Modeling nonlinear stochastic kinetic system and stochastic optimal control of microbial bioconversion process in batch culture

In this paper, we analyze a stochastic model representing batch fermentation in the process of glycerol bio-dissimilation to 1,3-propanediol by klebsiella pneumoniae. The stochasticity in the model is introduced by parameter perturbation which is a standard technique in stochastic population modelling. Thus, based on the nonlinear deterministic dynamical system of glycerol bioconversion to 1,3-propanediol in batch culture, we present the stochastic version of the batch fermentation process driven by a five-dimensional Brownian motion and Lipschitz coefficients, which is suitable for the factual fermentation. Subsequently, we study the existence and uniqueness of solutions for the stochastic system as well as the boundedness and Markov property of solutions. Moveover a stochastic optimal control model is constructed and the sufficient and necessary conditions for optimality are proved via dynamic programming principle. Finally we present computer simulation for the stochastic system by using Stochastic Euler–Maruyama scheme. Compared with the results from the deterministic system, numerical results reveal the peculiar role of stochasticity in the dynamical responses of the batch culture

Vector measure as controls for explicit nonlinear impulsive system of fed-batch culture

AbstractIn this paper, we consider an optimal control problem of microbial fermentation process in which glycerol is converted to 1,3-propanediol by Klebsiella pneumoniae in fed-batch culture. During the period of reaction, the variation of pH value is monitored to determine glycerol replenishment quantity, guaranteeing that microorganism can always keep growing fast under enough nutrition. Every time pH value is lower than seven, the quantity of glycerol added is such that pH value returns seven again. Glycerol is poured into reactor at discrete time instant and the quantity is controllable. The problem is to determine for each discrete time instant the glycerol quantity to add and maximize the final concentration of 1,3-propanediol. We present a controlled explicit nonlinear impulsive dynamical system of fed-batch culture with state independent vector measures as controls and study the existence, uniqueness, boundedness, continuous dependence and Gâteaux differentiability of its solution with respect to controls. We then propose a multiple objective programming model and demonstrate the regularity of cost functionals and weak compactness of admissible control set. Finally we discuss the existence of optimal control and implement a hybrid particle swarm optimization algorithm to solve the model optimally. Computational results are presented on a numerical example

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

Complex population dynamics in microbial systems

The study of spatial and temporal population dynamics has a long history in ecology, going back to the beginning of the 1900´s. Both intrinsic and extrinsic mechanisms are involved in determining the temporal and spatial occurrences of populations and species. Different dynamic patterns result from the strength and the interplay of the two mechanisms. The fact that in-trinsic driven population dynamics are woven together with extrinsic, often stochastic dynamics makes analyses of intrinsic mechanisms difficult and led to a controversial discussion about the relevance in nature. However, there is a gap between results from mathematical modelling showing the occurrence and meaning of intrinsically driven dynamics, and empirical proves. Recently, laboratory experiments under clearly defined and controlled conditions were shown to be a suitable tool to study intrinsic, deterministic population dynam-ics. Deterministic chaos is one type of dynamic behaviour exhibited by a change in one or more intrinsic parameters beside extinction, damped oscil-lations, and stable limit cycle. Most discussed is the relevance of chaotic be-haviour in population dynamics, due to the fact that empirical evidence is lim-ited to a simple one-species system. Furthermore, chaotic fluctuations are thought to lead to extinction of a population, because chaotic dynamics can obtain very small population sizes, even more vulnerable when mixed together with stochastic events. The question, if chaos occurs in the real world and under which circumstances chaos may be found in nature, is still open. Clearly defined laboratory experiments were established to analyse intrinsically driven dynamics in a multi-species system. Different dynamic behaviours were found in chemostat experiments with a two-prey-one-predator system of a bacterivorous ciliate as the predator and two bacteria strains as the prey organisms. The different population dynamics - extinction, damped oscillations, stable limit cycles and chaos - were triggered by a change in the dilution rate of the chemostat system and verified by calculations of the corresponding Lyapunov exponents. Therewith, chaos was shown in an experimental three-species system for the first time. The different dynamics in the microbial food web revealed a surprisingly short transition (4-7 days) to a different dynamic behaviour when the dilution rate as the control parameter was changed. All dynamics persisted in experiments when different local populations with different dynamics (chemostats with different dilution rates) were coupled. Experiments showed that the dynamic behaviours of the coupled populations were only triggered by the demographic parameter � in this case the dilution rate - and reacted independent of the constant inflow of organisms from populations with different dynamics. Here, we were able to shed more light on the question about the relevance of chaos in the real world. In conclusion spatio-temporal chaos might be more common in nature than generally assumed. Microbial communities with fast reproduction rates might be favoured candidates to show chaos and other complex dynamics in nature. Intrinsically driven dynamics might be persistent when perturbated by a constant fluctuating inflow of organisms and might lead to the establishment of chaos in habitats with constant flows (e.g. aquatic organisms in rivers and oceanic currents, and water drainage to groundwater). The fast transition to a different dynamic behaviour after a change in a control parameter shows how distinct intrinsic driven processes might be. A reason why chaotic dynamics in nature are not observed might be due too the large sampling intervals in most field studies