Persistence, Attractivity, and Delay in Facultative Mutualism

Permanence and global attractivity for facultative mutualism system with delay

“…Let (x 1 (t); x 2 (t)) denote any positive solution of system (1) Proof. We shall now prove the result using technique developed in Reference [3]. It is su cient to prove (a) and (c).…”

“…By developing persistence theory for inÿnite dimensional systems, Hale and Waltman [8] obtained a uniform persistence result for a two species competition delayed system. Persistence results for Lotka-Volterra type competition, cooperation and prey-predator systems with delays can be found in Lu and Takeuchi [4], 2 D. MUKHERJEE He and Gopalsamy [3] and Wang and Ma [7]. Freedman and Ruan [9] obtained conditions for uniform persistence of retarded functional di erential equations by applying Lyapunovlike function, Razumikhin technique and di erential inequalities.…”

“…He and Gopalsamy [3] studied the facultative mutualism model with same delay. But in case of di erent delays, they remarked that the global attractivity of the positive equilibrium can be shown by using Lyapunov functionals while for uniform persistence they have not discussed anything regarding its analysis.…”

“…Kuang [2] derived su cient conditions of a global atrractivity of a positive equilibrium in non-autonomous delay equation. He and Gopalsamy [3] obtained the conditions for global stability of the positive equilibrium of the delay system. Lu and Takeuchi [4] discussed the global attractivity of the positive equilibrium in a delayed competition system.…”

Dynamics of Gilpin-Ayala competition model with random perturbation

“…Since that time, many different forms of the Lotka-Volterra competition model have been studied (see, for example, [1], [2]). However, the Lotka-Volterra competition model is linear (i.e.…”