Local stability analysis global dynamics

As in the model of Section 2, the problem can be studied on its omega limit set with three rest points Eq,Ei,E2. A local stability analysis and, for some special cases, the asymptotic behavior of solutions were given in [E]. However, the populations cannot invade each other simultaneously El and E2 cannot be simultaneously unstable), so the persistence theory does not hold [E]. Moreover, for Michaelis-Menten dynamics, when one of the boundary rest points is locally stable and the other unstable, the locally stable one is globally stable [HWE]. In particular, the oscillation observed in the case of system (3.2) does not occur with (3.4). Indeed, the delayed system seems to behave much like the simple chemostat. 

As with the chemostat, the basic approach is to locate the rest points, analyze their local stability, and determine the global properties of the dynamical system. The first lemma gives some estimates of quantities that will be important in the analysis. It turns out to be easier to state the results in terms of rest points of the system (2.2) rather than those of the system (2.4). These results will be interpreted as needed for the system (2.4). 

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