Complex dynamics of an impulsive chemostat model

We propose a novel impulsive chemostat model with the substrate concentration as the basis for the implementation of control strategies, and then investigate the model’s global dynamics. The exact domains of the impulsive and phase sets are discussed in the light of phase portraits of the model, and then we define the Poincar´e map and study its complex properties. Furthermore, the existence and stability of the microorganism eradication periodic solution are addressed, and the analysis of a transcritical bifurcation reveals that an order-1 periodic solution is generated. We also provide the conditions for the global stability of an order-1 periodic solution and show the existence of order-k (k ≥ 2) periodic solutions. Moreover, the PRCC results and bifurcation analyses not only substantiate our results, but also indicate that the proposed system exists with complex dynamics. Finally, biological implications related to the theoretical results are discussed

Global dynamics of a state-dependent feedback control system

The main purpose is to develop novel analytical techniques and provide a comprehensive qualitative analysis of global dynamics for a state-dependent feedback control system arising from biological applications including integrated pest management. The model considered consists of a planar system of differential equations with state-dependent impulsive control. We characterize the impulsive and phase sets, using the phase portraits of the planar system and the Lambert W function to define the Poincaré map for impulsive point series defined in the phase set. The existence, local and global stability of an order-1 limit cycle and sharp sufficient conditions for the global stability of the boundary order-1 limit cycle have been provided. We further examine the flip bifurcation related to the existence of an order-2 limit cycle. We show that the existence of an order-2 limit cycle implies the existence of an order-1 limit cycle. We derive sufficient conditions under which any trajectory initiating from a phase set will be free from impulsive effects after finite state-dependent feedback control actions, and we also prove that order-k (k ≥ 3) limit cycles do not exist, providing a solution to an open problem in the integrated pest management community. We then investigate multiple attractors and their basins of attraction, as well as the interior structure of a horseshoe-like attractor. We also discuss implications of the global dynamics for integrated pest management strategy. The analytical techniques and qualitative methods developed in the present paper could be widely used in many fields concerning state-dependent feedback control

A STUDY ON DYNAMIC SYSTEMS RESPONSE OF THE PERFORMANCE CHARACTERISTICS OF SOME MAJOR BIOPHYSICAL SYSTEMS

Dynamic responses of biophysical systems - performance characteristic

Systems biology approaches to the dynamics of gene expression and chemical reactions

Systems biology is an emergent interdisciplinary field of study whose main goal is to understand the global properties and functions of a biological system by investigating its structure and dynamics [74]. This high-level knowledge can be reached only with a coordinated approach involving researchers with different backgrounds in molecular biology, the various omics (like genomics, proteomics, metabolomics), computer science and dynamical systems theory. The history of systems biology as a distinct discipline began in the 1960s, and saw an impressive growth since year 2000, originated by the increased accumulation of biological information, the development of high-throughput experimental techniques, the use of powerful computer systems for calculations and database hosting, and the spread of Internet as the standard medium for information diffusion [77]. In the last few years, our research group tried to tackle a set of systems biology problems which look quite diverse, but share some topics like biological networks and system dynamics, which are of our interest and clearly fundamental for this field. In fact, the first issue we studied (covered in Part I) was the reverse engineering of large-scale gene regulatory networks. Inferring a gene network is the process of identifying interactions among genes from experimental data (tipically microarray expression profiles) using computational methods [6]. Our aim was to compare some of the most popular association network algorithms (the only ones applicable at a genome-wide level) in different conditions. In particular we verified the predictive power of similarity measures both of direct type (like correlations and mutual information) and of conditional type (partial correlations and conditional mutual information) applied on different kinds of experiments (like data taken at equilibrium or time courses) and on both synthetic and real microarray data (for E. coli and S. cerevisiae). In our simulations we saw that all network inference algorithms obtain better performances from data produced with \u201cstructural\u201d perturbations (like gene knockouts at steady state) than with just dynamical perturbations (like time course measurements or changes of the initial expression levels). Moreover, our analysis showed differences in the performances of the algorithms: direct methods are more robust in detecting stable relationships (like belonging to the same protein complex), while conditional methods are better at causal interactions (e.g. transcription factor\u2013binding site interactions), especially in presence of combinatorial transcriptional regulation. Even if time course microarray experiments are not particularly useful for inferring gene networks, they can instead give a great amount of information about the dynamical evolution of a biological process, provided that the measurements have a good time resolution. Recently, such a dataset has been published [119] for the yeast metabolic cycle, a well-known process where yeast cells synchronize with respect to oxidative and reductive functions. In that paper, the long-period respiratory oscillations were shown to be reflected in genome-wide periodic patterns in gene expression. As explained in Part II, we analyzed these time series in order to elucidate the dynamical role of post-transcriptional regulation (in particular mRNA stability) in the coordination of the cycle. We found that for periodic genes, arranged in classes according either to expression profile or to function, the pulses of mRNA abundance have phase and width which are directly proportional to the corresponding turnover rates. Moreover, the cascade of events which occurs during the yeast metabolic cycle (and their correlation with mRNA turnover) reflects to a large extent the gene expression program observable in other dynamical contexts such as the response to stresses or stimuli. The concepts of network and of systems dynamics return also as major arguments of Part III. In fact, there we present a study of some dynamical properties of the so-called chemical reaction networks, which are sets of chemical species among which a certain number of reactions can occur. These networks can be modeled as systems of ordinary differential equations for the species concentrations, and the dynamical evolution of these systems has been theoretically studied since the 1970s [47, 65]. Over time, several independent conditions have been proved concerning the capacity of a reaction network, regardless of the (often poorly known) reaction parameters, to exhibit multiple equilibria. This is a particularly interesting characteristic for biological systems, since it is required for the switch-like behavior observed during processes like intracellular signaling and cell differentiation. Inspired by those works, we developed a new open source software package for MATLAB, called ERNEST, which, by checking these various criteria on the structure of a chemical reaction network, can exclude the multistationarity of the corresponding reaction system. The results of this analysis can be used, for example, for model discrimination: if for a multistable biological process there are multiple candidate reaction models, it is possible to eliminate some of them by proving that they are always monostationary. Finally, we considered the related property of monotonicity for a reaction network. Monotone dynamical systems have the tendency to converge to an equilibrium and do not present chaotic behaviors. Most biological systems have the same features, and are therefore considered to be monotone or near-monotone [85, 116]. Using the notion of fundamental cycles from graph theory, we proved some theoretical results in order to determine how distant is a given biological network from being monotone. In particular, we showed that the distance to monotonicity of a network is equal to the minimal number of negative fundamental cycles of the corresponding J-graph, a signed multigraph which can be univocally associated to a dynamical system

