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

J. M. Cushing

doi:10.1007/978-3-030-35502-9_8

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

J. M. Cushing

Mathematics

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

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 language	English (US)
Title of host publication	Difference Equations and Discrete Dynamical Systems with Applications - 24th ICDEA 2018
Editors	Martin Bohner, Stefan Siegmund, Roman Šimon Hilscher, Petr Stehlík
Publisher	Springer
Pages	201-211
Number of pages	11
ISBN (Print)	9783030355012
DOIs	https://doi.org/10.1007/978-3-030-35502-9_8
State	Published - 2020
Event	24th International Conference on Difference Equations and Applications, ICDEA 2018 - Dresden, Germany Duration: May 21 2018 → May 25 2018

Publication series

Name	Springer Proceedings in Mathematics and Statistics
Volume	312
ISSN (Print)	2194-1009
ISSN (Electronic)	2194-1017

Conference

Conference	24th International Conference on Difference Equations and Applications, ICDEA 2018
Country	Germany
City	Dresden
Period	5/21/18 → 5/25/18

Keywords

Allee effects
Bifurcation
Hysteresis
Leslie matrix models
Stability

ASJC Scopus subject areas

Mathematics(all)

Access to Document

10.1007/978-3-030-35502-9_8

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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

Equilibrium Stability and the Geometry of Bifurcation Graphs for a Class of Nonlinear Leslie Models. / Cushing, J. M.

Difference Equations and Discrete Dynamical Systems with Applications - 24th ICDEA 2018. ed. / Martin Bohner; Stefan Siegmund; Roman Šimon Hilscher; Petr Stehlík. Springer, 2020. p. 201-211 (Springer Proceedings in Mathematics and Statistics; Vol. 312).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

Cushing, JM 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. Springer Proceedings in Mathematics and Statistics, vol. 312, Springer, pp. 201-211, 24th International Conference on Difference Equations and Applications, ICDEA 2018, Dresden, Germany, 5/21/18. https://doi.org/10.1007/978-3-030-35502-9_8

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

Cushing, J. M. / Equilibrium Stability and the Geometry of Bifurcation Graphs for a Class of Nonlinear Leslie Models. Difference Equations and Discrete Dynamical Systems with Applications - 24th ICDEA 2018. editor / Martin Bohner ; Stefan Siegmund ; Roman Šimon Hilscher ; Petr Stehlík. Springer, 2020. pp. 201-211 (Springer Proceedings in Mathematics and Statistics).

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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.",

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note = "Funding Information: Acknowledgements The author gratefully acknowledges the support of the National Science Foundation (Mathematical Biology Program in the Division of Mathematical Sciences and the Population and Community Ecology Program in the Division of Environmental Biology) under NSF grant DMS 1407564. The author is also grateful to two anonymous reviewers for their careful reading of the manuscript and for their suggestions. Publisher Copyright: {\textcopyright} Springer Nature Switzerland AG 2020.; 24th International Conference on Difference Equations and Applications, ICDEA 2018 ; Conference date: 21-05-2018 Through 25-05-2018",

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N2 - 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.

AB - 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.

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

Abstract

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Keywords

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