Birth and death processes with random environments in continuous time

A generalization to continuous time is given for a discrete-time model of a birth and death process in a random environment. Some important properties of this process in the continuous-time setting are stated and proved including instability and extinction conditions, and when suitable absorbing barriers have been defined, methods are given for the calculation of extinction probabilities and the expected duration of the process.

Download Full-text

Birth and death processes in randomly fluctuating environments

Nagoya Mathematical Journal ◽

10.1017/s0027763000008278 ◽

2002 ◽

Vol 166 ◽

pp. 93-115

Author(s):

Kanji Ichihara

Keyword(s):

Random Environment ◽

Time Dependent ◽

Death Process ◽

Birth And Death Process ◽

Birth And Death Processes ◽

Fluctuating Environments

AbstractA birth and death process in a time-dependent random environment is introduced. We will discuss the recurrence and transience properties for the process.

Download Full-text

Extreme stability and almost periodicity in continuous and discrete neuronal models with finite delays

The ANZIAM Journal ◽

10.1017/s1446181100013936 ◽

2002 ◽

Vol 44 (2) ◽

pp. 261-282 ◽

Cited By ~ 4

Author(s):

S. Mohamad ◽

K. Gopalsamy

Keyword(s):

Discrete Time ◽

Continuous Time ◽

Model Neuron ◽

Sufficient Conditions ◽

External Stimulus ◽

Input Stimulus ◽

Time System ◽

Time Model ◽

Discrete Time Model ◽

Continuous Time System

We consider the dynamical characteristics of a continuous-time isolated Hopfield-type neuron subjected to an almost periodic external stimulus. The model neuron is assumed to be dissipative having finite time delays in the process of encoding the external input stimulus and recalling the encoded pattern associated with the external stimulus. By using non-autonomous Halanay-type inequalities we obtain sufficient conditions for the hetero-associative stable encoding of temporally non-uniform stimuli. A brief study of a discrete-time model derived from the continuous-time system is given. It is shown that the discrete-time model preserves the stability conditions of the continuous-time system.

Download Full-text

On Zeros of Discrete-Time Models for Collocated Mass-Damper-Spring Systems

Journal of Dynamic Systems Measurement and Control ◽

10.1115/1.1650383 ◽

2004 ◽

Vol 126 (1) ◽

pp. 205-210 ◽

Cited By ~ 3

Author(s):

Mitsuaki Ishitobi ◽

Shan Liang

Keyword(s):

Discrete Time ◽

Continuous Time ◽

Sampling Period ◽

Sufficient Condition ◽

Time System ◽

Time Model ◽

Discrete Time Model ◽

Mass Damper ◽

Approximate Expressions ◽

Continuous Time System

When a continuous-time system is discretized using the zero-order hold, there is no simple relation which shows how the zeros of the continuous-time system are transformed by sampling. In this paper, for a discrete-time model of a collocated mass-damper-spring system, the asymptotic behavior of the zeros is analyzed with respect to the sampling period and the linear approximate expressions are given. In addition, the linear approximate expressions lead to a sufficient condition for all the zeros of the discrete-time model to lie inside the unit circle for sufficiently small sampling periods. The sufficient condition is satisfied when a damping matrix is positive definite. Moreover, an example is shown to illustrate the validity of the linear approximations. Finally, a comment for a noncollocated system is presented.

Download Full-text

Family trees of continuous-time birth-and-death processes

Journal of Applied Probability ◽

10.1239/jap/1067436095 ◽

2003 ◽

Vol 40 (4) ◽

pp. 980-994 ◽

Cited By ~ 4

Author(s):

Johannes Müller ◽

Martin Möhle

Keyword(s):

Continuous Time ◽

Waiting Times ◽

Death Process ◽

Connected Components ◽

Birth And Death Process ◽

Birth And Death Processes ◽

Stochastic Graph ◽

Family Trees

We consider a stochastic graph generated by a continuous-time birth-and-death process with exponentially distributed waiting times. The vertices are the living particles, directed edges go from mothers to daughters. The size and the structure of the connected components are investigated. Furthermore, the number of connected components is determined.

Download Full-text

MAPPING DISCRETE-TIME MODELS FOR DESCRIPTOR-SYSTEMS WITH CONSISTENT INITIAL CONDITIONS

Transactions of the Canadian Society for Mechanical Engineering ◽

10.1139/tcsme-2016-0005 ◽

2016 ◽

Vol 40 (1) ◽

pp. 59-77 ◽

Cited By ~ 4

Author(s):

Shin Kawai ◽

Noriyuki Hori

Keyword(s):

Discrete Time ◽

Continuous Time ◽

Initial Conditions ◽

Descriptor Systems ◽

Descriptor System ◽

Initial Time ◽

Design Parameters ◽

Time Model ◽

Discrete Time Model ◽

Sinusoidal Input

Discretization of a regular continuous-time descriptor-system, whose initial condition is consistent with its input, is considered using a general mapping method presented in our previous paper. The proposed mapping discrete-time model is shown to be a proper discretization under the definition explained in the paper. This assures that the response of the mapping model approaches that of the continuous-time descriptor system as the sampling period approaches zero. The consistency of initial conditions for the discrete-time model is also studied and the long-standing issue of ambiguities surrounding irregularities of discrete-time responses at the initial time are clarified with a simple solution. A proper range of design parameters are investigated and their suitable choices suggested. To illustrate the use of the proposed method, a simple circuit that cannot be expressed in the ordinary state-space form is considered. Its responses to a sinusoidal input when started from the consistent and inconsistent initial conditions are simulated to show that the irregularities at the initial time can be overcome easily. The proposed technique provides a convenient simulation and design environment for handling discrete-time systems in a unified manner with consistency and ease.

Download Full-text

DISCRETE AND CONTINUOUS TIME POPULATION MODELS, A COMPARISON CONCERNING PROLIFERATION BY FISSION

Journal of Biological System ◽

10.1142/s0218339095000502 ◽

1995 ◽

Vol 03 (02) ◽

pp. 543-558 ◽

Cited By ~ 1

Author(s):

B.W. KOOI ◽

M.P. BOER

Keyword(s):

Discrete Time ◽

Continuous Time ◽

Matrix Theory ◽

Dimensional Torus ◽

Continuous Time Model ◽

Time Model ◽

Discrete Time Model ◽

Fixed Part ◽

Number Of Individuals ◽

Insight Into

We present two approaches, discrete time and continuous time models, for individuals which propagate through binary fission. The volumes of the two daughters are a fixed part of that of the mother, not necessarily the half, and their growth rates may differ. The discrete time approach gives more insight into the results obtained with the continuous time model. We define classes in the continuous time model such that the total number of individuals in these classes at specific moments in time is equal to the unknown number in a discrete time model. Then the discrete time model is homologous to the continuous one in the sense of having the same solutions at specific moments. Population matrix theory applies when the ratio of the inter-division times of the two daughters is rational. There is inter-class convergence but no intra-class convergence. The latter feature implies that there is no convergence of the size distribution in the continuous time model either. When the ratio is irrational the continuous time model holds and there is convergence but the rate of convergence can become infinitesimally small. This phenomenon is linked with quasi-periodicity on a 2-dimensional torus.

Download Full-text