Invariant manifold theory for impulsive functional differential equations with applications

The primary contribution of this thesis is a development of invariant manifold theory for impulsive functional differential equations. We begin with an in-depth analysis of linear systems, immersed in a nonautonomous dynamical systems framework. We prove a variation-of-constants formula, introduce appropriate generalizations of stable, centre and unstable subspaces, and develop a Floquet theory for periodic systems. Using the Lyapunov-Perron method, we prove the existence of local centre manifolds at a nonhyperbolic equilibrium of nonlinear impulsive functional differential equations. Using a formal differentiation procedure in conjunction with machinery from functional analysis -- specifically, contraction mappings on scales of Banach spaces -- we prove that the centre manifold is smooth in the state space. By introducing a coordinate system, we are able to prove that the coefficients of any Taylor expansion of the local centre manifold are unique and sufficiently regular in the time and lag arguments that they can be computed by solving an impulsive boundary-value problem. After proving a reduction principle, this leads naturally to explorations into bifurcation theory, where we establish generalizations of the classical fold and Hopf bifurcations for impulsive delay differential equations. Aside from the centre manifold, we demonstrate the existence and smoothness of stable and unstable manifolds and prove a linearized stability theorem. One of the applications of the theory above is an analysis of a SIR model with pulsed vaccination and finite temporary immunity modeled by a discrete delay. We determine an analytical stability criteria for the disease-free equilibrium and prove the existence of a transcritical bifurcation of periodic solutions at some critical vaccination coverage level for generic system parameters. Then, using numerical continuation and a monodromy operator discretization scheme, we track the bifurcating endemic periodic solution until a Hopf point is identifed. A cylinder bifurcation is observed; the periodic orbit expands into a cylinder in the extended phase space before eventually contracting onto a periodic orbit as the vaccination coverage vanishes. The other application is an impulsive stabilization method based on centre manifold reduction and optimization principles. Assuming a cost structure on the impulsive controller and a desired convergence rate target, we prove that under certain conditions there is always an impulsive controller that can stabilize a nonhyperbolic equilibrium with a trivial unstable subspace, robustly with respect to parameter perturbation, while guaranteeing a minimal cost. We then exploit the low-dimensionality of the centre manifold to develop a two-stage program that can be implemented to compute the optimal controller. To demonstrate the effectiveness of the two-stage program, which we call the centre probe method, we use the method to stabilize a complex network of 100 diffusively coupled nodes at a Hopf point. The cost structure is one that assigns higher cost to controlling of nodes that have more neighbours, while the jump functionals are required to be diagonal -- that is, they do not introduce further coupling. We also introduce a secondary goal, which is that the number of nodes that are controlled is minimized

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described