Nonautonomous chemostats with variable delays

The appearance of delay terms in a chemostat model can be fully justified since the future behavior of a dynamical system does not in general depend only on the present but also on its history. Sometimes only a short piece of history provides the relevant influence (bounded or finite delay), while in other cases it is the whole history that has to be taken into account (unbounded or infinite delay). In this paper a chemostat model with time variable delays and wall growth, hence a nonautonomous problem, is investigated. The analysis provides sufficient conditions for the asymptotic stability of nontrivial equilibria of the chemostat with variable delays, as well as for the existence of nonautonomous pullback attractors

Competitive Exclusion in a DAE Model for Microbial Electrolysis Cells

Microbial electrolysis cells (MECs) employ electroactive bacteria to perform extracellular electron transfer, enabling hydrogen generation from biodegradable substrates. In previous work, we developed and analyzed a differential-algebraic equation (DAE) model for MECs. The model resembles a chemostat with ordinary differential equations (ODEs) for concentrations of substrate, microorganisms, and an extracellular mediator involved in electron transfer. There is also an algebraic constraint for electric current and hydrogen production. Our goal is to determine the outcome of competition between methanogenic archaea and electroactive bacteria, because only the latter contribute to electric current and resulting hydrogen production. We investigate asymptotic stability in two industrially relevant versions of the model. An important aspect of chemostats models is the principle of competitive exclusion -- only microbes which grow at the lowest substrate concentration will survive as $t\to\infty$. We show that if methanogens grow at the lowest substrate concentration, then the equilibrium corresponding to competitive exclusion by methanogens is globally asymptotically stable. The analogous result for electroactive bacteria is not necessarily true. We show that local asymptotic stability of exclusion by electroactive bacteria is not guaranteed, even in a simplified version of the model. In this case, even if electroactive bacteria can grow at the lowest substrate concentration, a few additional conditions are required to guarantee local asymptotic stability. We also provide numerical simulations supporting these arguments. Our results suggest operating conditions that are most conducive to success of electroactive bacteria and the resulting current and hydrogen production in MECs. This will help identify when methane production or electricity and hydrogen production are favored

Threshold Dynamics of a Stochastic Chemostat Model with Two Nutrients and One Microorganism

A new stochastic chemostat model with two substitutable nutrients and one microorganism is proposed and investigated. Firstly, for the corresponding deterministic model, the threshold for extinction and permanence of the microorganism is obtained by analyzing the stability of the equilibria. Then, for the stochastic model, the threshold of the stochastic chemostat for extinction and permanence of the microorganism is explored. Difference of the threshold of the deterministic model and the stochastic model shows that a large stochastic disturbance can affect the persistence of the microorganism and is harmful to the cultivation of the microorganism. To illustrate this phenomenon, we give some computer simulations with different intensity of stochastic noise disturbance

Dynamical Models of Biology and Medicine

Mathematical and computational modeling approaches in biological and medical research are experiencing rapid growth globally. This Special Issue Book intends to scratch the surface of this exciting phenomenon. The subject areas covered involve general mathematical methods and their applications in biology and medicine, with an emphasis on work related to mathematical and computational modeling of the complex dynamics observed in biological and medical research. Fourteen rigorously reviewed papers were included in this Special Issue. These papers cover several timely topics relating to classical population biology, fundamental biology, and modern medicine. While the authors of these papers dealt with very different modeling questions, they were all motivated by specific applications in biology and medicine and employed innovative mathematical and computational methods to study the complex dynamics of their models. We hope that these papers detail case studies that will inspire many additional mathematical modeling efforts in biology and medicin

International Conference on Mathematical Analysis and Applications in Science and Engineering – Book of Extended Abstracts

The present volume on Mathematical Analysis and Applications in Science and Engineering - Book of Extended Abstracts of the ICMASC’2022 collects the extended abstracts of the talks presented at the International Conference on Mathematical Analysis and Applications in Science and Engineering – ICMA2SC'22 that took place at the beautiful city of Porto, Portugal, in June 27th-June 29th 2022 (3 days). Its aim was to bring together researchers in every discipline of applied mathematics, science, engineering, industry, and technology, to discuss the development of new mathematical models, theories, and applications that contribute to the advancement of scientific knowledge and practice. Authors proposed research in topics including partial and ordinary differential equations, integer and fractional order equations, linear algebra, numerical analysis, operations research, discrete mathematics, optimization, control, probability, computational mathematics, amongst others. The conference was designed to maximize the involvement of all participants and will present the state-of- the-art research and the latest achievements.info:eu-repo/semantics/publishedVersio

On human gut microbial ecosystem: In vitro experiment, in vivo study and mathematical modelling.

