Confidence domain in the stochastic competition chemostat model with feedback control

This paper studies a stochastically forced chemostat model with feedback control in which two organisms compete for a single growth-limiting substrate. In the deterministic counterpart, previous researches show that the coexistence of two competing organisms may be achieved as a stable positive equilibrium or a stable positive periodic solution by different feedback schedules. In the stochastic case, based on the stochastic sensitivity function technique,we construct the confidence domains for different feedback schedules which allow us to find the configurational arrangements of the stochastic attractors and analyze the dispersion of the random states of the stochastic model.     

具有时滞的周期非均匀竞争恒化器模型的空间动力学

本文提出了一个具有时滞的周期非均匀单种营养基——双种微生物的竞争恒化器模型,利用半群理论, 获得了该模型解的存在唯一性. 进一步, 建立了该模型的竞争排斥原理, 给出了两竞争物种共存的充分条件.     

Coexistence in a two species chemostat model with Markov switchings

This paper formulates a new switched two species chemostat model and discusses the coexistence behavior in the chemostat. A complete classification on the single-species chemostat is carried out firstly, where the stationary distribution with ergodicity is derived to exist and be unique. Then, based on the obtained stationary distribution and the comparison theorem, we put forward some sufficient conditions for the coexistence of microorganisms in the two species chemostat with Markov switchings. Moreover, when the species coexist in the deterministic chemostat for each state and have the same break-even concentrations for all states, they are proved to coexist still in the switched chemostat, which randomized the results of the classical deterministic chemostat. Results in this paper show that Markov switchings can contribute to coexistence of the two species.     

Dynamics and steady-state analysis of an unstirred chemostat model with internal storage and toxin mortality

This study proposes and analyzes a reaction–diffusion system describing the competition of two species for a single limiting nutrient that is stored internally in an unstirred chemostat, in which each species also produces a toxin that increases the mortality of its competitors. The possibility of coexistence and bistability for the model system is studied by the theory of uniform persistence and topological degree theory in cones, respectively. More precisely, the sharp a priori estimates for nonnegative solutions of the system are first established, which assure that all of nonnegative solutions belong to a special cone. Then it turns out that coexistence and bistability can be determined by the sign of the principal eigenvalues associated with specific nonlinear eigenvalue problems in the special positive cones. The local stability of two semi-trivial steady states cannot be studied via the technique of linearization since a singularity arises from the linearization around those steady states. Instead, we introduce a 1-homogeneous operator to rigorously investigate their local stability.     

Stability analysis of mathematical model of competition in a chain of chemostats in series with delay

We study a nonlinear system of differential delay equations describing a model of a chain of two chemostats, where one contains two microbial species in competition for a single limiting nutrient and receives an external input of the less advantaged competitor, which is cultivated in an external chemostat. We obtain sufficient conditions ensuring coexistence of all the species in competition which consist in upper delay bounds.     

一类未搅拌恒化器中食物网的共存态

本文讨论了一类在单种营养物输入的未搅拌恒化器中的简单食物网模型,该模型除了营养物以外,还包含有一个捕食者种群和两个竞争的食饵种群.应用Dancer不动点指数和度理论的知识,给出了该系统共存态存在的充分必要条件.     

Global stability for a model of competition in the chemostat with microbial inputs

We propose a model of competition of n species in a chemostat, with constant input of some species. We mainly emphasize the case that can lead to coexistence in the chemostat in a non-trivial way, i.e., where the n−1 less competitive species are in the input. We prove that if the inputs satisfy a constraint, the coexistence between the species is obtained in the form of a globally asymptotically stable (GAS) positive equilibrium, while a GAS equilibrium without the dominant species is achieved if the constraint is not satisfied. This work is round up with a thorough study of all the situations that can arise when having an arbitrary number of species in the chemostat inputs; this always results in a GAS equilibrium that either does or does not encompass one of the species that is not present in the input.     

Crowding effects promote coexistence in the chemostat

This paper deals with an almost-global stability result for a particular chemostat model. It deviates from the classical chemostat because crowding effects are taken into consideration. This model can be rewritten as a negative feedback interconnection of two systems which are monotone (as input/output systems). Moreover, these subsystems behave nicely when subject to constant inputs. This allows the use of a particular small-gain theorem which has recently been developed for feedback interconnections of monotone systems. Application of this theorem requires—at least approximate—knowledge of two gain functions associated to the subsystems. It turns out that for the chemostat model proposed here, these approximations can be obtained explicitly and this leads to a sufficient condition for almost-global stability. In addition, we show that coexistence occurs in this model if the crowding effects are large enough.     

