Distribution of Eigenvalues

2010 ◽  
pp. 146-159
2021 ◽  
Vol 31 (08) ◽  
pp. 2150114
Author(s):  
Ming Liu ◽  
Jun Cao ◽  
Xiaofeng Xu

In this paper, the dynamics of a phytoplankton–zooplankton system with delay and diffusion are investigated. The positivity and persistence are studied by using the comparison theorem and upper and lower solutions method. The stability of steady states and the existence of local Hopf bifurcation are obtained by analyzing the distribution of eigenvalues. And the global existence of positive periodic solutions is established by using the global Hopf bifurcation result given by Wu [1996]. Finally, some numerical simulations are carried out to illustrate the analytical results.


1988 ◽  
Vol 4 (1) ◽  
pp. 403-417 ◽  
Author(s):  
Jeffrey S. Geronimo ◽  
Evans M. Harrell ◽  
Walter Van Assche

1997 ◽  
Vol 341 ◽  
pp. 77-99 ◽  
Author(s):  
M. RIEUTORD ◽  
L. VALDETTARO

The structure and spectrum of inertial waves of an incompressible viscous fluid inside a spherical shell are investigated numerically. These modes appear to be strongly featured by a web of rays which reflect on the boundaries. Kinetic energy and dissipation are indeed concentrated on thin conical sheets, the meridional cross-section of which forms the web of rays. The thickness of the rays is in general independent of the Ekman number E but a few cases show a scaling with E1/4 and statistical properties of eigenvalues indicate that high-wavenumber modes have rays of width O(E1/3). Such scalings are typical of Stewartson shear layers. It is also shown that the web of rays depends on the Ekman number and shows bifurcations as this number is decreased.This behaviour also implies that eigenvalues do not evolve smoothly with viscosity. We infer that only the statistical distribution of eigenvalues may follow some simple rules in the asymptotic limit of zero viscosity.


1991 ◽  
Vol 44 (1) ◽  
pp. 42-53 ◽  
Author(s):  
Lawrence Barkwell ◽  
Peter Lancaster ◽  
Alexander S. Markus

AbstractEigenvalue problems for selfadjoint quadratic operator polynomials L(λ) = Iλ2 + Bλ+ C on a Hilbert space H are considered where B, C∈ℒ(H), C >0, and |B| ≥ kI + k-l C for some k >0. It is shown that the spectrum of L(λ) is real. The distribution of eigenvalues on the real line and other spectral properties are also discussed. The arguments rely on the well-known theory of (weakly) hyperbolic operator polynomials.


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