Univalent positive polynomial maps and the equilibrium state of chemical networks of reversible binding reactions

We consider a map f=(f1,…,fn):R⩾0n→R⩾0n given for x=(x1,…,xn)x=(x1,…,xn) byfi(x)=xiψi(xi)+∑α∈Iαiaαxα where I   is a finite subset of Z⩾0n, aαaα is a constant in R⩾0R⩾0 for each α∈Iα∈I, and ψ1,…,ψnψ1,…,ψn are differentiable order-preserving functions R⩾0→R>0R⩾0→R>0. We prove that f is a bijection. Surjectivity arises as a consequence of the Brouwer Fixed-Point Theorem. For injectivity, we show that the Jacobian matrix of f   is everywhere a P-matrix and we then apply the Gale–Nikaidô Global Univalence Theorem. With ψ1=⋯=ψn=1ψ1=⋯=ψn=1, f is a positive polynomial map of interest in the study of chemical networks of reversible binding reactions. For these, we propose notions of elementary and composite species and of normal and complete networks. Many networks in pharmacology and other fields fall in these classes. We prove that their equilibrium states and detailed-balanced states coincide and are unique with respect to total concentrations of elementary species. The map f gives rise to an equation that has a unique solution which gives the equilibrium state. We also prove that concentrations always converge to the equilibrium state, thereby settling for complete networks the Global Attractor Conjecture, which affirms this property for the larger class of complex-balancing networks.