The human gut microbiota is considered to be a highly specialized organ providing nourishment, regulating epithelial cell development, modulating innate immune responses and colonization resistances, and it significantly impacts human health and disease. Dispite of being extensively studied for several decades, the functionality of the microbiota colonization in the human gastrointestinal tract and the mechanisms of the interactions between the host and bacteria are still poorly understood. This research follows a novel and unique approach, which combines the complementary strengths of in vitro experiment, in vivo study and mathematical modelling. The work undertaken has three emphases: 1) probiotic strains and their impact on human health; 2) the development of gut microbiota in infants; 3) quantification of human gut microbial ecosystem at both the species level and the system level. In the first part of this research, a versatile anaerobic continuous culture platform was implemented following a novel and unique design, which allows easy and continuous sampling and monitoring of microbial growth. A number of carefully planned in vitro experiments have been conducted to investigate the growth and competition of probiotic strains under different culture conditions. These in vitro experiments improve the understanding for the growth behaviour of the specific probiotic strains. The second part of this project analyzed 50 faecal samples collected from 9 healthy infants with administration of probiotic strains and placebo. The analysis is based on the 454-pyrosequencing technology, which reveals the complete profiles of gut microbiota in these infants and confirmed the modulation effect of the specific probiotic strains. The last part of this research focused on the development of mathematical and computational models of human gut microbial ecosystem. The outcome from this part of the research includes: a) a new bacterial growth model that overcomes the parodox of competitative exclusion caused by previous models; b) a versatile computational framework to simulate in vitro fermentation experiments; and c) a comprehensive mathematical model for human gut and gut microbiota that is the first model for its nature

Mathematical analysis, modelling and simulation of microbial population dynamics

The physiology of unicellular organisms results from a central metabolism which input-output balance accounts for both the cells’ state and their culture medium’s abundance. When bacteria are cultivated in a locally fed fermenter and transported in a turbulent flow, they have to deal with concentration gradients throughout their trajectory in the reactor. Simulating this physics in a multiscale modelling approach requires taking into account not only the well-known laws of hydrodynamics, but also the cells’ biochemistry which is still ill-understood to date. Moreover, the prohibitive cost of the numerics forces to reduce the models to constrain the duration of the experiments to a few weeks. In this context, special consideration has been given to the biological phase. The bacteria population dynamics is given by an integro-differential transport-rupture equation in the space of the particles’ inner coordinates. Picking the most appropriate variables is of paramount importance to best report the time evolution of the cells’ state throughout their history in the fermenter, the latter being comparable to a markovian process. The microorganisms’ length testifies to their morphology and their progress in the cell cycle, whereas the uptake rate of the surrounding resources leads to an evaluation of the material transfer between the liquid and biotic phases. The result is the estimation of the source term in the organisms’ central metabolism which outputs are the apparent rate of anabolism and, if over-uptake, activation of peripheral reactions to combust the surplus in organic compounds. Beyond their own history, the individuals’ metabolic yields can be impacted by the substrate availability at their neighbourhood, which stems from the feeding and the level of mixing in the reactor. The state variables have a compact support, what raises the question of the mathematical problem’s wellposedness, similarly as solving a PDE over a bounded set is traditionally more difficult than over $\mathbb{R}^{n}$, $n \in \mathbb{N}$. It is shown that the Malthus eigenfunction associated with the transport-rupture equation is $\mathcal{C}^{1}$ as soon as fragmentation trumps cell growth near the right-hand edge of the size-distribution’s support. All in all, the solution is continuous at each time in the state space. These results allow the implementation of numerical codes to solve (in this work, by Monte-Carlo, Finite Volume, or Quadrature of MOMents methods) the well-posed problem, the algorithms being exploited to simulate five biochemical engineering experiments which conclusions are detailed in the literature

Novel strategies for process control based on hybrid semi-parametric mathematical systems

Tese de doutoramento. Engenharia Química. Universidade do Porto. Faculdade de Engenharia. 201