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We consider a periodic environment with favorable and unfavorable patches alternately arranged in one dimension. In such heterogeneous environments, invasive animals may undergo not only random diffusion but also directed movement toward favorable patches. Here we propose reaction–diffusion–advection models in which the diffusion term is of either Fickian or Fokker–Planck type, the advection term is given by a gradient-based taxis, and the growth rate depends on both the environmental conditions and the population density. We first present hypotheses for the existence of a periodic traveling wave and a method to derive its spreading speed. Then this method is applied to a case in which the environment is homogeneous within each patch, so that taxis occurs at interfaces between different patches, causing accumulated distributions in favorable patches with density jumps at the interfaces. We show how the Fickian or Fokker–Planck diffusion and the gradient-based taxis interplay to determine the spreading speed.

It is shown that the problem of eliminating a less-fit allele by allowing a mixture of genotypes whose densities satisfy a system of reaction-diffusion equations with population control to evolve in a reactor with impenetrable walls is approximately controllable.

It is shown that a trick introduced by H. R. Thieme [6] to study a one-species integral equation model with a nonmonotone operator can be used to show that some multispecies reaction-diffusion systems which are cooperative for small population densities but not for large ones have a spreading speed. The ideas are explained by considering a model for the interaction between ungulates and grassland.

Long-time asymptotic properties of the solutions of the system where f (u) is positive homogeneous of degree p>1, are studied. We also consider the corresponding linearly perturbed system It is shown that if A=α I, then the global existence of all solutions for one value of α implies that the same property holds for all α, and that all solutions converge to the origin when α

One may expect that if one adds a vector field which points toward the origin to the right-hand side of a first-order system of ordinary differential equations for which the origin is globally stable, then the origin is globally stable for the new system as well. The purpose of this note is to present an example which shows that this apparently obvious statement is, in fact, false. Copyright © 2004 John Wiley & Sons, Ltd.

We present an example of a predator-prey-like system with a prey-only state as a global attractor, and with the additional property that an attempt to control the prey by harvesting or poisoning both species produces solutions in which both populations blow up in finite time.

Conditions which must be satisfied by the steady flow of a rigid-plastic material through an extrusion die which minimizes dissipation power are derived.

One crucial measure of a species' invasiveness is the rate at which it spreads into a competitor's environment. A heuristic spread rate formula for a spatially explicit, two-species competition model relies on 'linear determinacy' which equates spread rate in the full nonlinear model with spread rate in the system linearized about the leading edge of the invasion. However, linear determinacy is not always valid for two-species competition; it has been shown numerically that the formula only works for certain values of model parameters when the model is diffusive Lotka-Volterra competition [2]. This paper derives a set of sufficient conditions for linear determinacy in spatially explicit two-species competition models. These conditions can be interpreted as requiring sufficiently large dispersal of the invader relative to dispersal of the out-competed resident and sufficiently weak interactions between the resident and the invader. When these conditions are not satisfied, spread rate may exceed linearly determined predictions. The mathematical methods rely on the application of results established in a companion paper [11].

Let | A | p , q be the norm induced on the matrix A with n rows and m columns by the Holder l p and l q norms on R n and R m (or C n and C m ), respectively. It is easy to find an upper bound for the ratio |A| r,s |A| p,q . In this paper we study the classes of matrices for which the upper bound is attained. We shall show that for fixed A , attainment of the bound depends only on the signs of r − p and s − q . Various criteria depending on these signs are obtained. For the special case p = q = 2, the set of all matrices for which the bound is attained is generated by means of singular value decompositions.

It is known [7] that dualizing a form of the Poisson summation formula yields a pair of linear transformations which map a function o of one variable into a function and its cosine transform in a generalized sense. The present work presents conditions on o for which the transform relation holds in the classical sense, and extends this result to a class of generalizations of the Poisson formula in any number of dimensions.

A backward–forward parabolic equation is considered in a strip which is infinite in the timelike direction, with boundary conditions on the sides of the strip. The unique bounded solution of the problem is given explicitly by separation of variables. In addition, a similar problem in a semi-infinite strip is treated, with initial data at the end of the strip. It is shown that the solution can be represented arbitrarily closely in the maximum norm by a sum of functions obtain by separation of variables.

It is shown that an ideal forming operation for an extrusion or drawing die shrinks not only the cross-sectional area but also the perimeter. It follows that for any billet shape there are product cross-sections which can be produced by extrusion or drawing dies but not by ideal forming operations.

A known result about the spectral radius of an irreducible nonnegative matrix is extended to all nonnegative matrices. By means of this result, it is shown that the structured singular value and the volumetric singular value of a class of nonnegative matrices can be approximated with arbitrary accuracy by the matrix norm induced by a weighted $\ell_2$ vector norm and in the simplest case by a weighted $\ell_p$ vector norm for any $p$.

The steady-state diffusion of heat in a fluid which flows rapidly but not unidirectionally in a semi-infinite channel leads to a singular perturbation of a backwards-forwards heat equation with Neumann conditions on the channel walls. The temperature is prescribed on the whole channel entrance for the perturbed problems, but only at the points where the fluid flow is inward for the limiting problem. Conditions are obtained for uniqueness, and the additional arbitrary constants needed to obtain a well-posed problem when uniqueness does not hold are characterized. Conditions for the convergence of the solution of the perturbed problem to that of the limiting problem are also found.

A well-known formula for the spreading speed of a discrete-time recursion model is extended to a class of problems for which its validity was previously unknown. These include migration models with moderately fat tails or fat tails. Examples of such models are given.

It is shown that as time approaches infinity, the solution of the initial value problem for a regularized one-dimensional scalar conservation law converges along rays to the solution of a certain Riemann problem for the hyperbolic conservation law, even when this conservation law is not genuinely nonlinear.

