27,599 research outputs found
Bounds on transient instability for complex ecosystems
Stability is a desirable property of complex ecosystems. If a community of
interacting species is at a stable equilibrium point then it is able to
withstand small perturbations to component species' abundances without
suffering adverse effects. In ecology, the Jacobian matrix evaluated at an
equilibrium point is known as the community matrix, which describes the
population dynamics of interacting species. A system's asymptotic short- and
long-term behaviour can be determined from eigenvalues derived from the
community matrix. Here we use results from the theory of pseudospectra to
describe intermediate, transient dynamics. We first recover the established
result that the transition from stable to unstable dynamics includes a region
of `transient instability', where the effect of a small perturbation to
species' abundances---to the population vector---is amplified before ultimately
decaying. Then we show that the shift from stability to transient instability
can be affected by uncertainty in, or small changes to, entries in the
community matrix, and determine lower and upper bounds to the maximum amplitude
of perturbations to the population vector. Of five different types of community
matrix, we find that amplification is least severe when predator-prey
interactions dominate. This analysis is relevant to other systems whose
dynamics can be expressed in terms of the Jacobian matrix. Our results will
lead to improved understanding of how multiple perturbations to a complex
system may irrecoverably break stability.Comment: 7 pages, two columns, 3 figures; text improved - Accepted for
publication on PLoS On
Eigenvalue Attraction
We prove that the complex conjugate (c.c.) eigenvalues of a smoothly varying
real matrix attract (Eq. 15). We offer a dynamical perspective on the motion
and interaction of the eigenvalues in the complex plane, derive their governing
equations and discuss applications. C.c. pairs closest to the real axis, or
those that are ill-conditioned, attract most strongly and can collide to become
exactly real. As an application we consider random perturbations of a fixed
matrix . If is Normal, the total expected force on any eigenvalue is
shown to be only the attraction of its c.c. (Eq. 24) and when is circulant
the strength of interaction can be related to the power spectrum of white
noise. We extend this by calculating the expected force (Eq. 41) for real
stochastic processes with zero-mean and independent intervals. To quantify the
dominance of the c.c. attraction, we calculate the variance of other forces. We
apply the results to the Hatano-Nelson model and provide other numerical
illustrations. It is our hope that the simple dynamical perspective herein
might help better understanding of the aggregation and low density of the
eigenvalues of real random matrices on and near the real line respectively. In
the appendix we provide a Matlab code for plotting the trajectories of the
eigenvalues.Comment: v1:15 pages, 12 figures, 1 Matlab code. v2: very minor changes, fixed
a reference. v3: 25 pages, 17 figures and one Matlab code. The results have
been extended and generalized in various ways v4: 26 pages, 10 figures and a
Matlab Code. Journal Reference Added.
http://link.springer.com/article/10.1007%2Fs10955-015-1424-
Sensitivity analysis of hydrodynamic stability operators
The eigenvalue sensitivity for hydrodynamic stability operators is investigated. Classical matrix perturbation techniques as well as the concept of epsilon-pseudoeigenvalues are applied to show that parts of the spectrum are highly sensitive to small perturbations. Applications are drawn from incompressible plane Couette, trailing line vortex flow and compressible Blasius boundary layer flow. Parametric studies indicate a monotonically increasing effect of the Reynolds number on the sensitivity. The phenomenon of eigenvalue sensitivity is due to the non-normality of the operators and their discrete matrix analogs and may be associated with large transient growth of the corresponding initial value problem
Spectral Theory of Sparse Non-Hermitian Random Matrices
Sparse non-Hermitian random matrices arise in the study of disordered
physical systems with asymmetric local interactions, and have applications
ranging from neural networks to ecosystem dynamics. The spectral
characteristics of these matrices provide crucial information on system
stability and susceptibility, however, their study is greatly complicated by
the twin challenges of a lack of symmetry and a sparse interaction structure.
In this review we provide a concise and systematic introduction to the main
tools and results in this field. We show how the spectra of sparse
non-Hermitian matrices can be computed via an analogy with infinite dimensional
operators obeying certain recursion relations. With reference to three
illustrative examples --- adjacency matrices of regular oriented graphs,
adjacency matrices of oriented Erd\H{o}s-R\'{e}nyi graphs, and adjacency
matrices of weighted oriented Erd\H{o}s-R\'{e}nyi graphs --- we demonstrate the
use of these methods to obtain both analytic and numerical results for the
spectrum, the spectral distribution, the location of outlier eigenvalues, and
the statistical properties of eigenvectors.Comment: 60 pages, 10 figure
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