Hans F. Weinberger

Long-Time Behavior of a Class of Biological Models

It is shown that many of the asymptotic properties of the Fisher model for population genetics and population ecology can also be derived for a class of models in which time is discrete and space may or may not be discrete. This allows one to discuss the behavior of models in which the data consist of occasional counts on survey tracts, as well as that of computer models.

Variational methods for eigenvalue approximation

Provides a common setting for various methods of bounding the eigenvalues of a self-adjoint linear operator and emphasizes their relationships.

NORM-PRESERVING DILATIONS AND THEIR APPLICATIONS TO OPTIMAL ERROR BOUNDS*

The problem is, given A, B, C, to find D such that

Spatial patterning of the spruce budworm

\left\| ( {\begin{array}{*{20}c} A & C \\ B & D \\ \end{array} )} \right\| \leqq \mu

Existence of traveling waves for integral recursions with nonmonotone growth functions.

; here we deal with Hilbert-space operators, A, B, and C ar...

On the spectrum of general second order operators

AbstractThe spatial and temporal distribution of the spruce budworm is modelled by a nonlinear diffusion equation. Two questions are considered:1.What is the critical size of a patch of forest which can support an outbreak?2.What is the width of an effective barrier to spread of an outbreak? Answers to these questions are obtained with the aid of comparison methods for nonlinear diffusion equations.

Lower Bounds for Vibration Frequencies of Elastically Supported Membranes and Plates

A class of integral recursion models for the growth and spread of a synchronized single-species population is studied. It is well known that if there is no overcompensation in the fecundity function, the recursion has an asymptotic spreading speed c*, and that this speed can be characterized as the speed of the slowest non-constant traveling wave solution. A class of integral recursions with overcompensation which still have asymptotic spreading speeds can be found by using the ideas introduced by Thieme (J Reine Angew Math 306:94–121, 1979) for the study of space-time integral equation models for epidemics. The present work gives a large subclass of these models with overcompensation for which the spreading speed can still be characterized as the slowest speed of a non-constant traveling wave. To illustrate our results, we numerically simulate a series of traveling waves. The simulations indicate that, depending on the properties of the fecundity function, the tails of the waves may approach the carrying capacity monotonically, may approach the carrying capacity in an oscillatory manner, or may oscillate continually about the carrying capacity, with its values bounded above and below by computable positive numbers.

Inequalities between dirichlet and Neumann eigenvalues

This result has been extended to other self ad joint problems for second order operators. See [2], [3], and [ó]. The purpose of this note is to show that the same technique locates the spectrum of a nonself adjoint problem in a half-plane. Such a result is of interest in investigating stability, where one needs to know whether there is any spectrum in the half-plane Re X ^ 0. In a bounded domain D we consider the differential equation

Spreading speeds of spatially periodic integro-difference models for populations with nonmonotone recruitment functions

If R is two-dimensional, the eigenvalues Xi(k) are proportional to the squares of the frequencies of a membrane covering R and elastically supported on the boundary B. If R is three-dimensional they are proportional to the squares of the frequencies of an acoustic resonator with elastic walls B. For k = so we have the boundary condition u = 0. In this case the eigenvalues are decreasing domain functionals so that lower bounds can be found immediately. This, however, is not true for finite k. In Section 2 we give the easily computed lower bound (2.11) for Xl(k) for any two-dimensional R and the analogue (2.15) for three-dimensional R. Equality is attained when R is a rectangle or parallelepiped. It is shown

Convergence to spatial-temporal clines in the Fisher equation with time-periodic fitnesses.

The purpose of this paper is to derive some inequalities of the form