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Published in:
2020 | OriginalPaper | Chapter
Equilibrium Stability and the Geometry of Bifurcation Graphs for a Class of Nonlinear Leslie Models
Author : J. M. Cushing
Published in: Difference Equations and Discrete Dynamical Systems with Applications
Publisher: Springer International Publishing
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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.