On circulant populations. I. The algebra of semelparity

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

Research output: Contribution to journalArticleAcademicpeer-review

26 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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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

VL - 398

SP - 185

EP - 243

JO - Linear algebra and its applications

JF - Linear algebra and its applications

SN - 0024-3795

ER -