Spectral properties of a non-compact operator in ecology

Abstract

Ecologists have recently used integral projection models (IPMs) to study fish and other animals which continue to grow throughout their lives. Such animals cannot shrink, since they have bony skeletons; a mathematical consequence of this is that the kernel of the integral projection operator T is unbounded, and the operator is not compact. To our knowledge, all theoretical work done on IPMs has assumed the operator is compact, and in particular has a bounded kernel. A priori, it is unclear whether these IPMs have an asymptotic growth rate $\lambda$, or a stable-stage distribution $\psi$. In the case of a compact operator, these quantities are its spectral radius and the associated eigenvector, respectively. Under biologically reasonable assumptions, we prove that the non-compact operators in these IPMs share some important traits with their compact counterparts: the operator T has a unique positive eigenvector $\psi$ corresponding to its spectral radius $\lambda$, this $\lambda$ is strictly greater than the supremum of the modulus of all other spectral values, and for any nonnegative initial population $\varphi _0$, there is a $c>0$ such that $T^n\varphi _0/\lambda ^n \rightarrow c \cdot \psi$.