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Session S12 - Delay and functional differential equations and applications

Wednesday, July 21, 19:10 ~ 19:45 UTC-3

Global stability and periodic solutions in delayed chemostat models

Gonzalo Robledo Veloso

Universidad de Chile, Chile   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

The chemostat is a continuous bioreactor, where a set of microbial species is cultivated in a homogeneous liquid medium and whose growth is due to the consumption of a limiting nutrient that is pumped into it continuously. The evolution of the variables (nutrient and microbial species) is described by differential equations. However, if we consider the time interval elapsed between the consumption of the nutrient and its metabolization, the corresponding models are described by differential delay equations (DDE).

In homogeneous conditions, the asymptotic behavior of the chemostat equations is described by the competitive exclusion principle, namely, the microbial species that need the minimal amount of nutrients to have a positive growth will be the unique survivor, while the other species will be driven to extinction. In this context, a great amount of research and modeling effort has been carried out in order to avoid competitive exclusion and in this talk we will revisit recent approaches.

First of all, we will study a chain of two chemostats described by an autonomous system of DDE, where a chemostat with two competitors receives a continuous feeding from a second chemostat cultivating only the less advantaged competitor. The coexistence of the competitors can be achieved as a positive equilibrium, whose global asymptotic stability is deduced by using a Lyapunov Krasovskii approach.

Secondly, we will study two $\omega$--periodically perturbed models with one species and prove the existence of an $\omega$--periodic solution and its uniqueness for small delays. The techniques used are a classical coincidence degree theorem and Poincaré map techniques tailored to the DDE framework. In spite of the importance of the result in the field bioprocesses, we also highlight that the adaptation of Poincaré map techniques to DDE equations is interesting from a mathematical point of view.

Amster P, Robledo G, Sepulveda D, Dynamics of a chemostat with periodic nutrient supply and delay in the growth. Nonlinearity 33 (2020), no. 11, 5839-–5860.

Amster P, Robledo G, Sepulveda D, Existence of ω-periodic solutions for a delayed chemostat with periodic inputs. Nonlinear Anal. Real World Appl. 55 (2020), 103134.

Mazenc F, Niculescu SI, Robledo G. Stability analysis of mathematical model of competition in a chain of chemostats in series with delay. Appl. Math. Model. 76 (2019), 311-–329.

Joint work with Frédéric Mazenc (L2S - CNRS - CentraleSupelec, France), Pablo Amster (Universidad de Buenos Aires, Argentina) and Daniel Sepúlveda (Universidad Tecnológica Metropolitana, Chile).

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