On circulant populations. I. The algebra of semelparity

N.V. Davydova, O. Diekmann, Stephanus A. van Gils

    Research output: Contribution to journalArticleAcademicpeer-review

    27 Citations (Scopus)

    Abstract

    We consider a class of nonlinear Leslie matrix models, describing the population dynamics of an age-structured semelparous species. Semelparous species are those whose individuals reproduce only once and die afterwards. Competitive interaction between individuals is modelled via a one-dimensional environmental quantity. Age classes are characterised by their impact on, and their sensitivity to, the environment. We do not restrict ourselves to some particular form of functional dependence and keep the model otherwise as general as possible. The system possesses a cyclic symmetry. Due to the symmetry it exhibits so-called vertical bifurcations, where a manifold filled with periodic orbits appears in the phase space for specific parameter combinations. This bifurcation serves as a switch between the main types of behaviour: coexistence of all year classes or a periodic regime with some year classes missing. In particular, the vertical bifurcation takes place when a certain circulant matrix is singular. We also analyse the local stability of the unique coexistence equilibrium state and derive a characteristic equation for it. The dynamics of populations with two and, especially, with three age classes are analysed in detail.
    Original languageUndefined
    Pages (from-to)185-243
    Number of pages59
    JournalLinear algebra and its applications
    Volume398
    DOIs
    Publication statusPublished - 15 Mar 2005

    Keywords

    • EWI-13995
    • Circulant
    • IR-74622
    • METIS-226016
    • Leslie matrix model
    • Neimark–Sacker bifurcation
    • Semelparous species
    • Age structure
    • Vertical bifurcation

    Cite this

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    title = "On circulant populations. I. The algebra of semelparity",
    abstract = "We consider a class of nonlinear Leslie matrix models, describing the population dynamics of an age-structured semelparous species. Semelparous species are those whose individuals reproduce only once and die afterwards. Competitive interaction between individuals is modelled via a one-dimensional environmental quantity. Age classes are characterised by their impact on, and their sensitivity to, the environment. We do not restrict ourselves to some particular form of functional dependence and keep the model otherwise as general as possible. The system possesses a cyclic symmetry. Due to the symmetry it exhibits so-called vertical bifurcations, where a manifold filled with periodic orbits appears in the phase space for specific parameter combinations. This bifurcation serves as a switch between the main types of behaviour: coexistence of all year classes or a periodic regime with some year classes missing. In particular, the vertical bifurcation takes place when a certain circulant matrix is singular. We also analyse the local stability of the unique coexistence equilibrium state and derive a characteristic equation for it. The dynamics of populations with two and, especially, with three age classes are analysed in detail.",
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    year = "2005",
    month = "3",
    day = "15",
    doi = "10.1016/j.laa.2004.12.020",
    language = "Undefined",
    volume = "398",
    pages = "185--243",
    journal = "Linear algebra and its applications",
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    On circulant populations. I. The algebra of semelparity. / Davydova, N.V.; Diekmann, O.; van Gils, Stephanus A.

    In: Linear algebra and its applications, Vol. 398, 15.03.2005, p. 185-243.

    Research output: Contribution to journalArticleAcademicpeer-review

    TY - JOUR

    T1 - On circulant populations. I. The algebra of semelparity

    AU - Davydova, N.V.

    AU - Diekmann, O.

    AU - van Gils, Stephanus A.

    PY - 2005/3/15

    Y1 - 2005/3/15

    N2 - We consider a class of nonlinear Leslie matrix models, describing the population dynamics of an age-structured semelparous species. Semelparous species are those whose individuals reproduce only once and die afterwards. Competitive interaction between individuals is modelled via a one-dimensional environmental quantity. Age classes are characterised by their impact on, and their sensitivity to, the environment. We do not restrict ourselves to some particular form of functional dependence and keep the model otherwise as general as possible. The system possesses a cyclic symmetry. Due to the symmetry it exhibits so-called vertical bifurcations, where a manifold filled with periodic orbits appears in the phase space for specific parameter combinations. This bifurcation serves as a switch between the main types of behaviour: coexistence of all year classes or a periodic regime with some year classes missing. In particular, the vertical bifurcation takes place when a certain circulant matrix is singular. We also analyse the local stability of the unique coexistence equilibrium state and derive a characteristic equation for it. The dynamics of populations with two and, especially, with three age classes are analysed in detail.

    AB - We consider a class of nonlinear Leslie matrix models, describing the population dynamics of an age-structured semelparous species. Semelparous species are those whose individuals reproduce only once and die afterwards. Competitive interaction between individuals is modelled via a one-dimensional environmental quantity. Age classes are characterised by their impact on, and their sensitivity to, the environment. We do not restrict ourselves to some particular form of functional dependence and keep the model otherwise as general as possible. The system possesses a cyclic symmetry. Due to the symmetry it exhibits so-called vertical bifurcations, where a manifold filled with periodic orbits appears in the phase space for specific parameter combinations. This bifurcation serves as a switch between the main types of behaviour: coexistence of all year classes or a periodic regime with some year classes missing. In particular, the vertical bifurcation takes place when a certain circulant matrix is singular. We also analyse the local stability of the unique coexistence equilibrium state and derive a characteristic equation for it. The dynamics of populations with two and, especially, with three age classes are analysed in detail.

    KW - EWI-13995

    KW - Circulant

    KW - IR-74622

    KW - METIS-226016

    KW - Leslie matrix model

    KW - Neimark–Sacker bifurcation

    KW - Semelparous species

    KW - Age structure

    KW - Vertical bifurcation

    U2 - 10.1016/j.laa.2004.12.020

    DO - 10.1016/j.laa.2004.12.020

    M3 - Article

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    SP - 185

    EP - 243

    JO - Linear algebra and its applications

    JF - Linear algebra and its applications

    SN - 0024-3795

    ER -