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).