Spectral properties of a non-compact operator in ecology

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