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.

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.

A stochastic model for two interacting populations

1970 ◽  
Vol 7 (3) ◽  
pp. 544-564 ◽  
Author(s):  
Niels G. Becker
Keyword(s):  
Stochastic Model ◽  
Linear Models ◽  
Death Process ◽  
Linear Component ◽  
Birth And Death Process ◽  
Linear Interaction ◽  
Interacting Populations ◽  
Non Linear ◽  
The Subject ◽  
Interaction Component

To explain the growth of interacting populations, non-linear models need to be proposed and it is this non-linearity which proves to be most awkward in attempts at solving the resulting differential equations. A model with a particular non-linear component, initially proposed by Weiss (1965) for the spread of a carrier-borne epidemic, was solved completely by different methods by Dietz (1966) and Downton (1967). Immigration parameters were added to the model of Weiss and the resulting model was made the subject of a paper by Dietz and Downton (1968). It is the aim here to further generalize the model by introducing birth and death parameters so that the result is a linear birth and death process with immigration for each population plus the non-linear interaction component.

Birth and death processes with random environments in continuous time

1981 ◽  
Vol 18 (01) ◽  
pp. 19-30 ◽  
Author(s):  
Robert Cogburn ◽  
William C. Torrez
Keyword(s):  
Discrete Time ◽  
Random Environment ◽  
Continuous Time ◽  
Death Process ◽  
Random Environments ◽  
Birth And Death Process ◽  
Time Model ◽  
Birth And Death Processes ◽  
Discrete Time Model ◽  
Extinction Probabilities

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.

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

scholarly journals Birth and death processes in randomly fluctuating environments

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.

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.

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.

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