49 research outputs found
Reconstruction of Cracks with Unknown Transmission Condition from Boundary Data
We examine the problem of Identifying both the location and constitutive law governing electrical current flow across a one-dimensional linear crack in a two dimensional region when the crack only partially blocks the flow of current. We develop a a constructive numerical procedure for solving the inverse problem and provide computational examples
One-Dimensional Birth-Death Process and Delbr\"{u}ck-Gillespie Theory of Mesoscopic Nonlinear Chemical Reactions
As a mathematical theory for the stochasstic, nonlinear dynamics of
individuals within a population, Delbr\"{u}ck-Gillespie process (DGP)
, is a birth-death system with state-dependent rates which
contain the system size as a natural parameter. For large , it is
intimately related to an autonomous, nonlinear ordinary differential equation
as well as a diffusion process. For nonlinear dynamical systems with multiple
attractors, the quasi-stationary and stationary behavior of such a birth-death
process can be underestood in terms of a separation of time scales by a
: a relatively fast, intra-basin diffusion
for and a much slower inter-basin Markov jump process for . In the present paper for one-dimensional systems, we study both
stationary behavior () in terms of invariant distribution
, and finite time dynamics in terms of the mean first passsage
time (MFPT) . We obtain an asymptotic expression of
MFPT in terms of the "stochastic potential" . We show in general no continuous diffusion process can provide
asymptotically accurate representations for both the MFPT and the
for a DGP. When and belong to two different basins of attraction,
the MFPT yields the in terms of . For systems with a saddle-node bifurcation and
catastrophe, discontinuous "phase transition" emerges, which can be
characterized by in the limit of . In terms of
time scale separation, the relation between deterministic, local nonlinear
bifurcations and stochastic global phase transition is discussed. The
one-dimensional theory is a pedagogic first step toward a general theory of
DGP.Comment: 32 pages, 3 figure
Circular Stochastic Fluctuations in SIS Epidemics with Heterogeneous Contacts Among Sub-populations
The conceptual difference between equilibrium and non-equilibrium steady
state (NESS) is well established in physics and chemistry. This distinction,
however, is not widely appreciated in dynamical descriptions of biological
populations in terms of differential equations in which fixed point, steady
state, and equilibrium are all synonymous. We study NESS in a stochastic SIS
(susceptible-infectious-susceptible) system with heterogeneous individuals in
their contact behavior represented in terms of subgroups. In the infinite
population limit, the stochastic dynamics yields a system of deterministic
evolution equations for population densities; and for very large but finite
system a diffusion process is obtained. We report the emergence of a circular
dynamics in the diffusion process, with an intrinsic frequency, near the
endemic steady state. The endemic steady state is represented by a stable node
in the deterministic dynamics; As a NESS phenomenon, the circular motion is
caused by the intrinsic heterogeneity within the subgroups, leading to a broken
symmetry and time irreversibility.Comment: 29 pages, 5 figure
Fixation, transient landscape and diffusion's dilemma in stochastic evolutionary game dynamics
Agent-based stochastic models for finite populations have recently received
much attention in the game theory of evolutionary dynamics. Both the ultimate
fixation and the pre-fixation transient behavior are important to a full
understanding of the dynamics. In this paper, we study the transient dynamics
of the well-mixed Moran process through constructing a landscape function. It
is shown that the landscape playing a central theoretical "device" that
integrates several lines of inquiries: the stable behavior of the replicator
dynamics, the long-time fixation, and continuous diffusion approximation
associated with asymptotically large population. Several issues relating to the
transient dynamics are discussed: (i) multiple time scales phenomenon
associated with intra- and inter-attractoral dynamics; (ii) discontinuous
transition in stochastically stationary process akin to Maxwell construction in
equilibrium statistical physics; and (iii) the dilemma diffusion approximation
facing as a continuous approximation of the discrete evolutionary dynamics. It
is found that rare events with exponentially small probabilities, corresponding
to the uphill movements and barrier crossing in the landscape with multiple
wells that are made possible by strong nonlinear dynamics, plays an important
role in understanding the origin of the complexity in evolutionary, nonlinear
biological systems.Comment: 34 pages, 4 figure
The Chemical Master Equation Approach to Nonequilibrium Steady-State of Open Biochemical Systems: Linear Single-Molecule Enzyme Kinetics and Nonlinear Biochemical Reaction Networks
We develop the stochastic, chemical master equation as a unifying approach to the dynamics of biochemical reaction systems in a mesoscopic volume under a living environment. A living environment provides a continuous chemical energy input that sustains the reaction system in a nonequilibrium steady state with concentration fluctuations. We discuss the linear, unimolecular single-molecule enzyme kinetics, phosphorylation-dephosphorylation cycle (PdPC) with bistability, and network exhibiting oscillations. Emphasis is paid to the comparison between the stochastic dynamics and the prediction based on the traditional approach based on the Law of Mass Action. We introduce the difference between nonlinear bistability and stochastic bistability, the latter has no deterministic counterpart. For systems with nonlinear bistability, there are three different time scales: (a) individual biochemical reactions, (b) nonlinear network dynamics approaching to attractors, and (c) cellular evolution. For mesoscopic systems with size of a living cell, dynamics in (a) and (c) are stochastic while that with (b) is dominantly deterministic. Both (b) and (c) are emergent properties of a dynamic biochemical network; We suggest that the (c) is most relevant to major cellular biochemical processes such as epi-genetic regulation, apoptosis, and cancer immunoediting. The cellular evolution proceeds with transitions among the attractors of (b) in a “punctuated equilibrium” manner
Thermodynamic Limit of a Nonequilibrium Steady-State: Maxwell-Type Construction for a Bistable Biochemical System
We show that the thermodynamic limit of a bistable
phosphorylation-dephosphorylation cycle has a selection rule for the "more
stable" macroscopic steady state. The analysis is akin to the Maxwell
construction. Based on the chemical master equation approach, it is shown that,
except at a critical point, bistability disappears in the stochastic model when
fluctuation is sufficiently low but unneglectable. Onsager's Gaussian
fluctuation theory applies to the unique macroscopic steady state. With initial
state in the basin of attraction of the "less stable" steady state, the
deterministic dynamics obtained by the Law of Mass Action is a metastable
phenomenon. Stability and robustness in cell biology are stochastic concepts.Comment: 12 pages, 2 figure
Markovian Dynamics on Complex Reaction Networks
Complex networks, comprised of individual elements that interact with each
other through reaction channels, are ubiquitous across many scientific and
engineering disciplines. Examples include biochemical, pharmacokinetic,
epidemiological, ecological, social, neural, and multi-agent networks. A common
approach to modeling such networks is by a master equation that governs the
dynamic evolution of the joint probability mass function of the underling
population process and naturally leads to Markovian dynamics for such process.
Due however to the nonlinear nature of most reactions, the computation and
analysis of the resulting stochastic population dynamics is a difficult task.
This review article provides a coherent and comprehensive coverage of recently
developed approaches and methods to tackle this problem. After reviewing a
general framework for modeling Markovian reaction networks and giving specific
examples, the authors present numerical and computational techniques capable of
evaluating or approximating the solution of the master equation, discuss a
recently developed approach for studying the stationary behavior of Markovian
reaction networks using a potential energy landscape perspective, and provide
an introduction to the emerging theory of thermodynamic analysis of such
networks. Three representative problems of opinion formation, transcription
regulation, and neural network dynamics are used as illustrative examples.Comment: 52 pages, 11 figures, for freely available MATLAB software, see
http://www.cis.jhu.edu/~goutsias/CSS%20lab/software.htm