On Allee effects in structured populations

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.