On Allee effects in structured populations
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- by Sebastian J. Schreiber PDF
- Proc. Amer. Math. Soc. 132 (2004), 3047-3053 Request permission
Abstract:
Maps $f(x)=A(x)x$ of the nonnegative cone $C$ of ${\mathbf R}^k$ into itself are considered where $A(x)$ are nonnegative, primitive matrices with nondecreasing entries and at least one increasing entry. Let $\lambda (x)$ denote the dominant eigenvalue of $A(x)$ and $\lambda (\infty )=\sup _{x\in C} \lambda (x)$. These maps are shown to exhibit a dynamical trichotomy. First, if $\lambda (0)\ge 1$, then $\lim _{n\to \infty } \|f^n(x)\|=\infty$ for all nonzero $x\in C$. Second, if $\lambda (\infty )\le 1$, then $\lim _{n\to \infty }f^n(x)=0$ for all $x\in C$. Finally, if $\lambda (0)<1$ and $\lambda (\infty )>1$, then there exists a compact invariant hypersurface $\Gamma$ separating $C$. For $x$ below $\Gamma$, $\lim _{n\to \infty }f^n(x)=0$, while for $x$ above, $\lim _{n\to \infty }\|f^n(x)\|=\infty$. An application to nonlinear Leslie matrices is given.References
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Additional Information
- Sebastian J. Schreiber
- Affiliation: Department of Mathematics, College of William and Mary, Williamsburg, Virginia 23187-8795
- Email: sjschr@wm.edu
- Received by editor(s): March 17, 2003
- Received by editor(s) in revised form: May 20, 2003
- Published electronically: May 12, 2004
- Additional Notes: This research was supported in part by National Science Foundation Grant DMS-0077986
- Communicated by: Carmen C. Chicone
- © Copyright 2004 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 132 (2004), 3047-3053
- MSC (2000): Primary 37N25, 92D25, 37C65
- DOI: https://doi.org/10.1090/S0002-9939-04-07406-4
- MathSciNet review: 2063126