scholarly journals Stochastic models for interference between searching insect parasites

1989 ◽  
Vol 30 (3) ◽  
pp. 268-277 ◽  
Author(s):  
Phil Diamond
Keyword(s):  
Stationary Distribution ◽  
Intraspecific Competition ◽  
Death Process ◽  
Confluent Hypergeometric Functions ◽  
Large Populations ◽  
The Mean ◽  
The One ◽  
Globally Stable ◽  
Insect Parasites ◽  
Birth Death

AbstractCompetition between a finite number of searching insect parasites is modelled by differential equations and birth-death processes. In the one species case of intraspecific competition, the deterministic equilibrium is globally stable and, for large populations, approximates the mean of the stationary distribution of the process. For two species, both inter- and intraspecific competition occurs and the deterministic equilibrium is globally stable. When the birth-death process is reversible, it is shown that the mean of the stationary distribution is approximated by the equilibrium. Confluent hypergeometric functions of two variables are important to the theory.

scholarly journals Large deviations and quasi-stationarity for density-dependent birth-death processes

1998 ◽  
Vol 40 (2) ◽  
pp. 238-256 ◽  
Author(s):  
Terence Chan
Keyword(s):  
Large Deviations ◽  
Stationary Distribution ◽  
Death Process ◽  
Large Deviations Principle ◽  
Density Dependent ◽  
Exponential Decay Rate ◽  
Finite State ◽  
Degenerate Distribution ◽  
The One ◽  
Birth Death

AbstractConsider a density-dependent birth-death process XN on a finite state space of size N. Let PN be the law (on D([0, T]) where T > 0 is arbitrary) of the density process XN/N and let πN be the unique stationary distribution (on[0,1]) of XN/N, if it exists. Typically, these distributions converge weakly to a degenerate distribution as N → ∞ so the probability of sets not containing the degenerate point will tend to 0; large deviations is concerned with obtaining the exponential decay rate of these probabilities. Friedlin-Wentzel theory is used to establish the large deviations behaviour (as N → ∞) of PN. In the one-dimensional case, a large deviations principle for the stationary distribution πN is obtained by elementary explicit computations. However, when the birth-death process has an absorbing state at 0 (so πN no longer exists), the same elementary computations are still applicable to the quasi-stationary distribution, and we show that the quasi-stationary distributions obey the same large deviations principle as in the recurrent case. In addition, we address some questions related to the estimated time to absorption and obtain a large deviations principle for the invariant distribution in higher dimensions by studying a quasi-potential.

Normal approximations to discrete unimodal distributions

1986 ◽  
Vol 23 (04) ◽  
pp. 1013-1018
Author(s):  
B. G. Quinn ◽  
H. L. MacGillivray
Keyword(s):  
Stationary Distribution ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Hypergeometric Distribution ◽  
Death Process ◽  
Normal Approximations ◽  
Discrete Random Variables ◽  
Birth Death

Sufficient conditions are presented for the limiting normality of sequences of discrete random variables possessing unimodal distributions. The conditions are applied to obtain normal approximations directly for the hypergeometric distribution and the stationary distribution of a special birth-death process.

Normal approximations to discrete unimodal distributions

1986 ◽  
Vol 23 (04) ◽  
pp. 1013-1018 ◽  
Author(s):  
B. G. Quinn ◽  
H. L. MacGillivray
Keyword(s):  
Stationary Distribution ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Hypergeometric Distribution ◽  
Death Process ◽  
Normal Approximations ◽  
Discrete Random Variables ◽  
Birth Death

Sufficient conditions are presented for the limiting normality of sequences of discrete random variables possessing unimodal distributions. The conditions are applied to obtain normal approximations directly for the hypergeometric distribution and the stationary distribution of a special birth-death process.

On the parity of individuals in a branching process

1976 ◽  
Vol 13 (02) ◽  
pp. 219-230 ◽  
Author(s):  
J. Gani ◽  
I. W. Saunders
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Branching Process ◽  
Yeast Cells ◽  
Death Process ◽  
Second Moments ◽  
The Mean ◽  
Watson Process ◽  
Number Of Cells ◽  
Birth Death

This paper is concerned with the parity of a population of yeast cells, each of which may bud, not bud or die. Two multitype models are considered: a Galton-Watson process in discrete time, and its analogous birth-death process in continuous time. The mean number of cells with parity 0, 1, 2, … is obtained in both cases; some simple results are also derived for the second moments of the two processes.

Evaluation of the decay parameter for some specialized birth-death processes

1992 ◽  
Vol 29 (4) ◽  
pp. 781-791 ◽  
Author(s):  
Masaaki Kijima
Keyword(s):  
State Space ◽  
Stationary Distribution ◽  
The State ◽  
Death Process ◽  
Decay Parameter ◽  
Speed Of Convergence ◽  
Tridiagonal Matrices ◽  
Decay Parameters ◽  
Quasi Stationary Distribution ◽  
Birth Death

Let N(t) be an exponentially ergodic birth-death process on the state space {0, 1, 2, ···} governed by the parameters {λn, μn}, where µ0 = 0, such that λn = λ and μn = μ for all n ≧ N, N ≧ 1, with λ < μ. In this paper, we develop an algorithm to determine the decay parameter of such a specialized exponentially ergodic birth-death process, based on van Doorn's representation (1987) of eigenvalues of sign-symmetric tridiagonal matrices. The decay parameter is important since it is indicative of the speed of convergence to ergodicity. Some comparability results for the decay parameters are given, followed by the discussion for the decay parameter of a birth-death process governed by the parameters such that limn→∞λn = λ and limn→∞µn = μ. The algorithm is also shown to be a useful tool to determine the quasi-stationary distribution, i.e. the limiting distribution conditioned to stay in {1, 2, ···}, of such specialized birth-death processes.

