On Allee effects in structured populations

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.