Some generalizations of Bailey's birth death and migration model

Davis, A. W.

doi:10.2307/3518345

Advances in Applied Probability

1970

DOI: 10.2307/3518345

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Some generalizations of Bailey's birth death and migration model

A. W. Davis¹

Abstract: Some results for a general Markov branching-diffusion process are presented, and applied to a model recently considered by Bailey. Moments of the limiting distributions of certain natural measures of the spatial location and dispersion of the population are shown to be expressible in terms of the Lauricella FD-type hypergeometric functions, when the population multiplies according to the simple birth and death process with λ > μ.

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1975

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Cited by 9 publications

References 12 publications

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Zur Bestimmung des Wanderungsprozesses

Tziafetas¹

1982

Biometrical J

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I n this paper we analyse the migration process in connection with the birth and desth process. We assume that the populations are distributed spatially and are subject to birth, death and migration to or from neighbouring populations. Furthermore, we distinguish between populations with infinite end finite number of colonies. By this way we calculate the cumulant of first order.First this problem wss studied by BAILEY and ADEE for time-dependent rates. Similarly PURI has specialized the problem for right migration (vij=O for i s j ) . Finally RENSHAW had the same result with other procedures.

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Zur Bestimmung des Wanderungsprozesses

Tziafetas¹

1982

Biometrical J

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Quasi-stationary distributions of a stochastic metapopulation model

Gyllenberg

Silvestrov

1994

J. Math. Biol.

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A stochastic metapopulation model which explicitly considers first order interactions between local populations is constructed. The model takes the spatial arrangement of patches into account and keeps track of which patches are occupied and which are empty. The time-evolution of the metapopulation is governed by a Markov chain with finite state space. We give a detailed description of the long term behaviour of the Markov chain. Many interesting biological issues can be addressed using the model. As an especially important example we discuss the so-called core and satellite species hypothesis in the light of the model.

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Consequences of limited seed dispersal within simulated annual populations

Bullock

1976

Oecologia

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Theoretical analyses of the consequences of dispersal have been almost entirely limited to studies of colonization. A simulation model is constructed and analyzed to study the effects of dispersal within local plant populations. Rules for the growth and reproduction of individuals are deterministic and related only to competition with other plants. Rules for the dispersal of individual seeds are stochastic; the generalized seed shadow is negatively exponential. The analysis has several interesting results. 1) In small, dense populations of annuals, mortality and growth behavior are significantly affected by small-scale changes in dispersal. 2) Stable population size, at any life stage, is not environmentally fixed (e.g. by the number of safe sites) but is also a function of reproductive capacity and dispersal. 3) There is a minimum productive capability which must be exceeded for a population to be able to respond to increased dispersal. The responsive range of the parameters may be related to the grain of safe site distribution. 4) When the total number of seeds per generation is large, high dispersal decreases the number of germlings that are able to establish. 5) For plants that reach the growing stage, however, expected reproduction can exceed that of genotypes with greater reproductive capacity but less dispersal. 6) An uneven distribution of seeds is essential to a spatial patterning of mortality which can deter competitive extinction.

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Some generalizations of Bailey's birth death and migration model

Cited by 9 publications

References 12 publications

Zur Bestimmung des Wanderungsprozesses

Zur Bestimmung des Wanderungsprozesses

Quasi-stationary distributions of a stochastic metapopulation model

Consequences of limited seed dispersal within simulated annual populations

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