具有质载和内抑制剂的非均匀恒化器模型共存解的多重性

本文研究一类具有质载和内抑制剂的非均匀恒化器竞争模型. 基于对第二类极限系统分歧的进一步分析, 我们研究了此模型共存解的多重性和稳定性, 从而在一定条件下得到了模型共存解分歧的确切形状, 验证了已有的数值结果.     

Random and stochastic disturbances on the input flow in chemostat models with wall growth

In this paper, we analyze a chemostat model with wall growth where the input flow is perturbed by two different stochastic processes: the well-known standard Wiener process, which leads into several drawbacks from the biological point of view, and a suitable Ornstein-Uhlenbeck process depending on some parameters which allow us to control the noise to be bounded inside some interval that can be fixed previously by practitioners. Thanks to this last approach, which has already proved to be very realistic when modeling other simplest chemostat models, it will be possible to prove the persistence and coexistence of the species in the model without needing the theory of random dynamical systems and pullback attractors needed when dealing with the Wiener process. This is an advantage since the theoretical framework in this paper is much less complicated and provides us much more information than the other.     

Existence and stability of coexistence states in a competition unstirred chemostat

In this paper, we consider the two similar competing species in a competition unstirred chemostat model with diffusion. The two competing species are assumed to be identical except for their maximal growth rates. In particular, we study the existence and stability of the coexistence states, and the semi-trivial equilibria or the unique coexistence state is the global attractor can be established under some suitable conditions. Our mathematical approach is based on Lyapunov–Schmidt reduction, the implicit function theory and spectral theory.     

Bottom up and top down effect on toxin producing phytoplankton and its consequence on the formation of plankton bloom

We consider a plankton-nutrient interaction model consisting of phytoplankton, zooplankton and dissolved limiting nutrient with general nutrient uptake functions and instantaneous nutrient recycling. In this model, it is assumed that phytoplankton releases toxic chemical for self defense against their predators. The model system is studied analytically and the threshold values for the existence and stability of various steady states are worked out. It is observed that if the maximal zooplankton conversion rate crosses a certain critical value, the system enters into Hopf bifurcation. Finally it is observed that to control the planktonic bloom and to maintain stability around the coexistence equilibrium we have to control the nutrient input rate specially caused by artificial eutrophication. In case if it is not possible to control the nutrient input rate, one could use toxic phytoplankton to prevent the recurrence bloom.     

Reduced order dead-beat observers for the chemostat

This paper studies the strong observability property and the reduced-order dead-beat observer design problem for a continuous bioreactor. New relationships between coexistence and strong observability, and checkable sufficient conditions for strong observability, are established for a chemostat with two competing microbial species. Furthermore, the dynamic output feedback stabilization problem is solved for the case of one species.     

一类具有抑制剂的非均匀恒化器模型的共存态

研究了一类具有抑制剂和Beddington-DeAngelis功能反应项的非均匀恒化器模型.根据单调动力系统理论得到了正平衡解的存在性.利用度理论、分歧理论以及摄动理论,分析了抑制剂对系统正平衡解及渐近行为的影响.结果表明当体现抑制作用的参数μ充分大时,此模型或者没有正解,并且一个半平凡晌非负解是全局吸引的;或者模型的所有正解均由一个极限问题决定.     

Multivariable fuzzy control applied to the physical–chemical treatment facility of a Cellulose factory

Feedback linearization is a well-known technique in nonlinear control in which known system nonlinearities are canceled by the control input leaving a linear control problem. Feedback linearization requires an exact model for the system. Fundamental and advanced developments in neuro-fuzzy synergy for modeling and control are used to apply the feedback linearization control law on second-order plants. In the models that are used, the nonlinear plant is decomposed on six fuzzy systems necessary to apply the control signal to allow the following of a reference value. A practical application is also presented using a waste water plant. This method can be extended to multiple input–multiple output (MIMO) plants based on input–output data pairs collected directly from the plant.     

