Global stability in a synthrophic chain model

A mathematical model formulated by G. Powell in 1986 is considered. This model describes a synthrophic chain of species Xi, 1 less than or equal to i less than or equal to n, in a continuous chemostat culture. Species Xi utilizes substrate Si and forms a product Si + 1. Substrate Si + 1 inhibits the growth of Xi and constitutes a growth-limiting substrate of Xi + 1. For this model, the conditions for coexistence of all species are improved, and the local stability results are extended to global stability. Some interesting cases are discussed.