DOCSLIB.ORG

An Introduction to Feedback Control in Systems Biology

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
An Introduction to Feedback Control in Systems Biology Carlo Cosentino, Declan Bates An Introduction to Feedback Control in Systems Biology CRC PRESS Boca Raton Ann Arbor London Tokyo To my parents, Rosa and Nicola, and to Cinzia. – C.C. To my parents, Br´ıd and Tony, and to Orlando and Lauren. – D.B. Contents 1 Introduction 7 1.1 Whatisfeedbackcontrol? . 7 1.2 Feedbackcontrolinbiologicalsystems . 10 1.2.1 The tryptophan operon feedback control system . 11 1.2.2 The polyamine feedback controlsystem . 12 1.2.3 Theheat shockfeedbackcontrolsystem . 14 1.3 Application of control theory to biological systems: a historical perspective ............................ 16 References ................................ 17 2 Linear Systems 23 2.1 Introduction ............................ 23 2.2 Statespacemodels ........................ 24 2.3 Linear time invariant systems and the frequency response .. 26 2.4 Fourieranalysis .......................... 32 2.5 Transfer functions and the Laplace transform . 36 2.6 Stability .............................. 39 2.7 Change of state variables and canonical representations ... 41 2.8 Characterising system dynamics in the time domain . 42 2.9 Characterising system dynamics in the frequency domain .. 46 2.10 Block diagram representations of interconnected systems ... 48 2.11 Case Study I: Characterising the frequency dependence of osmo–adaptation in Saccharomyces cerevisiae ......... 53 2.11.1 Introduction. 54 2.11.2 Frequencydomainanalysis . 54 2.11.3 Timedomainanalysis . 57 2.12 Case Study II: Characterising the dynamics of the Dictyostelium externalsignalreceptornetwork . 60 2.12.1 Introduction. 61 2.12.2 A generic structure for ligand/receptor interaction net- works ........................... 61 2.12.3 Structure of the ligand/receptor interaction network in aggregating Dictyostelium cells............ 63 2.12.4 Dynamic response of the Dictyostelium ligand/receptor interactionnetwork. 66 References ................................ 69 iii iv An Introduction to Feedback Control in Systems Biology 3 Nonlinear systems 73 3.1 Introduction ............................ 73 3.2 Equilibriumpoints ........................ 75 3.3 Linearisation around equilibrium points . 78 3.4 Stabilityandregionsofattractions . 84 3.4.1 Lyapunovstability . 84 3.4.2 Regionsofattraction. 87 3.5 Optimisation methods for nonlinear systems . 91 3.5.1 Localoptimisationmethods . 93 3.5.2 Global optimisation methods . 95 3.5.3 Linearmatrixinequalities . 97 3.6 Case Study III: Stability analysis of tumor dormancy equilib- rium ................................ 99 3.6.1 Introduction. .100 3.6.2 Modelofcancerdevelopment . 101 3.6.3 Stability of the equilibrium points . 102 3.6.4 Checking inclusion in the region of attraction . 103 3.6.5 Analysis of the tumor dormancy equilibrium . 106 3.7 Case Study IV: Global optimisation of a model of the trypto- phan control system against multiple experiment data . 111 3.7.1 Introduction. .112 3.7.2 Model of the tryptophan controlsystem . 112 3.7.3 Model analysis using global optimisation . 115 References ................................116 4 Negative feedback systems 121 4.1 Introduction ............................121 4.2 Stability of negative feedback systems . 125 4.3 Performanceofnegativefeedbacksystems . 128 4.4 Fundamental tradeoffs with negative feedback . 134 4.5 Case Study V: Analysis of stability and oscillations in the p53- Mdm2feedbacksystem . .138 4.6 Case Study VI: Perfect adaptation via integral feedback con- trolinbacterialchemotaxis . 143 4.6.1 A mathematical model of bacterial chemotaxis . 144 4.6.2 Analysis of the perfect adaptation mechanism . 147 4.6.3 Perfect adaptation requires demethylation only of ac- tivereceptors . .. .. .. .. .. .. .. .. .. .151 References ................................152 5 Positive feedback systems 155 5.1 Introduction ............................155 5.2 Bifurcations, bistability and limit cycles . 155 5.2.1 Bifurcations and bistability . 155 5.2.2 Limitcycles........................158 Table of Contents v 5.3 Monotonesystems ........................162 5.4 ChemicalReactionNetworkTheory . 165 5.4.1 Preliminaries on reaction network structure . 166 5.4.2 Networksofdeficiencyzero . 168 5.4.3 Networksofdeficiencyone. 170 5.5 Case Study VII: Positive feedback leads to multistability, bi- furcations, and hysteresis in a MAPK cascade . 