Spectral Theory of Differential Operators Optimal Harvesting of Fish

• Applicant: Professor Dr. Horst Behncke
• Subject Area: Mathematics
• Term: Funded in 2013
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 246371334

Projekt: I Spectral Theory of Differential Operators II Optimal Sustainable Harvesting of Fish I) In the last 10 years Professor D.B. Hinton and I have published 8 papers on higher order differential operators [e.g. B.H. 2011, 2011]. In these papers it is shown that the spectra are absolutely continuous, if the coefficients satisfy mild regularity assumptions. It is known from Sturm-Liouville operators that these assumptions are almost optional. The main techniques are asymptotic integration and the M-matrix. Recently Brown, Evans, and Plum have constructed the M-matrix for not necessarily selfadjoint operators, by using the method of Weyl circles. Applications to constant coefficient operators show that this technique is hardly useful for spectral theory. Thus Hinton and Ihave devised a new method, which will also permit to determine the spectral type. For Hamiltonian systems with almost constant coefficients this method seems to be promising. This, however, requires a detailed analysis of the characteristic polynomial P(lambda, z) and the algebraic curves P(lambda, z) = 0. The method of the M-matrix is typically based on one regular endpoint. Many problems of mathematical physiscs, however, lead to two sigular endpoints, infinity and a Bessel type singularity at 0. Generally this problem is treated by the decomposition method, leading to spectra of higher multiplicity [B.H. 2010]. If, however, the left partial operator has only discrete spectrum, this does not hold any more. In this case to total M-matrix is not any more of Nevanlinna type. Problems of this type will be studied separately. II) Some years ago I presented a talk on optimal harvesting of fish at the University of Tennessee, which takes into account the width of the fishing nets and age structure of the fish population. This led to a joint work with Prof. S. Lenhart and Dr. S. Ding. it is intended to continue this work in particular for cod and herring and to determine the optimal sustainable yield and analyze the corresponding optimal control problems.

• DFG Programme: Research Grants
• International Connection: USA
• Participating Persons: Professor Dr. Don B. Hinton; Professorin Dr. Suzanne Lenhart