Kolmogorov's differential equations and positive semigroups on first moment sequence spaces

Maia Martcheva, Horst Thieme, Thanate Dhirasakdanon

Research output: Contribution to journalArticle

4 Citations (Scopus)

Abstract

Spatially implicit metapopulation models with discrete patch-size structure and host-macroparasite models which distinguish hosts by their parasite loads lead to infinite systems of ordinary differential equations. In several papers, a this-related theory will be developed in sufficient generality to cover these applications. In this paper the linear foundations are laid. They are of own interest as they apply to continuous-time population growth processes (Markov chains). Conditions are derived that the solutions of an infinite linear system of differential equations, known as Kolmogorov's differential equations, induce a C 0-semigroup on an appropriate sequence space allowing for first moments. We derive estimates for the growth bound and the essential growth bound and study the asymptotic behavior. Our results will be illustrated for birth and death processes with immigration and catastrophes.

Original languageEnglish (US)
Pages (from-to)642-671
Number of pages30
JournalJournal of Mathematical Biology
Volume53
Issue number4
DOIs
StatePublished - Oct 2006

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Positive Semigroup
Moment Sequence
Kolmogorov Equation
Infinite Systems
Sequence Space
Differential equations
Parasite Load
Differential equation
Metapopulation
Birth and Death Process
Markov Chains
Immigration
Catastrophe
Population Growth
Growth Process
Emigration and Immigration
Growth
System of Differential Equations
System of Ordinary Differential Equations
Patch

ASJC Scopus subject areas

  • Agricultural and Biological Sciences (miscellaneous)
  • Mathematics (miscellaneous)

Cite this

Kolmogorov's differential equations and positive semigroups on first moment sequence spaces. / Martcheva, Maia; Thieme, Horst; Dhirasakdanon, Thanate.

In: Journal of Mathematical Biology, Vol. 53, No. 4, 10.2006, p. 642-671.

Research output: Contribution to journalArticle

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