zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
The extinction time of a birth, death and catastrophe process and of a related diffusion model. (English) Zbl 0551.92013
The author considers a Markov process with state space ${\bbfN}\sb+$ and generator $[q\sb{ij}]$ where $q\sb{ij}=i\lambda I\sb{\{i+1\}}(j)+i\mu I\sb{\{i-1\}}(j)+i\kappa d\sb{i-j}I\sb{(0,i)}(j)$ if $j\ge 1$ and $j\ne i$, $q\sb{i0}=i\kappa \sum\sp{\infty}\sb{k=i}d\sb k+\mu I\sb{\{i-1\}}(0)$ and $q\sb{ii}=-i(\lambda +\mu +\kappa)+i\kappa d\sb 0$. This defines a linear birth-death process modified to allow ’catastrophic’ decrements in the population size at a rate proportional to population size. This model, with specific catastrophe-size distributions $\{d\sb i\}$, has previously been examined by the author, {\it J. Gani} and {\it S. I. Resnick,} ibid. 14, 709-731 (1982; Zbl 0496.92007). Here the author is interested in the time T to extinction. He derives an expression for its probability generating function, criteria ensuring that $P\sb i(T<\infty)=1$ and an asymptotic expression for $P\sb i(T<\infty)$, when this is not unity, as $i\to \infty$. In addition he derives a generating function for $E\sb iT$ and obtains an asymptotic form of $E\sb iT$ for large i when the process drifts to the origin. Finally, he considers the analogous problems for the Feller continuous- state branching process modified to allow downward jumps at a rate proportional to the level of the process.
Reviewer: A.Pakes

92D25Population dynamics (general)
60J80Branching processes
60J85Applications of branching processes
Full Text: DOI