172 5.6 Case Study VIII: Coupled positive and negative feedback loops intheyeastgalactosepathway . 179 References ................................186 6 Model validation using robustness analysis 189 6.1 Introduction ............................189 6.2 Robustness analysis tools for model validation . 191 6.2.1 Bifurcationdiagrams . 191 6.2.2 Sensitivityanalysis . 192 6.2.3 µ-analysis .........................196 6.2.4 Optimisation-based robustness analysis . 199 6.2.5 Sum-of-Squarespolynomials . 200 6.2.6 MonteCarlosimulation . 202 6.3 New robustness analysis tools for biological systems . 203 6.4 Case Study IX: Validating models of cAMP oscillations in ag- gregating Dictyostelium cells ...................206 6.5 Case Study X: Validating models of the p53-Mdm2 System . 208 References ................................210 7 ReverseEngineeringBiomolecularNetworks 215 7.1 Introduction ............................215 7.2 Inferring network interactions using linear models . 215 7.2.1 Discrete-time vs Continuous-time model . 217 7.3 Leastsquares ...........................220 7.3.1 Leastsquaresfor dynamicalsystems . 225 7.3.2 Methods based on least squares regression . 228 7.4 Exploitingpriorknowledge . 230 7.4.1 Network inference via LMI–based optimisation . 231 7.4.2 MAX-PARSE: An algorithm for pruning a fully con- nected network according to maximum parsimony . 234 7.4.3 CORE-Net: A network growth algorithm using pref- erentialattachment. 235 7.5 Dealingwithmeasurementnoise . 236 7.5.1 TotalLeastSquares . 236 7.5.2 ConstrainedTotalLeastSquares . 238 7.6 Exploitingtime-varyingmodels . 240 7.7 Case Study XI: Inferring regulatory interactions in the innate immune systemfromnoisy measurements . 244 vi An Introduction to Feedback Control in Systems Biology 7.8 Case Study XII: Reverse engineering a cell-cycle regulatory subnetwork of Saccharomyces cerevisiae from experimental micro- arraydata .............................248 7.8.1 PACTLS: An algorithm for reverse engineering par- tially known networks from noisy data . 250 7.8.2 Results ..........................252 References ................................255 8 Stochastic effects in biological control systems 261 8.1 Introduction ............................261 8.2 Stochastic modelling and simulation . 262 8.3 A framework for analysing the effect of stochastic noise on stability ..............................265 8.3.1 The effective stability approximation . 266 8.3.2 A computationally efficient approximation of the dom- inantstochasticperturbation . 267 8.3.3 Analysis using the Nyquist stability criterion . 269 8.4 Case Study XIII: Stochastic effects on the stability of cAMP oscillations in aggregating Dictyostelium cells .........272 8.5 Case Study XIV: Stochastic effects on the robustness of cAMP oscillations in aggregating Dictyostelium cells .........277 References ................................282 Preface The field of Systems Biology encompasses scientists with extremely diverse backgrounds, from biologists, biochemists, clinicians and physiologists to math- ematicians, physicists, computer scientists and engineers. Although many of these researchers have recently become interested in control-theoretic ideas such as feedback, stability, noise and disturbance attenuation, and robust- ness, it is still unfortunately the case that only researchers with an engineering background will usually have received any formal training in control theory. Indeed, our initial motivation to write this book arose from the difficulty we found in recommending an introductory text on feedback control to colleagues who were not from an engineering background, but who needed to understand control engineering methods to analyse complex biological systems. This difficulty stems from the fact that the traditional audience for control textbooks is made up of electrical, mechanical, process and aerospace engi- neers who require formal training in control system design methods for their respective applications. Systems Biologists, on the other hand, are more in- terested in the fundamental concepts and ideas which may be used to analyse the effects of feedback in evolved biological control systems. Researchers with a biological sciences background may often also lack the expertise in physical systems modelling (Newtonian mechanics, Kirchhoff’s electrical circuit laws, etc) that is typically assumed in the examples used in standard texts on feed- back control theory. The type of “control applications” in which a Systems Biologist is