scholarly journals Contributions of Covariance: Decomposing the Components of Stochastic Population Growth in Cypripedium calceolus

2013 ◽  
Vol 181 (3) ◽  
pp. 410-420 ◽  
Author(s):  
Raziel Davison ◽  
Florence Nicolè ◽  
Hans Jacquemyn ◽  
Shripad Tuljapurkar
Keyword(s):  
Population Growth ◽  
Stochastic Population Growth ◽  
Cypripedium Calceolus

scholarly journals Stochastic population growth in spatially heterogeneous environments

2012 ◽  
Vol 66 (3) ◽  
pp. 423-476 ◽  
Author(s):  
Steven N. Evans ◽  
Peter L. Ralph ◽  
Sebastian J. Schreiber ◽  
Arnab Sen
Keyword(s):  
Population Growth ◽  
Heterogeneous Environments ◽  
Stochastic Population Growth

The Martin boundary for Polya's urn scheme, and an application to stochastic population growth

1964 ◽  
Vol 1 (2) ◽  
pp. 284-296 ◽  
Author(s):  
David Blackwell ◽  
David Kendall
Keyword(s):  
Markov Process ◽  
Population Growth ◽  
Transition Probability ◽  
Martin Boundary ◽  
Transition Probability Matrix ◽  
Stochastic Population Growth ◽  
Urn Scheme ◽  
Pólya’S Urn ◽  
Image Position ◽  
Positive Integers

1. In 1923 Eggenberger and Pólya introduced the following ‘urn scheme’ as a model for the development of a contagious phenomenon. A box contains b black and r red balls, and a ball is drawn from it at random with ‘double replacement’ (i.e. whatever ball is drawn, it is returned to the box together with a fresh ball of the same colour); the procedure is then continued indefinitely. A slightly more complicated version with m-fold replacement is sometimes discussed, but it will be sufficient for our purposes to keep m = 2 and it will be convenient further to simplify the scheme by taking b = r = 1 as the initial condition. We shall however generalise the scheme in another direction by allowing an arbitrary number k(≧2) of colours. Thus initially the box will contain k differently coloured balls and successive random drawings will be followed by double replacement as before. We write sn (a k-vector with jth component ) for the numerical composition of the box immediately after the nth replacement, so that and we observe that is a Markov process for which the state-space consists of all ordered k-ads of positive integers, the (constant) transition-probability matrix having elements determined by where Sn is the sum of the components of sn and (e(i))j = δij. We shall calculate the Martin boundary for this Markov process, and point out some applications to stochastic models for population growth.

The Martin boundary for Polya's urn scheme, and an application to stochastic population growth

1964 ◽  
Vol 1 (02) ◽  
pp. 284-296 ◽  
Author(s):  
David Blackwell ◽  
David Kendall
Keyword(s):  
Markov Process ◽  
Population Growth ◽  
Transition Probability ◽  
Martin Boundary ◽  
Transition Probability Matrix ◽  
Stochastic Population Growth ◽  
Urn Scheme ◽  
Pólya’S Urn ◽  
Image Position ◽  
Positive Integers

1. In 1923 Eggenberger and Pólya introduced the following ‘urn scheme’ as a model for the development of a contagious phenomenon. A box containsbblack andrred balls, and a ball is drawn from it at random with ‘double replacement’ (i.e. whatever ball is drawn, it is returned to the box together with a fresh ballof the same colour); the procedure is then continued indefinitely. A slightly more complicated version with m-fold replacement is sometimes discussed, but it will be sufficient for our purposes to keepm= 2 and it will be convenient further to simplify the scheme by takingb=r= 1 as the initial condition. We shall however generalise the scheme in another direction by allowing an arbitrary numberk(≧2) of colours. Thus initially the box will containkdifferently coloured balls and successive random drawings will be followed by double replacement as before. We writesn(ak-vector withjth component) for the numerical composition of the box immediately after the nth replacement, so thatand we observe thatis a Markov process for which the state-space consists of all orderedk-ads of positive integers, the (constant) transition-probability matrix having elements determined bywhere Snis the sum of the components of snand (e(i))j= δij. We shall calculate the Martin boundary for this Markov process, and point out some applications to stochastic models for population growth.

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