A general analysis of exact lumping in chemical kinetics

A general analysis of exact lumping is presented. This analysis can be applied to any reaction system with n species described by a set of first order ordinary differential equations dy/dt = f (y), where y is an n-dimensional vector; f (y) is an arbitrary n-dimensional function vector. Here we consider lumping by means of an n̂ × n real constant matrix M with rank n̂ (n̂ < n). It is found that a reaction systems is exactly lumpable if and only if there exist nontrivial fixed invariant subspaces M of the transpose of the Jacobian matrix J^T (y) of f(y), no mater what value y takes, and the corresponding eignevalues are the same for J^T(y) and J^T (M̄ My). Here the rows of M are the basis vectors of M and M̄ is any generalized inverse of M satisfying MM̄ = I_n̂ with I_n̂ being the _n̂-identity matrix. The fixed invariant subspaces of J^T(y) can be obtained either from the simultaneously invariant subspaces of all A_k, where the A_k's form the basis of the decomposition of J^T(y), or by determining the fixed Ker {Π_i(J^T(y)-λ_iI_n) ^r_iΠ_j[(σ² _j + τ² _j)I_n - 2σ_jJ^T(y) +(J^T(y))²]^r_j}, where λ_i, σ_j±_iτ_j are the real and nonreal eigenvalues of J^T (y) and λ_i, σ _j and τ_j are usually functions of y;r_i,r_j are nonnegative integers. The kinetic equations of the lumped system can be described as dŷ/dt=Mf(M̄ŷ). This method is illustrated by some simple examples.