Mutation-Selection Balance: Ancestry, Load, and Maximum Principle

We show how concepts from statistical physics, such as order parameter, thermodynamic limit, and quantum phase transition, translate into biological concepts in mutation-selection models for sequence evolution and can be used there. The article takes a biological point of view within a population genetics framework, but contains an appendix for physicists, which makes this correspondence clear. We analyze the equilibrium behavior of deterministic haploid mutation-selection models. Both the forward and the time-reversed evolution processes are considered. The stationary state of the latter is called the ancestral distribution, which turns out as a key for the study of mutation-selection balance. We find that it determines the sensitivity of the equilibrium mean fitness to changes in the fitness values and discuss implications for the evolution of mutational robustness. We further show that the difference between the ancestral and the population mean fitness, termed mutational loss, provides a measure for the sensitivity of the equilibrium mean fitness to changes in the mutation rate. For a class of models in which the number of mutations in an individual is taken as the trait value, and fitness is a function of the trait, we use the ancestor formulation to derive a simple maximum principle, from which the mean and variance of fitness and the trait may be derived; the results are exact for a number of limiting cases, and otherwise yield approximations which are accurate for a wide range of parameters. These results are applied to (error) threshold phenomena caused by the interplay of selection and mutation. They lead to a clarification of concepts, as well as criteria for the existence of thresholds.Comment: 54 pages, 15 figures; to appear in Theor. Pop. Biol. 61 or 62 (2002

Ordinary Differential Equations -- Lecture Notes 2014-2015

In these notes we study the basic theory of ordinary differential equations, with emphasis on initial value problems, together with some modelling aspects. The following topics are treated: 1. Models and Explicit Solutions, 2. Existence and Uniqueness, 3. Linear Systems, 4. Stability and Linearization, 5. Some Models in two and three dimensions, 6. Quantitative Stability Estimates, 7. Boundary Value Problem

Fractional Calculus and the Future of Science

Newton foresaw the limitations of geometry’s description of planetary behavior and developed fluxions (differentials) as the new language for celestial mechanics and as the way to implement his laws of mechanics. Two hundred years later Mandelbrot introduced the notion of fractals into the scientific lexicon of geometry, dynamics, and statistics and in so doing suggested ways to see beyond the limitations of Newton’s laws. Mandelbrot’s mathematical essays suggest how fractals may lead to the understanding of turbulence, viscoelasticity, and ultimately to end of dominance of the Newton’s macroscopic world view.Fractional Calculus and the Future of Science examines the nexus of these two game-changing contributions to our scientific understanding of the world. It addresses how non-integer differential equations replace Newton’s laws to describe the many guises of complexity, most of which lay beyond Newton’s experience, and many had even eluded Mandelbrot’s powerful intuition. The book’s authors look behind the mathematics and examine what must be true about a phenomenon’s behavior to justify the replacement of an integer-order with a noninteger-order (fractional) derivative. This window into the future of specific science disciplines using the fractional calculus lens suggests how what is seen entails a difference in scientific thinking and understanding

Dynamical Systems; Proceedings of an IIASA Workshop, Sopron, Hungary, September 9-13, 1985

The investigation of special topics in systems dynamics -- uncertain dynamic processes, viability theory, nonlinear dynamics in models for biomathematics, inverse problems in control systems theory -- has become a major issue at the System and Decision Sciences Research Program of IIASA. The above topics actually reflect two different perspectives in the investigation of dynamic processes. The first, motivated by control theory, is concerned with the properties of dynamic systems that are stable under variations in the systems' parameters. This allows us to specify classes of dynamic systems for which it is possible to construct and control a whole "tube" of trajectories assigned to a system with uncertain parameters and to resolve some inverse problems of control theory within numerically stable solution schemes. The second perspective is to investigate generic properties of dynamic systems that are due to nonlinearity (as bifurcations theory, chaotic behavior, stability properties, and related problems in the qualitative theory of differential systems). Special stress is given to the applications of nonlinear dynamic systems theory to biomathematics and ecology. The proceedings of a workshop on the "Mathematics of Dynamic Processes", dealing with these topics is presented in this volume

Spatio-temporal stochastic hybrid models of biological excitable membranes

A large number of biological systems are intrinsically random, in particular, biological excitable membranes, such as neuronal membranes, cardiac tissue or models for calcium dynamics. The present thesis is concerned with hybrid stochastic models of spatio-temporal dynamics of biological excitable membranes using Piecewise Deterministic Markov Processes (PDMPs). This class of processes allows a precise mathematical description of the internal noise structure of excitable membranes. Overall the aim of the thesis is two-fold: On the one hand, we establish a general hybrid modelling framework for biological excitable membranes and, on the other hand, we are interested in a general advance of PDMP theory which the former necessitates. Regarding the first aim we exemplify the modelling framework on the classical Hodgkin-Huxley model of a squid giant axon. Regarding the latter we present a general PDMP theory incorporating spatial dynamics and present tools for their analysis. Here we focus on two aspects. Firstly, we consider the approximation of PDMPs by deterministic models or continuous stochastic processes. To this end we derive as general theoretical tools a law of large numbers for PDMPs and martingale central limit theorems. The former establishes a connection of stochastic hybrid models to deterministic models given, e.g., by systems of partial differential equations. Whereas the latter connects the stochastic fluctuations in the hybrid models to diffusion processes. Furthermore, these limit theorems provide the basis for a general Langevin approximation to PDMPs, i.e., certain stochastic partial differential equations that are expected to be similar in their dynamics to PDMPs. Secondly, we also address the question of numerical simulation of PDMPs. We present and analyse the convergence in the pathwise sense of a class of simulation methods for PDMPs in Euclidean space.Engineering and Physics Research Council (EPRC) EP/E03635X/1ANR project MANDy (ANR-09-BLAN-0008-01)BC/DAAD ARC project Nr 1349/50021880ARC project Nr. 133

A Markov Model of a Limit Order Book: Thresholds, Recurrence, and Trading Strategies

We analyze a tractable model of a limit order book on short time scales, where the dynamics are driven by stochastic fluctuations between supply and demand. We establish the existence of a limiting distribution for the highest bid, and for the lowest ask, where the limiting distributions are confined between two thresholds. We make extensive use of fluid limits to establish recurrence properties of the model. We use the model to analyze various high-frequency trading strategies, and comment on the Nash equilibria that emerge between high-frequency traders when a market in continuous time is replaced by frequent batch auctions.The second author’s research was partially supported by the National Science Foundation [Grant DMS-1204311] and NSF Graduate Research Fellowship