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Population-dependent branching processes with a threshold

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  • Klebaner, Fima C.

Abstract

A branching process model where offspring distributions depend on the threshold as well as on the population size is introduced. Behaviour of such models is related to the behaviour of the corresponding deterministic models, whose behaviour is known from the chaos theory. Asymptotic behaviour of such branching processes is obtained when the population threshold is large. If the initial population size is comparable to the threshold then the size of the nth generation relative to the threshold has a normal distribution with the mean being the nth iterate of the one-step mean function. If the initial population size is negligible when compared to the threshold and the offspring distributions converge then the size of any fixed generation approaches that of a size-dependent branching process. These results are supported by a simulation study.

Suggested Citation

  • Klebaner, Fima C., 1993. "Population-dependent branching processes with a threshold," Stochastic Processes and their Applications, Elsevier, vol. 46(1), pages 115-127, May.
    • Handle: RePEc:eee:spapps:v:46:y:1993:i:1:p:115-127
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    Cited by:

    1. Klebaner, F. C. & Liptser, R., 1999. "Moderate deviations for randomly perturbed dynamical systems," Stochastic Processes and their Applications, Elsevier, vol. 80(2), pages 157-176, April.
    2. Ramanan, Kavita & Zeitouni, Ofer, 1999. "The quasi-stationary distribution for small random perturbations of certain one-dimensional maps," Stochastic Processes and their Applications, Elsevier, vol. 84(1), pages 25-51, November.
    3. Högnäs, Göran, 1997. "On the quasi-stationary distribution of a stochastic Ricker model," Stochastic Processes and their Applications, Elsevier, vol. 70(2), pages 243-263, October.
    4. Christine Jacob, 2010. "Branching Processes: Their Role in Epidemiology," IJERPH, MDPI, vol. 7(3), pages 1-19, March.

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