@inproceedings{b0be8187103c4e98b1183db682ca6c7e,

title = "Equilibrium Stability and the Geometry of Bifurcation Graphs for a Class of Nonlinear Leslie Models",

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.",

keywords = "Allee effects, Bifurcation, Hysteresis, Leslie matrix models, Stability",

author = "Cushing, {J. M.}",

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",

year = "2020",

doi = "10.1007/978-3-030-35502-9_8",

language = "English (US)",

isbn = "9783030355012",

series = "Springer Proceedings in Mathematics and Statistics",

publisher = "Springer",

pages = "201--211",

editor = "Martin Bohner and Stefan Siegmund and {{\v S}imon Hilscher}, Roman and Petr Stehl{\'i}k",

booktitle = "Difference Equations and Discrete Dynamical Systems with Applications - 24th ICDEA 2018",

}