Carlos R. Takaessu Jr

arXiv2026

Random chemostats with competition and different kinetics to investigate the growth of the gut microbiome

We investigate some chemostat models incorporating wall growth, competition, random fluctuations on the dilution rate, and different consumption functions (Monod and Haldane). We analyze the asymptotic behavior of the solutions of the corresponding random differential systems to establish conditions on the model parameters under which the microbes persist in the gut or disappear. Moreover, several numerical simulations are presented to support the theoretical results and illustrate their biological interpretation.

math.DS↗

arXiv2025

Shadowing for Infinite Dimensional Dynamical Systems

In this paper we extend to an infinite dimensional setting some results on the shadowing property that are known on finite dimensional compact manifolds without border and in $\mathbb{R}^n$. In fact, we show that if $\{\T(t):t\ge 0\}$ is a Morse-Smale semigroup defined in a Hilbert space with a global attractor $\mathcal{A}$ and non-wandering set given only by its equilibria, then $\T(1)|_{\mathcal{A}}:\mathcal{A}\to \mathcal{A} $ admits the Lipschitz Shadowing property. Moreover, for any positively invariant bounded neighborhood $\U\supset\mathcal{A}$ of the global attractor, the map $\T(1)|_{\U}:\U\to \U$ has the Hölder-Shadowing property. We obtain results related to the structural stability of Morse-Smale semigroups, that were only known on finite dimension and continuity of global attractors.

math.DS↗