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Spectral Properties of a Non-Compact Operator in Ecology

Matthew Reichenbach, University of Nebraska - Lincoln


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.

Subject Area

Applied Mathematics|Mathematics|Ecology

Recommended Citation

Reichenbach, Matthew, "Spectral Properties of a Non-Compact Operator in Ecology" (2020). ETD collection for University of Nebraska - Lincoln. AAI28258139.