interested are systems such as metabolic and gene-regulatory net- works, not aircraft, robots or engines, and the type of mathematical models they are familiar with are typically derived from classical reaction kinetics, not classical mechanics. Another significant problem for Systems Biologists is that current under- graduate books on control theory (which introduce the basic concepts at great length) are uniformly restricted to linear systems, while nonlinear systems are usually only considered by specialist post-graduate texts which require advanced mathematical skills. Although it will always be appropriate to in- troduce basic ideas in control using linear systems, biological systems are in general highly nonlinear, and thus a clear understanding
Load more

Recommended publications
  • Thinking About Evolutionary Mechanisms: Natural Selection
    Studies in History and Philosophy of Biological and Biomedical Sciences Stud. Hist. Phil. Biol. & Biomed. Sci. 36 (2005) 327–347 www.elsevier.com/locate/shpsc Thinking about evolutionary mechanisms: natural selection Robert A. Skipper Jr. a, Roberta L. Millstein b a Department of Philosophy, ML 0374, University of Cincinnati, Cincinnati, OH 45221-0374, USA b Department of Philosophy, California State University, East Bay, CA 94542-3042, USA Abstract This paper explores whether natural selection, a putative evolutionary mechanism, and a main one at that, can be characterized on either of the two dominant conceptions of mecha- nism, due to Glennan and the team of Machamer, Darden, and Craver, that constitute the Ônew mechanistic philosophyÕ. The results of the analysis are that neither of the dominant con- ceptions of mechanism adequately captures natural selection. Nevertheless, the new mechanis- tic philosophy possesses the resources for an understanding of natural selection under the rubric. Ó 2005 Elsevier Ltd. All rights reserved. Keywords: Mechanism; Natural selection; Evolution 1. Introduction This paper explores whether natural selection—a putative evolutionary mecha- nism, and a main one at that—can be adequately characterized under the rubric of the Ônew mechanistic philosophyÕ. What we call the new mechanistic philosophy is comprised of two dominant conceptions of mechanism, one due to Glennan E-mail addresses: [email protected] (R.A. Skipper Jr.), [email protected] (R.L. Millstein). 1369-8486/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.shpsc.2005.03.006 328 R.A. Skipper Jr., R.L. Millstein / Stud.
    [Show full text]
  • What Is a Complex Adaptive System?
    PROJECT GUTS What is a Complex Adaptive System? Introduction During the last three decades a leap has been made from the application of computing to help scientists ‘do’ science to the integration of computer science concepts, tools and theorems into the very fabric of science. The modeling of complex adaptive systems (CAS) is an example of such an integration of computer science into the very fabric of science; models of complex systems are used to understand, predict and prevent the most daunting problems we face today; issues such as climate change, loss of biodiversity, energy consumption and virulent disease affect us all. The study of complex adaptive systems, has come to be seen as a scientific frontier, and an increasing ability to interact systematically with highly complex systems that transcend separate disciplines will have a profound affect on future science, engineering and industry as well as in the management of our planet’s resources (Emmott et al., 2006). The name itself, “complex adaptive systems” conjures up images of complicated ideas that might be too difficult for a novice to understand. Instead, the study of CAS does exactly the opposite; it creates a unified method of studying disparate systems that elucidates the processes by which they operate. A complex system is simply a system in which many independent elements or agents interact, leading to emergent outcomes that are often difficult (or impossible) to predict simply by looking at the individual interactions. The “complex” part of CAS refers in fact to the vast interconnectedness of these systems. Using the principles of CAS to study these topics as related disciplines that can be better understood through the application of models, rather than a disparate collection of facts can strengthen learners’ understanding of these topics and prepare them to understand other systems by applying similar methods of analysis (Emmott et al., 2006).
