Asymptotic behaviour of Markov population processes by asymptotically linear rate of change

1994 ◽  
Vol 31 (3) ◽  
pp. 614-625 ◽  
Author(s):  
F. C. Klebaner
Keyword(s):  
Markov Processes ◽  
Growth Rates ◽  
Continuous Time ◽  
Polynomial Growth ◽  
Rate Of Change ◽  
Asymptotically Linear ◽  
Linear Rate ◽  
Type Distribution ◽  
Linear Differential ◽  
Markov Population Processes

Multidimensional Markov processes in continuous time with asymptotically linear mean change per unit of time are studied as randomly perturbed linear differential equations. Conditions for exponential and polynomial growth rates with stable type distribution are given. From these conditions results on branching models of populations with stabilizing reproduction for near-supercritical and near-critical cases follow.

Asymptotic behaviour of Markov population processes by asymptotically linear rate of change

1994 ◽  
Vol 31 (03) ◽  
pp. 614-625
Author(s):  
F. C. Klebaner
Keyword(s):  
Markov Processes ◽  
Growth Rates ◽  
Continuous Time ◽  
Polynomial Growth ◽  
Rate Of Change ◽  
Asymptotically Linear ◽  
Linear Rate ◽  
Type Distribution ◽  
Linear Differential ◽  
Markov Population Processes

Multidimensional Markov processes in continuous time with asymptotically linear mean change per unit of time are studied as randomly perturbed linear differential equations. Conditions for exponential and polynomial growth rates with stable type distribution are given. From these conditions results on branching models of populations with stabilizing reproduction for near-supercritical and near-critical cases follow.

Equivalence of two absorption problems with Markovian transitions and continuous or discrete time parameters

1959 ◽  
Vol 55 (2) ◽  
pp. 177-180 ◽  
Author(s):  
R. A. Sack
Keyword(s):  
Finite Number ◽  
Markov Processes ◽  
Discrete Time ◽  
Continuous Time ◽  
Time Parameter ◽  
Rate Of Change ◽  
Matrix Form ◽  
Constant Matrix ◽  
Time Parameters ◽  
Image Position

1. Introduction. Ledermann(1) has treated the problem of calculating the asymptotic probabilities that a system will be found in any one of a finite number N of possible states if transitions between these states occur as Markov processes with a continuous time parameter t. If we denote by pi(t) the probability that at time t the system is in the ith state and by aij ( ≥ 0) the constant probability per unit time for transitions from the jth to the ith state, the rate of change of pi is given bywhere the sum is to be taken over all j ≠ i. This set of equations can be written in matrix form aswhere P(t) is the vector with components pi(t) and the constant matrix A has elements

scholarly journals Local Limit Approximations for Markov Population Processes

2009 ◽  
Vol 46 (3) ◽  
pp. 690-708
Author(s):  
Sanda N. Socoll ◽  
A.D. Barbour
Keyword(s):  
Poisson Distribution ◽  
Markov Processes ◽  
Continuous Time ◽  
Equilibrium Distribution ◽  
Local Limit ◽  
Moment Condition ◽  
Factor Log ◽  
The Difference ◽  
Time Density ◽  
Markov Population Processes

In this paper we are concerned with the equilibrium distribution ∏n of the nth element in a sequence of continuous-time density-dependent Markov processes on the integers. Under a (2+α)th moment condition on the jump distributions, we establish a bound of order O(n-(α+1)/2√logn) on the difference between the point probabilities of ∏n and those of a translated Poisson distribution with the same variance. Except for the factor √logn, the result is as good as could be obtained in the simpler setting of sums of independent, integer-valued random variables. Our arguments are based on the Stein-Chen method and coupling.

A description of the long-term behaviour of absorbing continuous-time Markov chains using a centre manifold

1990 ◽  
Vol 22 (1) ◽  
pp. 111-128 ◽  
Author(s):  
P. K. Pollett ◽  
A. J. Roberts
Keyword(s):  
Differential Equations ◽  
Markov Chains ◽  
Invariant Manifold ◽  
Markov Processes ◽  
Continuous Time ◽  
Transition Probabilities ◽  
Centre Manifold ◽  
Continuous Time Markov Chains ◽  
Linear Differential

We use the notion of an invariant manifold to describe the long-term behaviour of absorbing continuous-time Markov processes with a denumerable infinity of states. We show that there exists an invariant manifold for the forward differential equations and we are able to describe the evolution of the state probabilities on this manifold. Our approach gives rise to a new method for calculating conditional limiting distributions, one which is also appropriate for dealing with processes whose transition probabilities satisfy a system of non-linear differential equations.

