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 [Formula: see text], or a stable-stage distribution [Formula: see text]. 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 [Formula: see text] corresponding to its spectral radius [Formula: see text], this [Formula: see text] is strictly greater than the supremum of the modulus of all other spectral values, and for any nonnegative initial population [Formula: see text], there is a [Formula: see text] such that [Formula: see text].
Keywords: Essential spectrum; Indeterminate growth; Integral projection models; Measures of non-compactness; Perron–Frobenius theorem; Population dynamics; Positive operators.