    [Show full text]
  • Closed Loop Transfer Function
    Lecture 4 Transfer Function and Block Diagram Approach to Modeling Dynamic Systems System Analysis Spring 2015 1 The Concept of Transfer Function Consider the linear time-invariant system defined by the following differential equation : ()nn ( 1) ay01 ay ann 1 y ay ()mm ( 1) bu01 bu bmm 1 u b u() n m Where y is the output of the system, and x is the input. And the Laplace transform of the equation is, nn1 ()()aS01 aS ann 1 S a Y s mm1 ()()bS01 bS bmm 1 S b U s System Analysis Spring 2015 2 The Concept of Transfer Function The ratio of the Laplace transform of the output (response function) to the Laplace Transform of the input (driving function) under the assumption that all initial conditions are Zero. mm1 Ys() bS01 bS bmm 1 S b Transfer Function G() s nn1 Us() aS01 aS ann 1 S a Ps() ()nm Qs() System Analysis Spring 2015 3 Comments on Transfer Function 1. A mathematical model. 2. Property of system itself. - Independent of the input function and initial condition - Denominator of the transfer function is the characteristic polynomial, - TF tells us something about the intrinsic behavior of the model. 3. ODE equivalence - TF is equivalent to the ODE. We can reconstruct ODE from TF. 4. One TF for one input-output pair. : Single Input Single Output system. - If multiple inputs affect Obtain TF for each input x 6207()3()xx f t g t -If multiple outputs xxy 32 yy xut 3() System Analysis Spring 2015 4 Comments on Transfer Function 5.
    [Show full text]
  • Coping with the Bounds: a Neo-Clausewitzean Primer / Thomas J
    About the CCRP The Command and Control Research Program (CCRP) has the mission of improving DoD’s understanding of the national security implications of the Information Age. Focusing upon improving both the state of the art and the state of the practice of command and control, the CCRP helps DoD take full advantage of the opportunities afforded by emerging technologies. The CCRP pursues a broad program of research and analysis in information superiority, information operations, command and control theory, and associated operational concepts that enable us to leverage shared awareness to improve the effectiveness and efficiency of assigned missions. An important aspect of the CCRP program is its ability to serve as a bridge between the operational, technical, analytical, and educational communities. The CCRP provides leadership for the command and control research community by: • articulating critical research issues; • working to strengthen command and control research infrastructure; • sponsoring a series of workshops and symposia; • serving as a clearing house for command and control related research funding; and • disseminating outreach initiatives that include the CCRP Publication Series. This is a continuation in the series of publications produced by the Center for Advanced Concepts and Technology (ACT), which was created as a “skunk works” with funding provided by the CCRP under the auspices of the Assistant Secretary of Defense (NII). This program has demonstrated the importance of having a research program focused on the national security implications of the Information Age. It develops the theoretical foundations to provide DoD with information superiority and highlights the importance of active outreach and dissemination initiatives designed to acquaint senior military personnel and civilians with these emerging issues.
    [Show full text]
  • Designing Biological Systems
    Downloaded from genesdev.cshlp.org on October 4, 2021 - Published by Cold Spring Harbor Laboratory Press REVIEW Designing biological systems David A. Drubin,1 Jeffrey C. Way,2 and Pamela A. Silver1,3 1Department of Systems Biology, Harvard Medical School, Boston, Massachusetts 02115, USA; 2EMD Lexigen Research Center, Billerica, Massachusetts 01821, USA The design of artificial biological systems and the under- chemical systems that mimic living systems, especially standing of their natural counterparts are key objectives in regard to replication and Darwinian selection. This of the emerging discipline of synthetic biology. Toward review will primarily focus on accomplishments of the both ends, research in synthetic biology has primarily first strategy; however, readers are referred to several fine focused on the construction of simple devices, such as reviews on the generation of artificial, life-emulating transcription-based oscillators and switches. Construc- systems (Szostak et al. 2001; Benner 2003; Benner and tion of such devices should provide us with insight on Sismour 2005). the design of natural systems, indicating whether our According to the synthetic biology paradigm, a syn- understanding is complete or whether there are still gaps thetically programmed cell should be composed of many in our knowledge. Construction of simple biological sys- subsystems that operate successfully as a result of ex- tems may also lay the groundwork for the construction tensive characterization and educated design. System of more complex systems that have practical utility. To construction is the product of iterative cycles of com- realize its full potential, biological systems design bor- puter modeling, biological assembly, and testing. In this rows from the allied fields of protein design and meta- way, much from the field of systems biology can be ap- bolic engineering.
