By a complex chemical reaction we mean a set of reaction steps

, where

are chemical species (molecules, radicals, ions, etc.), the non-negative integers

and

are the stoichiometric coefficients or molecularities, and the formal linear combinations

and

are the reactant and product complexes, respectively. The vectors

and

, that is,

and

, corresponding to the reaction complexes, are the complex vectors, the reactant complex vector, and the product complex vector, respectively. If a complex vector is the null vector, then the corresponding reaction complex is the empty complex, denoted by 0. The reaction step vector expressing the change caused by the

reaction step is the difference of complex vectors:

. The matrix

with

as its

column is the stoichiometric matrix.

One may identify the

formal reaction mechanism with the four ordered elements:

, where

is the set of formal chemical species,

is the set of reaction steps, with

, and

are the matrices of type

with column vectors

and

, respectively.

By the continuous time and continuous state place deterministic model (CCD) of the formal reaction mechanism

we mean the following differential equation:

, where the function

is sometimes called the kinetic, and the most common type of

is when it is mass-action kinetic, that is, there are

positive real constants, such that for any

,

. (

is the concentration of the chemical component

at time

.) One may call the differential equation above the induced kinetic differential equation of the formal reaction mechanism

.

P. Érdi and J. Tóth,

*Mathematical Models of Chemical Reactions: Theory and Applications of Deterministic and Stochastic Models*, Manchester: Manchester University Press, 1989.