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  • 1
    Electronic Resource
    Electronic Resource
    Springer
    Journal of mathematical biology 26 (1988), S. 519-533 
    ISSN: 1432-1416
    Keywords: Inbreeding ; Regular mating ; Cubic ancestral growth ; Random walks ; Graph of finitely presented semigroups
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A specific regular inbreeding system of quadruple half-second cousin mating is considered. A regular inbreeding system can be thought of as a graph which satisfies certain natural homogeneity properties. Random walks X n and Y n are introduced on the nodes of the graph; the event {X n=Yn} is a renewal event by the homogeneity properties. In Arzberger (1985) it is shown that 1) graphs associated with left cancellative semigroups are regular, and 2) for regular systems, the population becomes genetically uniform if and only if the event {X n=Yn} is recurrent. In Arzberger (1986) the system of quadruple half-second cousin mating is associated with a cancellative semigroup, thus the system is regular. In this paper we show that 1) An is asymptotically of the form cn 3, where A n is the number of ancestors n generations into the past, 2) {X n=Yn} is not recurrent (this is shown by associating (X n, Y n) with a random walk in Z 3, 3) P[X 3n =Y 3n ] is asymptotically of the form cn −3/2. Thus, in this example, genetic heterogeneity is maintained, with a cubic rate of growth for An, not by an exponential growth rate, as in all previous examples of regular inbreeding systems in which genetic heterogeneity is maintained.
    Type of Medium: Electronic Resource
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  • 2
    Electronic Resource
    Electronic Resource
    Springer
    Journal of mathematical biology 22 (1985), S. 175-197 
    ISSN: 1432-1416
    Keywords: Inbreeding ; Regular Mating Systems ; Markov Chains Martingales ; Renewal Events ; Graph of Finitely Presented Semigroups
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Summary A probabilistic and algebraic treatment of regular inbreeding systems is presented. Regular inbreeding systems can be thought of as graphs which have certain natural homogeneity properties. Random walks Xn and Yn are introduced on the nodes of the graphs; the event {Xn = Yn} is a renewal event by the homogeneity property. We show that in such regular inbreeding systems the population becomes genetically uniform if and only if the event {Xn = Yn} is recurrent, which happens if ∑ 1/ An diverges, where An is the number of ancestors n generations into the past. We give two counterexamples to show the converse is false in general, but we verify the converse in the case of the graphs of certain finitely presented semigroups.
    Type of Medium: Electronic Resource
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  • 3
    Electronic Resource
    Electronic Resource
    Springer
    Journal of mathematical biology 26 (1988), S. 535-550 
    ISSN: 1432-1416
    Keywords: Inbreeding ; Regular mating systems ; Markov chains ; Renewal events ; Graphs of finitely presented semigroups
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A probabilistic and algebraic treatment of regular inbreeding systems was introduced in Arzberger (1985). In that paper it was shown that (1) regular inbreeding systems can be thought of as graphs of certain semigroups, (2) these graphs reflect a certain natural homogeneity property, (3) a sufficient condition for the population to become genetically uniform is GS 1/A r diverges, where A r is the number of ancestors r generations into the past. In this paper, a specific class of inbreeding systems is studied. For this class, the results of the previous paper are extended to generalized regular inbreeding systems in which overlapping generations in the mating scheme are allowed. A new result about the structure of the set of ancestors of two individuals is presented.
    Type of Medium: Electronic Resource
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  • 4
    Publication Date: 2022-05-25
    Description: Author Posting. © American Institute of Biological Sciences, 2005. This article is posted here by permission of American Institute of Biological Sciences for personal use, not for redistribution. The definitive version was published in BioScience 55 (2005): 501–510, doi:10.1641/0006-3568(2005)055[0501:CIEACM]2.0.CO;2.
    Description: Creative approaches at the interface of ecology, statistics, mathematics, informatics, and computational science are essential for improving our understanding of complex ecological systems. For example, new information technologies, including powerful computers, spatially embedded sensor networks, and Semantic Web tools, are emerging as potentially revolutionary tools for studying ecological phenomena. These technologies can play an important role in developing and testing detailed models that describe real-world systems at multiple scales. Key challenges include choosing the appropriate level of model complexity necessary for understanding biological patterns across space and time, and applying this understanding to solve problems in conservation biology and resource management. Meeting these challenges requires novel statistical and mathematical techniques for distinguishing among alternative ecological theories and hypotheses. Examples from a wide array of research areas in population biology and community ecology highlight the importance of fostering synergistic ties across disciplines for current and future research and application.
    Description: This paper is the result of a National Science Foundation (NSF) workshop on quantitative environmental and integrative biology (DEB-0092081). J. L. G. would like to acknowledge financial support from the NSF (DEB-0107555).
    Keywords: Ecological complexity ; Quantitative conservation biology ; Cyberinfrastructure ; Metadata ; Semantic Web
    Repository Name: Woods Hole Open Access Server
    Type: Article
    Format: 577104 bytes
    Format: application/pdf
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