    [Show full text]
  • Warren Mcculloch and the British Cyberneticians
    Warren McCulloch and the British cyberneticians Article (Accepted Version) Husbands, Phil and Holland, Owen (2012) Warren McCulloch and the British cyberneticians. Interdisciplinary Science Reviews, 37 (3). pp. 237-253. ISSN 0308-0188 This version is available from Sussex Research Online: http://sro.sussex.ac.uk/id/eprint/43089/ This document is made available in accordance with publisher policies and may differ from the published version or from the version of record. If you wish to cite this item you are advised to consult the publisher’s version. Please see the URL above for details on accessing the published version. Copyright and reuse: Sussex Research Online is a digital repository of the research output of the University. Copyright and all moral rights to the version of the paper presented here belong to the individual author(s) and/or other copyright owners. To the extent reasonable and practicable, the material made available in SRO has been checked for eligibility before being made available. Copies of full text items generally can be reproduced, displayed or performed and given to third parties in any format or medium for personal research or study, educational, or not-for-profit purposes without prior permission or charge, provided that the authors, title and full bibliographic details are credited, a hyperlink and/or URL is given for the original metadata page and the content is not changed in any way. http://sro.sussex.ac.uk Warren McCulloch and the British Cyberneticians1 Phil Husbands and Owen Holland Dept. Informatics, University of Sussex Abstract Warren McCulloch was a significant influence on a number of British cyberneticians, as some British pioneers in this area were on him.
    [Show full text]
  • Control Theory
    Control theory S. Simrock DESY, Hamburg, Germany Abstract In engineering and mathematics, control theory deals with the behaviour of dynamical systems. The desired output of a system is called the reference. When one or more output variables of a system need to follow a certain ref- erence over time, a controller manipulates the inputs to a system to obtain the desired effect on the output of the system. Rapid advances in digital system technology have radically altered the control design options. It has become routinely practicable to design very complicated digital controllers and to carry out the extensive calculations required for their design. These advances in im- plementation and design capability can be obtained at low cost because of the widespread availability of inexpensive and powerful digital processing plat- forms and high-speed analog IO devices. 1 Introduction The emphasis of this tutorial on control theory is on the design of digital controls to achieve good dy- namic response and small errors while using signals that are sampled in time and quantized in amplitude. Both transform (classical control) and state-space (modern control) methods are described and applied to illustrative examples. The transform methods emphasized are the root-locus method of Evans and fre- quency response. The state-space methods developed are the technique of pole assignment augmented by an estimator (observer) and optimal quadratic-loss control. The optimal control problems use the steady-state constant gain solution. Other topics covered are system identification and non-linear control. System identification is a general term to describe mathematical tools and algorithms that build dynamical models from measured data.
