Asymptotic stability of a non-weakly reversible biochemical reaction network composed of cardiac hypertrophy factors

Abstract

We consider an autonomous system of ordinary differential equations, which is derived from a network composed of biochemical reactions among 25 kinds of cardiac hypertrophy factors by using the mass action kinetics and has definitely been proposed by Ito and Yamamoto (Adv Math Sci Appl 23:1–33, 2013). The main objective of the present paper is to show any global-in-time solution is bounded if the nonnegative initial values are chosen from a suitable bounded subset $${\mathbb {R}}^{25}$$ and converges to an equilibrium point, which is uniquely determined, as time goes to infinity. It is widely known that Feinberg’s Deficiency Zero Theorem is a very powerful tool for proving the stability of equilibrium points of chemical reaction networks. However, we cannot directly apply this theorem to our system because the system dose not satisfy the weak reversibility, one of the conditions for the theorem. In order to overcome this difficulty, we decompose our network into two weakly reversible subnetworks, which enables us to apply the Deficiency Zero Theorem to prove the stability of equilibrium points.