A description of the long-term behaviour of absorbing continuous-time Markov chains using a centre manifold

1990 ◽  
Vol 22 (01) ◽  
pp. 111-128 ◽  
Author(s):  
P. K. Pollett ◽  
A. J. Roberts
Keyword(s):  
Differential Equations ◽  
Markov Chains ◽  
Invariant Manifold ◽  
Markov Processes ◽  
Continuous Time ◽  
Transition Probabilities ◽  
Centre Manifold ◽  
Continuous Time Markov Chains ◽  
Linear Differential

We use the notion of an invariant manifold to describe the long-term behaviour of absorbing continuous-time Markov processes with a denumerable infinity of states. We show that there exists an invariant manifold for the forward differential equations and we are able to describe the evolution of the state probabilities on this manifold. Our approach gives rise to a new method for calculating conditional limiting distributions, one which is also appropriate for dealing with processes whose transition probabilities satisfy a system of non-linear differential equations.

scholarly journals Local Limit Approximations for Markov Population Processes

2009 ◽  
Vol 46 (03) ◽  
pp. 690-708
Author(s):  
Sanda N. Socoll ◽  
A.D. Barbour
Keyword(s):  
Poisson Distribution ◽  
Markov Processes ◽  
Continuous Time ◽  
Equilibrium Distribution ◽  
Local Limit ◽  
Moment Condition ◽  
Factor Log ◽  
The Difference ◽  
Time Density ◽  
Markov Population Processes

In this paper we are concerned with the equilibrium distribution ∏ n of the nth element in a sequence of continuous-time density-dependent Markov processes on the integers. Under a (2+α)th moment condition on the jump distributions, we establish a bound of order O(n -(α+1)/2√logn) on the difference between the point probabilities of ∏ n and those of a translated Poisson distribution with the same variance. Except for the factor √logn, the result is as good as could be obtained in the simpler setting of sums of independent, integer-valued random variables. Our arguments are based on the Stein-Chen method and coupling.

scholarly journals Growth and zinc homeostasis in the severely Zn-deficient rat

1984 ◽  
Vol 52 (3) ◽  
pp. 545-560 ◽  
Author(s):  
R. Giugliano ◽  
D. J. Millward
Keyword(s):  
Body Weight ◽  
Growth Rates ◽  
Control Diet ◽  
Rate Of Change ◽  
Zinc Homeostasis ◽  
Whole Body ◽  
Zn Deficiency ◽  
Control Groups ◽  
Body Weights ◽  
Cyclic Changes

1. Male weanling rats were fed on diets either adequate (55 mg/kg), or severely deficient (0.4 mg/kg) in zinc, either ad lib. or in restricted amounts in four experiments. Measurements were made of growth rates and Zn contents of muscle and several individual tissues.2. Zn-deficient rats exhibited the expected symptoms of deficiency including growth retardation, cyclic changes in food intake and body-weight.3. Zn deficiency specifically reduced whole body and muscle growth rates as indicated by the fact that (a) growth rates were lower in ad lib.-fed Zn-deficient rats compared with rats pair-fed on the control diet in two experiments, (b) Zn supplementation increased body-weights of Zn-deficient rats given a restricted amount of diet at a level at which they maintained weight if unsupplemented, (c) Zn supplementation maintained body-weights of Zn-deficient rats fed a restricted amount of diet at a level at which they lost weight if unsupplemented (d) since the ratio, muscle mass:body-weight was lower in the Zn-deficient rats than in the pair-fed control groups, the reduction in muscle mass was greater than the reduction in body-weight.4. Zn concentrations were maintained in muscle, spleen and thymus, reduced in comparison to some but not all control groups in liver, kidney, testis and intestine, and markedly reduced in plasma and bone. In plasma, Zn concentrations varied inversely with the rate of change of body-weight during the cyclic changes in body-weight.5. Calculation of the total Zn in the tissues examined showed a marked increase in muscle Zn with a similar loss from bone, indicating that Zn can be redistributed from bone to allow the growth of other tissues.6. The magnitude of the increase in muscle Zn in the severely Zn-deficient rat, together with the magnitude of the total losses of muscle tissue during the catabolic phases of the cycling, indicate that in the Zn-deficient rat Zn may be highly conserved in catabolic states.

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