ABSTRACT
This paper revisits a recently introduced chemostat model of one-species with a periodic input of a single nutrient which is described by a system of delay differential equations. Previous results provided sufficient conditions ensuring the existence and uniqueness of a periodic solution for arbitrarily small delays. This paper partially extends these results by proving-with the construction of Lyapunov-like functions-that the evoked periodic solution is globally asymptotically stable when considering Monod uptake functions and a particular family of nutrient inputs.
ABSTRACT
We present a model of single species fishery which alternates closed seasons with pulse captures. The novelty is that the length of a closed season is determined by the remaining stock size after the last capture. The process is described by a new type of impulsive differential equations recently introduced. The main result is a fishing effort threshold which determines either the sustainability of the fishery or the extinction of the resource.
Subject(s)
Conservation of Natural Resources/statistics & numerical data , Fisheries/statistics & numerical data , Fishes , Models, Biological , Animals , Conservation of Natural Resources/economics , Extinction, Biological , Fisheries/economics , Population DynamicsABSTRACT
We apply basic tools of control theory to a chemostat model that describes the growth of one species of microorganisms that consume a limiting substrate. Under the assumption that available measurements of the model have fixed delay t>0, we design a family of feedback control laws with the objective of stabilizing the limiting substrate concentration in a fixed level. Effectiveness of this control problem is equivalent to global attractivity of a family of differential delay equations. We obtain sufficient conditions (upper bound for delay t>0 and properties of the feedback control) ensuring global attractivity and local stability. Illustrative examples are included.
