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On Allee effects in structured populations

Author: Sebastian J. Schreiber
Journal: Proc. Amer. Math. Soc. 132 (2004), 3047-3053
MSC (2000): Primary 37N25, 92D25, 37C65
Published electronically: May 12, 2004
MathSciNet review: 2063126
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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 $\lim_{n\to\infty} \Vert f^n(x)\Vert=\infty$ 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, $\lim_{n\to\infty}\Vert f^n(x)\Vert=\infty$. An application to nonlinear Leslie matrices is given.

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Additional Information

Sebastian J. Schreiber
Affiliation: Department of Mathematics, College of William and Mary, Williamsburg, Virginia 23187-8795

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
Article copyright: © Copyright 2004 American Mathematical Society

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