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Spectral Properties of a Non-Compact Operator in Ecology
Ecologists have 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. A priori, it is unclear whether these IPMs have an asymptotic growth rate λ, or a stable-stage distribution ψ. 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 important spectral properties with their compact counterparts. Specifically, we show that the operator T has a unique positive eigenvector ψ corresponding to its spectral radius λ, the spectral radius λ is strictly greater than the supremum of all other spectral values, and for any nonnegative initial population ϕ0, there is a c > 0 such that Tnϕ0/λn → c · ψ. We also show that the zeros of certain functions defined by sums of compact operators can be used to approximate the spectral radius λ of the non-compact operator T. In the final chapter, we give some simulations showing the long-term behavior of a density-dependent IPM.
Reichenbach, Matthew, "Spectral Properties of a Non-Compact Operator in Ecology" (2020). ETD collection for University of Nebraska - Lincoln. AAI28258139.