752 research outputs found
Periodic Oscillations in a Chemostat Model with Two Discrete Delays
Periodic oscillations of solutions of a chemostat-type model in which a species feeds on a limiting nutrient are considered. The model incorporates two discrete delays representing the lag in nutrient recycling and nutrient conversion. Through the study of characteristic equation associated with the linearized system, a unique positive equilibrium is found and proved to be locally asymptotically stable under some conditions. Meanwhile, a Hopf bifurcation causing periodic solutions is also obtained. Numerical simulations illustrate the theoretical results
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
Dynamics of a Stochastic Functional System for Wastewater Treatment
The dynamics of a delayed stochastic model simulating wastewater treatment process are studied. We assume that there are stochastic fluctuations in the concentrations of the nutrient and microbes around a steady state, and introduce two distributed delays to the model describing, respectively, the times involved in nutrient recycling and the bacterial reproduction response to nutrient uptake. By constructing Lyapunov functionals, sufficient conditions for the stochastic stability of its positive equilibrium are obtained. The combined effects of the stochastic fluctuations and delays are displayed
Design of a Cascade Observer for a Model of Bacterial Batch Culture with Nutrient Recycling
International audienc
Population growth and persistence in a heterogeneous environment: the role of diffusion and advection
The spatio-temporal dynamics of a population present one of the most
fascinating aspects and challenges for ecological modelling. In this article we
review some simple mathematical models, based on one dimensional
reaction-diffusion-advection equations, for the growth of a population on a
heterogeneous habitat. Considering a number of models of increasing complexity
we investigate the often contrary roles of advection and diffusion for the
persistence of the population. When it is possible we demonstrate basic
mathematical techniques and give the critical conditions providing the survival
of a population, in simple systems and in more complex resource-consumer models
which describe the dynamics of phytoplankton in a water column.Comment: Introductory review of simple conceptual models. 45 pages, 15 figures
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Importance of allelopathy as peudo-mixotrophy for the dynamics and diversity of phytoplankton
Understanding the dynamics and diversity of marine phytoplankton is essential for predicting oceanic primary production, oxygen generation and carbon sequestration. Several top-down and bottom-up factors lead to complex phytoplankton dynamics. Complexities further arise
from inter-species interactions within phytoplankton communities. Consequently, some of the basic questions on phytoplankton diversity, identified long ago, still puzzle the ecologists: for example, what regulates the diversity in simple systems where species compete for limiting
resources? In this context, allelopathic interaction among phytoplankton species has been identified as a potential driver of their dynamics and regulator of their diversity. This chapter deals with the importance of allelopathy in regulating the outcome of nutrient competition
among phytoplankton species, through analysis of a resource-competition model. It demonstrates that, through the mechanism of pseudo-mixotrophy - proposed earlier by the author - allelopathy provides essential growth advantage to weaker competitors, and stabilizes resource competition, which ensures the coexistence of two phytoplankton on a single nutrient.
In simple nutrient-phytoplankton interactions where higher-trophic influences are negligible, this mechanism theoretically promotes phytoplankton diversity, and can potentially support high diversity in natural phytoplankton communities
Design of a Cascade Observer for a Model of Bacterial Batch Culture with Nutrient Recycling
International audienceA mathematical model of microbial growth on a single limited substrate in batch culture is proposed as an extension of the Monod's one. This model takes into account cell mortality, non-viable cell accumulation and cell recycling. We consider that only substrate concentration and total biomass are measured on-line. The parameters of the model are not identifiable at steady state, but we propose a design of an observer that reconstructs both parameters and state variable with a practical convergence, from any initial condition away from the equilibrium. The observer is build as a coupling of two non-linear observers in cascade with different time scales
Vertical distribution and composition of phytoplankton under the influence of an upper mixed layer
The vertical distribution of phytoplankton is of fundamental importance for
the dynamics and structure of aquatic communities. Here, using an
advection-reaction-diffusion model, we investigate the distribution and
competition of phytoplankton species in a water column, in which inverse
resource gradients of light and a nutrient can limit growth of the biomass.
This problem poses a challenge for ecologists, as the location of a production
layer is not fixed, but rather depends on many internal parameters and
environmental factors. In particular, we study the influence of an upper mixed
layer (UML) in this system and show that it leads to a variety of dynamic
effects: (i) Our model predicts alternative density profiles with a maximum of
biomass either within or below the UML, thereby the system may be bistable or
the relaxation from an unstable state may require a long-lasting transition.
(ii) Reduced mixing in the deep layer can induce oscillations of the biomass;
we show that a UML can sustain these oscillations even if the diffusivity is
less than the critical mixing for a sinking phytoplankton population. (iii) A
UML can strongly modify the outcome of competition between different
phytoplankton species, yielding bistability both in the spatial distribution
and in the species composition. (iv) A light limited species can obtain a
competitive advantage if the diffusivity in the deep layers is reduced below a
critical value. This yields a subtle competitive exclusion effect, where the
oscillatory states in the deep layers are displaced by steady solutions in the
UML. Finally, we present a novel graphical approach for deducing the
competition outcome and for the analysis of the role of a UML in aquatic
systems.Comment: 20 pages, 8 figure
Dynamics Analysis and Biomass Productivity Optimisation of a Microbial Cultivation Process through Substrate Regulation
A microbial cultivation process model with variable biomass yield, control of substrate concentration, and biomass recycle is formulated, where the biochemical kinetics follows an extension of the Monod and Contois models. Control of substrate concentration allows for indirect monitoring of biomass and dissolved oxygen concentrations and consequently obtaining high yield and productivity of biomass. Dynamics analysis of the proposed model is carried out and the existence of order-1 periodic solution is deduced with a formulation of the period, which provides a theoretical possibility to convert the state-dependent control to a periodic one while keeping the dynamics unchanged. Moreover, the stability of the order-1 periodic solution is verified by a geometric method. The stability ensures a certain robustness of the adopted control; that is, even with an inaccurately detected substrate concentration or a deviation, the system will be always stable at the order-1 periodic solution under the control. The simulations are carried out to complement the theoretical results and optimisation of the biomass productivity is presented
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