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Population Processes with Immigration

Published on May 29, 2016
· DOI :10.1007/978-3-319-65313-6_16
Dan Han1
Estimated H-index: 1
(UNCC: University of North Carolina at Charlotte),
Stanislav Molchanov28
Estimated H-index: 28
(UNCC: University of North Carolina at Charlotte),
Joseph M. Whitmeyer13
Estimated H-index: 13
(UNCC: University of North Carolina at Charlotte)
Abstract
The paper contains a complete analysis of the Galton–Watson models with immigration, including the processes in the random environment, stationary or nonstationary ones. We also study the branching random walk on \(Z^{d}\) with immigration and prove the existence of the limits for the first two correlation functions.
  • References (6)
  • Citations (3)
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References6
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#1Yaqin Feng (UNCC: University of North Carolina at Charlotte)H-Index: 1
#2Stanislav Molchanov (UNCC: University of North Carolina at Charlotte)H-Index: 28
Last. Joseph M. Whitmeyer (UNCC: University of North Carolina at Charlotte)H-Index: 13
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The central result of this paper is the existence of limiting distributions for two classes of critical homogeneous-in-space branching processes with heavy tails spatial dynamics in dimension d = 2. In dimension d ≥ 3, the same results are true without any special assumptions on the underlying (non-degenerated) stochastic dynamics.
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In a great variety of fields, e.g., biology, epidemic theory, physics, and chemistry, ordinary differential equations are used to give continuous deterministic models for dynamic processes which are actually discrete and random in their development. Perhaps the simplest example is the differential equation used to describe a number of processes including radioactive decay and population growth.
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In the Ehrenfest model with continuous time one considers two urns and N balls distributed in the urns. The system is said to be in state i if there are i balls in urn I, N − i balls in urn II. Events occur at random times and the time intervals T between successive events are independent random variables all with the same negative exponential distribution When an event occurs a ball is chosen at random (each of the N balls has probability 1/ N to be chosen), removed from its urn, and then place...
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The following scheme of generating particles is considered. Each particle existing at a given time turns into k particles with the probability \delta _{k1} + p_k \Delta t + o(\Delta t)in time \Delta t \to 0independently of its age and origin and the history of the other particles. Moreover, k particles arise with the probability \delta _{k0} + q_k \Delta t + o(\Delta t)in time \Delta t \to 0no matter how many other particles may be present. Here $\sum\nolimits_{k = 1}^\infty {p_k } =...
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#1William FellerH-Index: 46
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Cited By3
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#1Elena Chernousova (MIPT: Moscow Institute of Physics and Technology)H-Index: 2
#2Ostap Hryniv (Durham University)H-Index: 7
Last. Stanislav Molchanov (UNCC: University of North Carolina at Charlotte)H-Index: 28
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ABSTRACTIn a population model in continuous space, individuals evolve independently as branching random walks subject to immigration. If the underlying branching mechanism is subcritical, the model...
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#1Elena ChernousovaH-Index: 2
#2Yaqin FengH-Index: 1
Last. Joseph M. WhitmeyerH-Index: 13
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We consider the subcritical contact branching random walk on Zd in continuous time with the arbitrary number of offspring and with immigration. We prove the existence of the steady state (statistical equilibrium).
#1Dan Han (UNCC: University of North Carolina at Charlotte)H-Index: 1
#2Yulia Makarova (MSU: Moscow State University)
Last. Elena Yarovaya (MSU: Moscow State University)H-Index: 9
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The paper contains several results on the existence of limits for the first two moments of the popular model in the population dynamics: continuous-time branching random walks on the multidimensional lattice \(\mathbb Z^d\), \(d\ge 1\), with immigration and infinite number of initial particles. Additional result concerns the Lyapunov stability of the moments with respect to small perturbations of the parameters of the model such as mortality rate, the rate of the birth of \((n-1)\) offsprings an...
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