The asymptotic behaviour of a divergent linear birth and death process

1978 ◽  
Vol 15 (1) ◽  
pp. 187-191 ◽  
Author(s):  
John Haigh
Keyword(s):  
Asymptotic Behaviour ◽  
Death Rate ◽  
High Probability ◽  
Birth Rate ◽  
Mean Value ◽  
Death Process ◽  
Applied Probability ◽  
Birth And Death Process ◽  
Initial Size

A recent paper in Advances in Applied Probability (Siegel (1976)) considered the duration of the time Tmn for a linear birth and death process to grow from a (large) initial size m to a larger size n. The main aim was to show that, when the birth rate exceeds the death rate, Tmn is close to its mean value, log n/m, with high probability. This paper establishes this result using much simpler techniques.

The asymptotic behaviour of a divergent linear birth and death process

1978 ◽  
Vol 15 (01) ◽  
pp. 187-191 ◽  
Author(s):  
John Haigh
Keyword(s):  
Asymptotic Behaviour ◽  
Death Rate ◽  
High Probability ◽  
Birth Rate ◽  
Mean Value ◽  
Death Process ◽  
Applied Probability ◽  
Birth And Death Process ◽  
Initial Size

A recent paper in Advances in Applied Probability (Siegel (1976)) considered the duration of the time Tmn for a linear birth and death process to grow from a (large) initial size m to a larger size n. The main aim was to show that, when the birth rate exceeds the death rate, Tmn is close to its mean value, log n/m, with high probability. This paper establishes this result using much simpler techniques.

A linear birth-and-death predator–prey process

1995 ◽  
Vol 32 (01) ◽  
pp. 274-277
Author(s):  
John Coffey
Keyword(s):  
Death Rate ◽  
Birth Rate ◽  
Prey Population ◽  
Death Process ◽  
Predator Population ◽  
Birth And Death Process ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model

A new stochastic predator-prey model is introduced. The predator population X(t) is described by a linear birth-and-death process with birth rate λ 1 X and death rate μ 1 X. The prey population Y(t) is described by a linear birth-and-death process in which the birth rate is λ 2 Y and the death rate is . It is proven that and iff

A linear birth-and-death predator–prey process

1995 ◽  
Vol 32 (1) ◽  
pp. 274-277 ◽  
Author(s):  
John Coffey
Keyword(s):  
Death Rate ◽  
Birth Rate ◽  
Prey Population ◽  
Death Process ◽  
Predator Population ◽  
Birth And Death Process ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model

A new stochastic predator-prey model is introduced. The predator population X(t) is described by a linear birth-and-death process with birth rate λ1X and death rate μ1X. The prey population Y(t) is described by a linear birth-and-death process in which the birth rate is λ2Y and the death rate is . It is proven that and iff

A model for blue-green algae and gorillas

1977 ◽  
Vol 14 (04) ◽  
pp. 675-688 ◽  
Author(s):  
Byron J. T. Morgan ◽  
B. Leventhal
Keyword(s):  
Exact Solutions ◽  
Green Algae ◽  
Death Rate ◽  
Death Process ◽  
Birth And Death Process ◽  
Mean Values ◽  
Blue Green Algae ◽  
Supercritical Case ◽  
The Mean ◽  
Runge Kutta

The linear birth-and-death process is elaborated to allow the elements of the process to live as members of linear clusters which have the possibility of breaking up. For the supercritical case, expressions, based on an approximation, are derived for the mean numbers of clusters of the various sizes as time → ∞. These expressions compare very well with exact solutions obtained by the method of Runge-Kutta. Exact solutions for the mean values for all time are given for when the death rate is zero.

A model for blue-green algae and gorillas

1977 ◽  
Vol 14 (4) ◽  
pp. 675-688 ◽  
Author(s):  
Byron J. T. Morgan ◽  
B. Leventhal
Keyword(s):  
Exact Solutions ◽  
Green Algae ◽  
Death Rate ◽  
Death Process ◽  
Birth And Death Process ◽  
Mean Values ◽  
Blue Green Algae ◽  
Supercritical Case ◽  
The Mean ◽  
Runge Kutta

The linear birth-and-death process is elaborated to allow the elements of the process to live as members of linear clusters which have the possibility of breaking up. For the supercritical case, expressions, based on an approximation, are derived for the mean numbers of clusters of the various sizes as time → ∞. These expressions compare very well with exact solutions obtained by the method of Runge-Kutta. Exact solutions for the mean values for all time are given for when the death rate is zero.

