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On Allee effects in structured populations

Authors:	Sebastian J Schreiber

Institution:	Department of Mathematics, College of William and Mary, Williamsburg, Virginia 23187-8795

Abstract:	Maps of the nonnegative cone of ${\mathbf R}^k$ into itself are considered where are nonnegative, primitive matrices with nondecreasing entries and at least one increasing entry. Let $\lambda(x)$ denote the dominant eigenvalue of 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 . For below $\Gamma$ , $\lim_{n\to\infty}f^n(x)=0$ , while for above, $\lim_{n\to\infty}\Vert f^n(x)\Vert=\infty$ . An application to nonlinear Leslie matrices is given.

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