Equilibrium Stability and the Geometry of Bifurcation Graphs for a Class of Nonlinear Leslie Models

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Abstract

For nonlinear scalar difference equations that arise in population dynamics the geometry of the graph obtained by plotting the population growth rate as a function of inherent fertility leads to information about the number of positive equilibria and about the local stability of positive equilibria. Specifically, equilibria on decreasing segments of this graph are always unstable. Equilibria on increasing segments are stable in two circumstances: when the equilibrium is sufficiently close either to 0 or to a critical point on the graph. These geometric criteria are shown to hold for a class of nonlinear Leslie models in which (age-specific) survival rates are population density independent and fertilities are dependent on a weighted total population size. Examples are given to show how this geometric method can be used to identity strong Allee and hysteresis effects in these models.

Original languageEnglish (US)
Title of host publicationDifference Equations and Discrete Dynamical Systems with Applications - 24th ICDEA 2018
EditorsMartin Bohner, Stefan Siegmund, Roman Šimon Hilscher, Petr Stehlík
PublisherSpringer
Pages201-211
Number of pages11
ISBN (Print)9783030355012
DOIs
StatePublished - Jan 1 2020
Event24th International Conference on Difference Equations and Applications, ICDEA 2018 - Dresden, Germany
Duration: May 21 2018May 25 2018

Publication series

NameSpringer Proceedings in Mathematics and Statistics
Volume312
ISSN (Print)2194-1009
ISSN (Electronic)2194-1017

Conference

Conference24th International Conference on Difference Equations and Applications, ICDEA 2018
CountryGermany
CityDresden
Period5/21/185/25/18

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Keywords

  • Allee effects
  • Bifurcation
  • Hysteresis
  • Leslie matrix models
  • Stability

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Cushing, J. M. (2020). Equilibrium Stability and the Geometry of Bifurcation Graphs for a Class of Nonlinear Leslie Models. In M. Bohner, S. Siegmund, R. Šimon Hilscher, & P. Stehlík (Eds.), Difference Equations and Discrete Dynamical Systems with Applications - 24th ICDEA 2018 (pp. 201-211). (Springer Proceedings in Mathematics and Statistics; Vol. 312). Springer. https://doi.org/10.1007/978-3-030-35502-9_8