vignes as an experimental statistical method to estimate the accuracy on the result of numerical program. Later an abstract formalization of the theory called Stochastic Arithmetic has been developed and many of its algebraic properties have been studied. Here a brief presentation of stochastic arithmetic, of its main properties and of the different software existing for its implementation are given. Then it is demonstrated that the use of stochastic arithmetic in the solver of a differential system can easily reveal whether the computed solution is stable or not. Moreover the stability can be studied with respect to the coefficients of the system or with respect to the initial conditions. At the end it is also pointed out that the same method can be used to detect instabilities due to the used solver. Some examples taken from the biological literature are given.