Divergent population processes and mammal outbreaks
- 1 November 1967
- journal article
- Published by Cambridge University Press (CUP) in Journal of Applied Probability
- Vol. 4 (1) , 1-8
- https://doi.org/10.2307/3212295
Abstract
Birth and death processes, i.e. Markov chains {X(t)}, 0 ≦t≦ ∞, taking values in the setIof non-negative integers, for which the infinitesimal transition probabilities are have found wide application in mathematical ecology . In order that the infinite system of differential Equations (1) should have solutions such thatPij(t) ≦ 0, alltand it is necessary and sufficient that . The conditions ensure that with probability one, the sample paths ofX(t) have no discontinuities worse than jumps, and in this case the solution is unique. If (3) does not hold then for somet1at least, there is a set of sample paths of positive probability, each of which hast1as a limit point of jumps. For pure birth processes, (in which μj= 0) this clearly impliesX(t)→ ∞ ast→t1.Keywords
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