Interparticle dependence in a linear death process subjected to a random environment

1990 ◽  
Vol 27 (3) ◽  
pp. 491-498 ◽  
Author(s):  
Claude Lefèvre ◽  
György Michaletzky
Keyword(s):  
Random Environment ◽  
Death Rate ◽  
Monotone Function ◽  
Death Process ◽  
Birth And Death Process ◽  
Survival Times ◽  
Current State ◽  
Non Linear

Recently, Ball and Donnelly (1987) investigated the nature of the interparticle dependence in a death process with non-linear rates. In this note, after some remarks on their result, a similar problem is examined for a linear death process where the death rate per particle is a monotone function of the current state of a random environment. It is proved that if the exterior process involved is a homogeneous birth-and-death process valued in ℕ, then the survival times of any subset of particles are positively upper orthant dependent. A simple example shows that this property is not valid for general exterior processes.

The asymptotic behavior of a divergent linear birth and death process

1976 ◽  
Vol 8 (02) ◽  
pp. 315-338 ◽  
Author(s):  
Martha J. Siegel
Keyword(s):  
Asymptotic Behavior ◽  
High Probability ◽  
Arrival Time ◽  
Death Process ◽  
Birth And Death Process ◽  
Time Zero ◽  
First Arrival ◽  
First Arrival Time

We consider a birth and death process with Q-matrix of rates q m,m + 1 = mβ, q m,m − 1 = mδ, qm,m = – m(β + δ) and qm,n = 0 otherwise. We assume that 0 < δ < β and β – δ = 1. The asymptotic behavior of first-arrival time at state n given that the process is at state m at time zero is expressed in terms of polynomials and it is shown that if m < n and m is large that the first-arrival time is close to log n/m with high probability.

Interparticle dependence in a linear death process subjected to a random environment

1990 ◽  
Vol 27 (03) ◽  
pp. 491-498 ◽  
Author(s):  
Claude Lefèvre ◽  
György Michaletzky
Keyword(s):  
Random Environment ◽  
Death Rate ◽  
Monotone Function ◽  
Death Process ◽  
Birth And Death Process ◽  
Survival Times ◽  
Current State ◽  
Non Linear

Recently, Ball and Donnelly (1987) investigated the nature of the interparticle dependence in a death process with non-linear rates. In this note, after some remarks on their result, a similar problem is examined for a linear death process where the death rate per particle is a monotone function of the current state of a random environment. It is proved that if the exterior process involved is a homogeneous birth-and-death process valued in ℕ, then the survival times of any subset of particles are positively upper orthant dependent. A simple example shows that this property is not valid for general exterior processes.

The asymptotic behavior of a divergent linear birth and death process

1976 ◽  
Vol 8 (2) ◽  
pp. 315-338 ◽  
Author(s):  
Martha J. Siegel
Keyword(s):  
Asymptotic Behavior ◽  
High Probability ◽  
Arrival Time ◽  
Death Process ◽  
Birth And Death Process ◽  
Time Zero ◽  
First Arrival ◽  
First Arrival Time

We consider a birth and death process with Q-matrix of rates qm,m + 1 = mβ, qm,m − 1 = mδ, qm,m = – m(β + δ) and qm,n = 0 otherwise. We assume that 0 < δ < β and β – δ = 1. The asymptotic behavior of first-arrival time at state n given that the process is at state m at time zero is expressed in terms of polynomials and it is shown that if m < n and m is large that the first-arrival time is close to log n/m with high probability.

The asymptotic behaviour of birth and death and some related processes

1975 ◽  
Vol 7 (01) ◽  
pp. 28-43 ◽  
Author(s):  
Andrew D. Barbour
Keyword(s):  
Asymptotic Behaviour ◽  
Central Limit ◽  
Markov Processes ◽  
Continuous Time ◽  
Death Process ◽  
Transition Rates ◽  
Birth And Death Process ◽  
Iterated Logarithm ◽  
The Matrix ◽  
Positive Integers

The paper examines those continuous time Markov processes Z(·) on the positive integers which have the ‘skip free upwards’ property, with regard to their asymptotic behaviour in the event of Z(t) tending to infinity. The behaviour is characterised in terms of the convergence or divergence of an appropriate function of Z(t), and the description is improved by central limit and iterated logarithm theorems. The conditions of the theorems are expressed entirely in terms of the matrix Q of instantaneous transition rates for Z(·). The method is applied, by way of example, to the super-critical linear birth and death process.

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