## Mathematics, Department of

#### First Advisor

Richard Rebarber

#### Second Advisor

Brigitte Tenhumberg

#### Date of this Version

Fall 12-3-2020

#### Citation

M. Reichenbach, "*Spectral properties of a non-compact operator in ecology,*" Ph.D. dissertation, 2020. University of Nebraska-Lincoln

#### Abstract

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 φ 0 , there is a c > 0 such that T^n φ_0 /λ_n → 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.

Adviser: Richard Rebarber and Brigitte Tenhumberg

## Comments

A DISSERTATION Presented to the Faculty of The Graduate College at the University of Nebraska In Partial Fulfillment of Requirements For the Degree of Doctor of Philosophy, Major: Mathematics, Under the Supervision of Professors Richard Rebarber and Brigitte Tenhumberg. Lincoln, Nebraska: December, 2020

Copyright © 2020 Matthew Reichenbach