Global Stability in Delay Differential Systems without Dominating Instantaneous Negative Feedbacks

Abstract

Most of the global stability or convergence results appearing so far for delayed Lotka-Volterra-type systems or other delay differential systems require that the instantaneous negative feedbacks dominate both delayed feedback and interspecific interactions. Such a requirement is rarely met in real systems since feedbacks are generally delayed. This leads to the long standing question: Under what conditions will the global stability of a nonnegative steady state of a differential delay system persist when time delays involved in some part of the negative feedbacks are small enough? This paper presents a partial answer to this open question. Roughly, our results show that if the delay differential system remains dissipative when delays are introduced, then the above open question has an affirmative answer. Our results also improve some recent findings.