Dynamical Systems and Stochastic Programming: To Ordinary Differential Equations and Back

In this paper we focus on the relation between models of biological systems consisting of ordinary differential equations (ODE) and models written in a stochastic and concurrent paradigm (sCCP stochastic Concurrent Constraint Programming). In particular, we define a method to associate a set of ODE's to an sCCP program and a method converting ODE's into sCCP programs. Then we study the properties of these two translations. Specifically, we show that the mapping from sCCP to ODE's preserves rate semantics for the class of biochemical models (i.e. chemical kinetics is maintained) and we investigate the invertibility properties of the two mappings. Finally, we concentrate on the question of behavioral preservation, i.e if the models obtained applying the mappings have the same dynamics. We give a convergence theorem in the direction from ODE's to sCCP and we provide several well-known examples in which this property fails in the inverse direction, discussing them in detail.