A bivariate birth-death process which approximates to the spread of a disease involving a vector

1972 ◽  
Vol 9 (01) ◽  
pp. 65-75 ◽  
Author(s):  
D. A. Griffiths
Keyword(s):  
Simple Model ◽  
Probability Distribution ◽  
General Class ◽  
Large Population ◽  
Vector Model ◽  
Death Process ◽  
The Mean ◽  
Probability Of Extinction ◽  
Over Time ◽  
Birth Death

A simple model for a bivariate birth-death process is proposed. This model approximates to the host-vector epidemic situation. An investigation of the transient process is made and the mean behaviour over time is explicitly found. The probability of extinction and the behaviour of the process conditional upon extinction are examined and the probability distribution of the cumulative population size to extinction is found. Appropriate circumstances are suggested under which the model might possibly be applied to malaria. The host-vector model is classified within a general class of models which represent large population approximations to epidemics involving two types of infectives.

A bivariate birth-death process which approximates to the spread of a disease involving a vector

1972 ◽  
Vol 9 (1) ◽  
pp. 65-75 ◽  
Author(s):  
D. A. Griffiths
Keyword(s):  
Simple Model ◽  
Probability Distribution ◽  
General Class ◽  
Large Population ◽  
Vector Model ◽  
Death Process ◽  
The Mean ◽  
Probability Of Extinction ◽  
Over Time ◽  
Birth Death

A simple model for a bivariate birth-death process is proposed. This model approximates to the host-vector epidemic situation. An investigation of the transient process is made and the mean behaviour over time is explicitly found. The probability of extinction and the behaviour of the process conditional upon extinction are examined and the probability distribution of the cumulative population size to extinction is found. Appropriate circumstances are suggested under which the model might possibly be applied to malaria. The host-vector model is classified within a general class of models which represent large population approximations to epidemics involving two types of infectives.

Evaluation of the decay parameter for some specialized birth-death processes

1992 ◽  
Vol 29 (04) ◽  
pp. 781-791 ◽  
Author(s):  
Masaaki Kijima
Keyword(s):  
State Space ◽  
Stationary Distribution ◽  
The State ◽  
Death Process ◽  
Decay Parameter ◽  
Speed Of Convergence ◽  
Tridiagonal Matrices ◽  
Decay Parameters ◽  
Quasi Stationary Distribution ◽  
Birth Death

Let N(t) be an exponentially ergodic birth-death process on the state space {0, 1, 2, ···} governed by the parameters {λn, μn }, where µ 0 = 0, such that λn = λ and μn = μ for all n ≧ N, N ≧ 1, with λ &lt; μ. In this paper, we develop an algorithm to determine the decay parameter of such a specialized exponentially ergodic birth-death process, based on van Doorn's representation (1987) of eigenvalues of sign-symmetric tridiagonal matrices. The decay parameter is important since it is indicative of the speed of convergence to ergodicity. Some comparability results for the decay parameters are given, followed by the discussion for the decay parameter of a birth-death process governed by the parameters such that lim n→∞ λn = λ and lim n→∞ µn = μ. The algorithm is also shown to be a useful tool to determine the quasi-stationary distribution, i.e. the limiting distribution conditioned to stay in {1, 2, ···}, of such specialized birth-death processes.

scholarly journals On regularly varying distributions generated by birth-death process

2006 ◽  
Vol 19 (1) ◽  
pp. 109-131 ◽  
Author(s):  
Jaakko Astola ◽  
Eduard Danielian
Keyword(s):  
Stationary Distribution ◽  
Power Law ◽  
Regular Variation ◽  
Death Process ◽  
Skewed Distributions ◽  
Biomolecular Systems ◽  
Moderate Growth ◽  
Regularly Varying ◽  
Birth Death

Skewed distributions generated by birth-death process with different particular forms of intensivities? moderate growth are used in biomolecular systems and various non-mathematical fields. Based on datasets of biomolecular systems such distributions have to exhibit the power law like behavior at infinity, i.e. regular variation. In the present paper for the standard birth-death process with most general than before assumptions on moderate growth of intensivities the following problems are solved. 1. The stationary distribution varies regularly if the sequence of intensivities varies regularly. 2. The slowly varying component and the exponent of regular variation of stationary distribution are found.

scholarly journals Population size and the length of the chromosome blocks identical by descent over generations

2015 ◽  
Author(s):  
Mathieu Tiret ◽  
Frederic Hospital
Keyword(s):  
Theoretical Prediction ◽  
Population Size ◽  
Genetic Drift ◽  
Population Model ◽  
The Other ◽  
Identical By Descent ◽  
Large Populations ◽  
The Mean ◽  
The One ◽  
Over Time

In all populations, as the time runs, crossovers break apart ancestor haplotypes, forming smaller blocks at each generation. Some blocks, and eventually all of them, become identical by descent because of the genetic drift. We have in this paper developed and benchmarked a theoretical prediction of the mean length of such blocks and used it to study a simple population model assuming panmixia, no selfing and drift as the only evolutionary pressure. Besides, we have on the one hand derived, for any user defined error threshold, the range of the parameters this prediction is reliable for, and on the other hand shown that the mean length remains constant over time in ideally large populations.

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