    [Show full text]
  • An Introduction to Control Systems; K. Warwick
    An Introduction to Control Systems; K. Warwick 362 pages; World Scientific, 1996; K. Warwick; 9810225970, 9789810225971; 1996; An Introduction to Control Systems; This significantly revised edition presents a broad introduction to Control Systems and balances new, modern methods with the more classical. It is an excellent text for use as a first course in Control Systems by undergraduate students in all branches of engineering and applied mathematics. The book contains: A comprehensive coverage of automatic control, integrating digital and computer control techniques and their implementations, the practical issues and problems in Control System design; the three-term PID controller, the most widely used controller in industry today; numerous in-chapter worked examples and end-of-chapter exercises. This second edition also includes an introductory guide to some more recent developments, namely fuzzy logic control and neural networks. file download wici.pdf The Breakthrough in Artificial Intelligence; While horror films and science fiction have repeatedly warned of robots running amok, Kevin Warwick takes the threats out of the realm of fiction and into the real world, truly; Computers; K. Warwick; 307 pages; ISBN:0252072235; 1997; March of the Machines Control Bruce O. Watkins; Introduction to control systems; UOM:39015002007683; Technology & Engineering; 625 pages; 1969 Robot Control; ISBN:0863411282; Jan 1, 1988; K. Warwick, Alan Pugh; Technology & Engineering; 238 pages; Theory and Applications Automatic control; ISBN:0750622989; Davinder K. Anand; 730 pages; Since the second edition of this classic text for students and engineers appeared in 1984, the use of computer-aided design software has become an important adjunct to the; Introduction to Control Systems; Jan 1, 1995 An Introduction to Control Systems pdf download 596 pages; Mar 18, 1993; STANFORD:36105004050907; based on the proceedings of a conference on Robotics, applied mathematics and computational aspects; K.
    [Show full text]
  • Observation of Nonlinear Verticum-Type Systems Applied to Ecological Monitoring
    2nd Reading June 4, 2012 13:53 WSPC S1793-5245 242-IJB 1250051 International Journal of Biomathematics Vol. 5, No. 6 (November 2012) 1250051 (15 pages) c World Scientific Publishing Company DOI: 10.1142/S1793524512500519 OBSERVATION OF NONLINEAR VERTICUM-TYPE SYSTEMS APPLIED TO ECOLOGICAL MONITORING S. MOLNAR´ Institute of Mathematics and Informatics Szent Istv´an University P´ater K. u. 1., H-2103 Godollo, Hungary [email protected] M. GAMEZ´ ∗ and I. LOPEZ´ † Department of Statistics and Applied Mathematics University of Almer´ıa La Ca˜nada de San Urbano 04120 Almer´ıa, Spain ∗[email protected] †[email protected] Received 4 November 2011 Accepted 13 January 2012 Published 7 June 2012 In this paper the concept of a nonlinear verticum-type observation system is introduced. These systems are composed from several “subsystems” connected sequentially in a particular way: a part of the state variables of each “subsystem” also appears in the next “subsystem” as an “exogenous variable” which can also be interpreted as a con- trol generated by an “exosystem”. Therefore, these “subsystems” are not observation systems, but formally can be considered as control-observation systems. The problem of observability of such systems can be reduced to rank conditions on the “subsystems”. Indeed, under the condition of Lyapunov stability of an equilibrium of the “large”, verticum-type system, it is shown that the Kalman rank condition on the linearization of the “subsystems” implies the observability of the original, nonlinear verticum-type system. For an illustration of the above linearization result, a stage-structured fishery model with reserve area is considered.
    [Show full text]
  • The Effects of Feedback Interventions on Performance: a Historical Review, a Meta-Analysis, and a Preliminary Feedback Intervention Theory
    Psychological Bulletin Copyright 1996 by the American Psychological Association, Inc. 1996, Vol. II9, No. 2, 254-284 0033-2909/96/S3.00 The Effects of Feedback Interventions on Performance: A Historical Review, a Meta-Analysis, and a Preliminary Feedback Intervention Theory Avraham N. Kluger Angelo DeNisi The Hebrew University of Jerusalem Rutgers University Since the beginning of the century, feedback interventions (FIs) produced negative—but largely ignored—effects on performance. A meta-analysis (607 effect sizes; 23,663 observations) suggests that FIs improved performance on average (d = .41) but that over '/3 of the FIs decreased perfor- mance. This finding cannot be explained by sampling error, feedback sign, or existing theories. The authors proposed a preliminary FI theory (FIT) and tested it with moderator analyses. The central assumption of FIT is that FIs change the locus of attention among 3 general and hierarchically organized levels of control: task learning, task motivation, and meta-tasks (including self-related) processes. The results suggest that FI effectiveness decreases as attention moves up the hierarchy closer to the self and away from the task. These findings are further moderated by task characteristics that are still poorly understood. To relate feedback directly to behavior is very confusing. Results able has been historically disregarded by most FI researchers. This are contradictory and seldom straight-forward. (Ilgen, Fisher, & disregard has led to a widely shared assumption that FIs consis- Taylor, 1979, p. 368) tently improve performance. Fortunately, several FI researchers The effects of manipulation of KR [knowledge of results] on motor (see epigraphs) have recently recognized that FIs have highly vari- learning.
