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Singular perturbation of a single species model with time-delay

Said Achchab


The aim of this work is to extend approximate aggregation methods for ordinary differential equations to delayed differential equations (DDEs). Approximate aggregation consists of describing the dynamics of a general system involving many coupled variables by means of the dynamics of a reduced system with a few global variables. A delayed two-stage population model in a multi-patch environment is considered. The existence of two different time scales is assumed: the migration process takes place on the behavioral level of, and is thus much faster than the population dynamics. This is the case for some aquatic population for example.
We study afterwards the asymptotic behavior of aggregated model, and prove that it has a globally asymptotically stable steady state. We define the total carrying capacity K time-delay dependent, we show that there exists an optimal time delay that maximizes the total population. Finally, we prove under certain assumptions that initial models are of the monotone type.

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