Modelling a hormone-inspired controller for individual- and multi-modular robotic systems

For all living organisms, the ability to regulate internal homeostasis is a crucial feature. This ability to control variables around a set point is found frequently in the physiological networks of single cells and of higher organisms. Also, nutrient allocation and task selection in social insect colonies can be interpreted as homeostatic processes of a super-organism. And finally, behaviour can also represent such a control scheme. We show how a simple model of hormone regulation, inspired by simple biological organisms, can be used as a novel method to control the behaviour of autonomous robots. We demonstrate the formulation of such an artificial homeostatic hormone system (AHHS) by a set of linked difference equations and explain how the homeostatic control of behaviour is achieved by homeostatic control of the internal ‘hormonal’ state of the robot. The first task that we used to check the quality of our AHHS controllers was a very simple one, which is often a core functionality in controller programmes that are used in autonomous robots: obstacle avoidance. We demonstrate two implementations of such an AHHS controller that performs this task in differing levels of quality. Both controllers use the concept of homeostatic control of internal variables (hormones) and they extend this concept to also include the outside world of the robots into the controlling feedback loops: As they try to regulate internal hormone levels, they are forced to keep a homeostatic control of sensor values in a way that the desired goal ‘obstacle avoidance’ is achieved. Thus, the created behaviour is also a manifestation of the acts of homeostatic control. The controllers were evaluated using a stock-and-flow model that allowed sensitivity analysis and stability tests. Afterwards, we have also tested both controllers in a multi-agent simulation tool, which allowed us to predict the robots' behaviours in various habitats and group sizes. Finally, we demonstrate how this novel AHHS controller is suitable to control a multi-cellular robotic organism in an evolutionary robotics approach, which is used for self-programming in a gait-learning task. These examples shown in this article represent the first step in our research towards autonomous aggregation and coordination of robots to higher-level modular robotic organisms that consist of several joined autonomous robotic units. Finally, we plan to achieve such aggregation patterns and to control complex-shaped robotic organisms using AHHS controllers, as they are described here.     

Asymptotic behavior of an unstirred chemostat model with internal inhibitor

This paper deals with a chemostat model with an internal inhibitor. First, the elementary stability and asymptotic behavior of solutions of the system are determined. Second, the effects of the inhibitor are considered. It turns out that the parameter μ, which measures the effect of the inhibitor, plays a very important role in deciding the stability and longtime behavior of solutions of the system. The results show that if μ is sufficiently large, this model has no coexistence solution and one of the semitrivial equilibria is a global attractor when the maximal growth rate a of the species u lies in certain range; but when a belongs to another range, all positive solutions of this model are governed by a limit problem, and two semitrivial equilibria are bistable. The main tools used here include monotone system theory, degree theory, bifurcation theory and perturbation technique.     

Asymptotic behaviour of the chemostat model with delayed response in growth

The asymptotic behaviour of solutions of the model for competition between three microorganisms in the chemostat is studied. The model incorporates time delays that allow for cellular components of each competing species to be structured to include unassimilated or stored nutrient and allow uptake functions to be nonmonotone. By introducing three auxiliary functions and using a lemma, it is shown that at most one competitor survives, and the substrate and the surviving competitor (if two exist) approach limiting values.     

A competition model for two resources in un-stirred chemostat

This paper studies a un-stirred chemostat with two species competing for two growth-limiting, non-reproducing resources. We determine the conditions for positive steady states of the two species, and then consider the global attractors of the model. In addition, we obtain the conditions under which the two populations uniformly strongly persist or go to extinction. Since the diffusion mechanism with homogeneous boundary conditions inhibits the growth of the organism species, it can be understood that the coexistence will be ensured by proportionally smaller diffusions for the two species. In particular, it is found that both instability and bi-stability subcases of the two semitrivial steady states are included in the coexistence region. The two populations will go to extinction when both possess large diffusion rates. If just one of them spreads faster with the other one diffusing slower, then the related semitrivial steady state will be globally attracting. The techniques used for the above results consist of the degree theory, the semigroup theory, and the maximum principle.     

Nutrient-phytoplankton-zooplankton models with a toxin

Models of nutrient-plankton interaction with a toxic substance that inhibits either the growth rate of phytoplankton, zooplankton or both trophic levels are proposed and studied. For simplicity, it is assumed that both nutrient and the toxin have the same constant input and washout rate as the chemostat system. The effects of the toxin upon the existence, magnitude, and stability of the steady states are examined. Numerical simulations demonstrate that the system can have multiple attractors when the phytoplankton’s nutrient uptake rate is inhibited by the toxin.