    [Show full text]
  • Behavioral Responses of Zooplankton to Predation
    BULLETIN OF MARINE SCIENCE, 43(3): 530-550, 1988 BEHAVIORAL RESPONSES OF ZOOPLANKTON TO PREDATION M. D. Ohman ABSTRACT Many behavioral traits of zooplankton reduce the probability of successful consumption by predators, Prey behavioral responses act at different points of a predation sequence, altering the probability of a predator's success at encounter, attack, capture or ingestion. Avoidance behavior (through spatial refuges, diel activity cycles, seasonal diapause, locomotory behavior) minimizes encounter rates with predators. Escape responses (through active motility, passive evasion, aggregation, bioluminescence) diminish rates of attack or successful capture. Defense responses (through chemical means, induced morphology) decrease the probability of suc- cessful ingestion by predators. Behavioral responses of individuals also alter the dynamics of populations. Future efforts to predict the growth of prey and predator populations will require greater attention to avoidance, escape and defense behavior. Prey activities such as occupation of spatial refuges, aggregation responses, or avoidance responses that vary ac- cording to the behavioral state of predators can alter the outcome of population interactions, introducing stability into prey-predator oscillations. In variable environments, variance in behavioral traits can "spread the risk" (den Boer, 1968) of local extinction. At present the extent of variability of prey and predator behavior, as well as the relative contributions of genotypic variance and of phenotypic plasticity,
    [Show full text]
  • Digital Signal Processing Module 3 Z-Transforms Objective
    Digital Signal Processing Module 3 Z-Transforms Objective: 1. To have a review of z-transforms. 2. Solving LCCDE using Z-transforms. Introduction: The z-transform is a useful tool in the analysis of discrete-time signals and systems and is the discrete-time counterpart of the Laplace transform for continuous-time signals and systems. The z-transform may be used to solve constant coefficient difference equations, evaluate the response of a linear time-invariant system to a given input, and design linear filters. Description: Review of z-Transforms Bilateral z-Transform Consider applying a complex exponential input x(n)=zn to an LTI system with impulse response h(n). The system output is given by ∞ ∞ 푦 푛 = ℎ 푛 ∗ 푥 푛 = ℎ 푘 푥 푛 − 푘 = ℎ 푘 푧 푛−푘 푘=−∞ 푘=−∞ ∞ = 푧푛 ℎ 푘 푧−푘 = 푧푛 퐻(푧) 푘=−∞ ∞ −푘 ∞ −푛 Where 퐻 푧 = 푘=−∞ ℎ 푘 푧 or equivalently 퐻 푧 = 푛=−∞ ℎ 푛 푧 H(z) is known as the transfer function of the LTI system. We know that a signal for which the system output is a constant times, the input is referred to as an eigen function of the system and the amplitude factor is referred to as the system’s eigen value. Hence, we identify zn as an eigen function of the LTI system and H(z) is referred to as the Bilateral z-transform or simply z-transform of the impulse response h(n). 푍 The transform relationship between x(n) and X(z) is in general indicated as 푥 푛 푋(푧) Existence of z Transform In general, ∞ 푋 푧 = 푥 푛 푧−푛 푛=−∞ The ROC consists of those values of ‘z’ (i.e., those points in the z-plane) for which X(z) converges i.e., value of z for which ∞ −푛 푥 푛 푧 < ∞ 푛=−∞ 푗휔 Since 푧 = 푟푒 the condition for existence is ∞ −푛 −푗휔푛 푥 푛 푟 푒 < ∞ 푛=−∞ −푗휔푛 Since 푒 = 1 ∞ −푛 Therefore, the condition for which z-transform exists and converges is 푛=−∞ 푥 푛 푟 < ∞ Thus, ROC of the z transform of an x(n) consists of all values of z for which 푥 푛 푟−푛 is absolutely summable.
    [Show full text]