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Three stage semelparous Leslie models.

J M Cushing1

  • 1Department of Mathematics and the Interdisciplinary Program in Applied Mathematics, University of Arizona, 617 N Santa Rita, Tucson, AZ 85721, USA. cushing@math.arizona.edu

Journal of Mathematical Biology
|September 9, 2008
PubMed
Summary
This summary is machine-generated.

Nonlinear Leslie matrix models reveal distinct population dynamics for semelparous species. Strong competition promotes oscillations, while weak competition leads to stable population equilibration.

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Area of Science:

  • Population Dynamics
  • Mathematical Biology
  • Theoretical Ecology

Background:

  • Nonlinear Leslie matrix models are used for semelparous species dynamics.
  • These models exhibit a transcritical bifurcation at R(0) = 1, but differ from general theory due to higher co-dimension.
  • This leads to a dichotomy: equilibration with overlapping age classes or oscillations with missing age classes, relevant for periodical insect outbreaks.

Purpose of the Study:

  • To thoroughly analyze the bifurcation at R(0) = 1 in three-dimensional nonlinear Leslie models.
  • To investigate the occurrence and properties of bifurcating invariant loops on the boundary of the positive cone.
  • To determine the stability of equilibria and invariant loops, linking conditions to inter-age class competition.

Main Methods:

  • Bifurcation theory techniques applied to three-dimensional nonlinear Leslie models.
  • Analysis of planar monotone maps and average Lyapunov functions.
  • Classification and stability analysis of invariant loops and positive equilibria.

Main Results:

  • Detailed accounting of the bifurcation at R(0) = 1 in the 3D case.
  • Identification and classification of three types of invariant loops, including three-cycles and heteroclinic orbits.
  • Stability analysis of equilibria and loops, dependent on model parameters and competition strength/symmetry.

Conclusions:

  • Strong inter-age class competition favors oscillations with separated life stages.
  • Weak competition promotes stable equilibration with overlapping life stages.
  • The study provides global dynamics on the cone boundary and clarifies conditions for population oscillations or stability.