October 4, 2008
Classical models for the growth and spread of introduced species
track the front of an expanding wave of population density. Models
are typically parabolic partial differential equations and related integral
formulations. One method to infer the speed of the expanding wave is
to equate the speed of spread of the nonlinear system with the speed of
spread of a related linear system. When these two speeds coincide we say that the
spread rate is linearly predictable. While many spread rates are linearly
predictable, some notable cases are not, such as those involving competition between
multiple species.
Hans Weinberger's work has impacted the theory of linear predictability,
both for single-species and for multi-species models. I will review
some of this theory, from the perspective of a mathematical ecologist
interested in applying the theory to biology. In my talk I will apply some of the results
to real biological problems, including species competition, spread of disease and
population dynamics of stream ecosystems.
In this talk we will survey several papers (listed
below) by
Hans Weinberger dealing with numerical and approximation
issues. We have
divided them into three categories: (i) approximation of
eigenvalues; (ii)
approximation theory issues; and (iii) error bounds for
iterative methods for
matrix inversion.
The seven papers listed are only a small part of Hans’ work—but
they
were very influential. We, of course, cannot discuss any of
these papers
in detail, but will instead concentrate on those results that
are especially
insightful and elegant.
References
Approximation of Eigenvalues
[1] Upper and lower bounds for eigenvalues by finite difference
methods.
Communications on Pure and Appl. Math. 9 (1956), pp. 613-623.
[2] Lower bounds for higher eigenvalues by finite difference
methods. Pacific
J. Math. 8 (1958), pp. 339-368.
Approximation Theory Issues
[3] Optimal approximations and error bounds (joint with M.
Golumb). In
Proc. Symposium on Numerical Approximation, Univ. of Wisconsin
Press, 1959, pp. 117-190.
[4] Optimal approximation for functions prescribed at equally
spaced points.
Nat. Bureau of Standards J. of Research 65B, 2 (1961), pp.
99-104.
[5] On optimal numerical solution of partial differential
equations. SIAM J.
Numer. Anal. 9 (1972), pp. 182-198.
[6] Optimal numerical approximation of a linear operator.
Linear Alg. and
its Appl. 52/53 (1983), pp. 717-737.
Error Bounds for Iterative Methods for Matrix Inversion
[7] A posteriori error bounds in iterative matrix inversion. In
Numerical
Treatment of Partial Differential Equations, Academic Press
1965,
pp. 153-163. Proceedings of Symposium on Numerical Solution of
Partial
Differential Equations, held at the Univ. of Maryland in 1965
(Edited by
J. Bramble).