Modelling effects of high product and substrate inhibition on oscillatory behavior in continuous bioreactors

In this study we consider a model for continuous bioreactors which incorporates the effects of high product and substrate inhibition on the kinetics as well as biomass and product yields. We theoretically investigate the possibility of various dynamic behaviors in the bioreactor over different ranges of operating parameters to determine the delineating process conditions which may lead to oscillatory behavior. Application of the singular perturbation technique allows us to derive explicit conditions on the system parameters which specifically ascertain the existence of limit cycles composed of concatenations of catastrophic transitions occurring at different speeds. We discover further that the interactions between the limiting substrate and the growing microorganisms can give rise to high frequency oscillations which can arise during the transients toward the attractor or during the low-frequency cycle. Such a study not only can describe more fully the kinetics in a fermentor but also assist in formulating optimum fermentor operating conditions and in developing control strategy for maintaining optimum productivity.     

Modeling of bone formation and resorption mediated by parathyroid hormone: response to estrogen/PTH therapy

Bone, a major reservoir of body calcium, is under the hormonal control of the parathyroid hormone (PTH). Several aspects of its growth, turnover, and mechanism, occur in the absence of gonadal hormones. Sex steroids such as estrogen, nonetheless, play an important role in bone physiology, and are extremely essential to maintain bone balance in adults. In order to provide a basis for understanding the underlying mechanisms of bone remodeling as it is mediated by PTH, we propose here a mathematical model of the process. The nonlinear system model is then utilized to study the temporal effect of PTH as well as the action of estrogen replacement therapy on bone turnover. Analysis of the model is done on the assumption, supported by reported clinical evidence, that the process is characterized by highly diversified dynamics, which warrants the use of singular perturbation arguments. The model is shown to exhibit limit cycle behavior, which can develop into chaotic dynamics for certain ranges of the system's parametric values. Effects of estrogen and PTH administrations are then investigated by extending on the core model. Analysis of the model seems to indicate that the paradoxical observation that intermittent PTH administration causes net bone deposition while continuous administration causes net bone loss, and certain other reported phenomena may be attributed to the highly diversified dynamics which characterizes this nonlinear remodeling process.     

Modeling insulin kinetics: responses to a single oral glucose administration or ambulatory-fed conditions

This paper presents a nonlinear mathematical model of the glucose-insulin feedback system, which has been extended to incorporate the beta-cells' function on maintaining and regulating plasma insulin level in man. Initially, a gastrointestinal absorption term for glucose is utilized to effect the glucose absorption by the intestine and the subsequent release of glucose into the bloodstream, taking place at a given initial rate and falling off exponentially with time. An analysis of the model is carried out by the singular perturbation technique in order to derive boundary conditions on the system parameters which identify, in particular, the existence of limit cycles in our model system consistent with the oscillatory patterns often observed in clinical data. We then utilize a sinusoidal term to incorporate the temporal absorption of glucose in order to study the responses in the patients under ambulatory-fed conditions. A numerical investigation is carried out in this case to construct a bifurcation diagram to identify the ranges of parametric values for which chaotic behavior can be expected, leading to interesting biological interpretations.     

A Simple,Realistic Stochastic Model of Gastric Emptying

Several models of Gastric Emptying (GE) have been employed in the past to represent the rate of delivery of stomach contents to the duodenum and jejunum. These models have all used a deterministic form (algebraic equations or ordinary differential equations), considering GE as a continuous, smooth process in time. However, GE is known to occur as a sequence of spurts, irregular both in size and in timing. Hence, we formulate a simple stochastic process model, able to represent the irregular decrements of gastric contents after a meal. The model is calibrated on existing literature data and provides consistent predictions of the observed variability in the emptying trajectories. This approach may be useful in metabolic modeling, since it describes well and explains the apparently heterogeneous GE experimental results in situations where common gastric mechanics across subjects would be expected.     

Nonlinear stability analyses of vegetative pattern formation in an arid environment

The development of spontaneous stationary vegetative patterns in an arid isotropic homogeneous environment is investigated by means of various weakly nonlinear stability analyses applied to the appropriate governing equation for this phenomenon. In particular, that process can be represented by a fourth-order partial differential time-evolution logistic equation for the total plant biomass per unit area divided by the carrying capacity of its territory and defined on an unbounded flat spatial domain. Those patterns that consist of parallel stripes, labyrinth-like mazes, rhombic arrays of rectangular patches, and hexagonal distributions of spots or gaps are generated by the balance between the effects of short-range facilitation and long-range competition. Then those theoretical predictions are compared with both relevant observational evidence and existing numerical simulations as well as placed in the context of the results from some recent nonlinear pattern formation studies.     

Coexistence of competing microbial species in a chemostat where one population feeds on another

In this paper, we consider a model for a chemostat in which two microbial species compete for a single rate-limiting nutrient, while one of the species feeds on another. Under certain simplifying hypotheses, such a chemostat can be described by a system of three nonlinear ordinary differential equations. A theoretical study is conducted to characterize the possible types of solutions. A limit cycle solution was obtained for some parametric values of the system indicating that coexistence of the two species is possible in a significant range of the operating parameters.     

Relating damped oscillations to sustained limit cycles describing real and ideal batch fermentation processes

A batch fermentation model is presented in which the specific growth rate and yield functions are chosen such that sustained oscillations in both the cell and substrate concentration occur. This phenomenon is shown to be a Hopf bifurcation in the underlying system of non-linear ordinary differential equations which comprises the model. It is shown that for oscillations in the substrate concentration to occur it is necessary for the yield term to depend on both the cell and substrate levels.     

Direction of the sustained oscillation trajectory in the cell-product phase plane describing product inhibition on continuous fermentation systems

A criterion for the direction (clockwise or counterclockwise) of the limit cycle in the cell massproduct phase plane is developed for a product inhibition model of a continuous microbial culture. It is then hypothesized that the model will not admit clockwise oscillations.     

Bifurcation analysis of a product inhibition model of a continuous fermentation process

Summary A product inhibition model of a continuous fermentation process is considered. If the yield term is a variable function of ethanol concentration, oscillation in the cell and ethanol concentrations is shown to be a Hopf bifurcation in the underlying system of nonlinear, ordinary differential equations which comprises the model.     

The effect of the yield expression on the existence of oscillatory behavior in a three-variable model of a continuous fermentation system subject to product inhibition

A three-variable model of a continuous fermentation process characterised by product inhibition is studied. It is shown that if the cell to substrate yield is constant, the system cannot have periodic solutions. If, on the other hand, the yield term is a variable function of substrate concentration, the model will exhibit oscillations in the cells, substrate and product concentrations in the form of Hopf bifurcation in the underlying system of three nonlinear, ordinary differential equations which comprise the model.