A necessary condition for realization by a decentralized finite-dimensional mechanism is proved. The concepts of the realization of a time sequence of allocations by a decentralized temporal process and by the particular more realistic class of decentralized evolutionary temporal processes are defined. The above condition shows that the optimal allocation in a simple model for intertemporal production and consumption can be realized by a decentralized temporal process but not by a decentralized evolutionary process which uses either a finite number of verification conditions or a finite-dimensional message space.

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.

An idea used by Thieme (J. Math. Biol. 8, 173–187, 1979) is extended to show that a class of integro-difference models for a periodically varying habitat has a spreading speed and a formula for it, even when the recruitment function R(u, x) is not nondecreasing in u, so that overcompensation occurs. Numerical simulations illustrate the behavior of solutions of the recursion whose initial values vanish outside a bounded set.

This work presents an example of a cooperative system of truncated linear recursions in which the interaction between species causes one of the species to have an anomalous spreading speed. By this we mean that this species spreads at a speed which is strictly greater than its spreading speed in isolation from the other species and the speeds at which all the other species actually spread. An ecological implication of this example is discussed in Sect. 5. Our example shows that the formula for the fastest spreading speed given in Lemma 2.3 of our paper (Weinberger et al. in J Math Biol 45:183–218, 2002) is incorrect. However, we find an extra hypothesis under which the formula for the faster spreading speed given in (Weinberger et al. in J Math Biol 45:183–218, 2002) is valid. We also show that the hypotheses of all but one of the theorems of (Weinberger et al. in J Math Biol 45:183–218, 2002) whose proofs rely on Lemma 2.3 imply this extra hypothesis, so that all but one of the theorems of (Weinberger et al. in J Math Biol 45:183–218, 2002) and all the examples given there are valid as they stand.

It is well known that in many scalar models for the spread of a fitter phenotype or species into the territory of a less fit one, the asymptotic spreading speed can be characterized as the lowest speed of a suitable family of traveling waves of the model. Despite a general belief that multi-species (vector) models have the same property, we are unaware of any proof to support this belief. The present work establishes this result for a class of multi-species model of a kind studied by Lui [Biological growth and spread modeled by systems of recursions. I: Mathematical theory, Math. Biosci. 93 (1989) 269] and generalized by the authors [Weinberger et al., Analysis of the linear conjecture for spread in cooperative models, J. Math. Biol. 45 (2002) 183; Lewis et al., Spreading speeds and the linear conjecture for two-species competition models, J. Math. Biol. 45 (2002) 219]. Lui showed the existence of a single spreading speed c ∗ for all species. For the systems in the two aforementioned studies by the authors, which include related continuous-time models such as reaction-diffusion systems, as well as some standard competition models, it sometimes happens that different species spread at different rates, so that there are a slowest speed c ∗ and a fastest speed c f ∗ . It is shown here that, for a large class of such multi-species systems, the slowest spreading speed c ∗ is always characterized as the slowest speed of a class of traveling wave solutions.

It is shown that the methods previously used by the author [Wei82] and by R. Lui [Lui89] to obtain asymptotic spreading results and sometimes the existence of traveling waves for a discrete-time recursion with a translation invariant order preserving operator can be extended to a recursion with a periodic order preserving operator. The operator can be taken to be the time-one map of a continuous time reaction-diffusion model, or it can be a more general model of time evolution in population genetics or population ecology in a periodic habitat. Methods of estimating the speeds of spreading in various directions will also be presented.

The discrete-time recursion system $\u_{n+1}=Q[\u_n]$ with $\u_n(x)$ a vector of population distributions of species and $Q$ an operator which models the growth, interaction, and migration of the species is considered. Previously known results are extended so that one can treat the local invasion of an equilibrium of cooperating species by a new species or mutant. It is found that, in general, the resulting change in the equilibrium density of each species spreads at its own asymptotic speed, with the speed of the invader the slowest of the speeds. Conditions on $Q$ are given which insure that all species spread at the same asymptotic speed, and that this speed agrees with the more easily calculated speed of a linearized problem for the invader alone. If this is true we say that the recursion has a single speed and is linearly determinate. The conditions are such that they can be verified for a class of reaction-diffusion models.

If f(x) and g(x) are a Fourier cosine transform pair, then the Poisson summation formula can be written as 2sumfrominfinityn = 1g(n) + g(0) = 2sumfrominfinityn = 1f(n) + f(0). The concepts of linear transformation theory lead to the following dual of this classical relation. Let phi(x) and gamma(x) = phi(1/x)/x have absolutely convergent integrals over the positive real line. Let F(x) = sumfrominfinityn = 1phi(n/x)/x - integralinfinity0phi(t)dt and G(x) = sumfrominfinityn = 1gamma (n/x)/x - integralinfinity0 gamma(t)dt. Then F(x) and G(x) are a Fourier cosine transform pair. We term F(x) the "discrepancy" of phi because it is the error in estimating the integral phi of by its Riemann sum with the constant mesh spacing 1/x.

The proof of Lemma 2.4 is correct when the habitat is one-dimensional. However, the fact that the function ρ1 defined by formula (6.14) approaches zero on the part of the cone |x| = ct where x1 = 0 invalidates the argument that ρ1 ≤ ρ1 for |x| ≤ ct when the habitat is multidimensional. This gap is easily repaired by replacing the function coshμx1 in (6.14) by a radially symmetric positive solution (|x|) of the equation ∇2 = μ2 . When the habitat has dimension 2, one can let := I0(μ|x|) where I0 is the usual Bessel function with imaginary argument. In three dimensions, one can let := [sinhμ|x|]/[μ|x|].

This is a set of lecture notes for a very short course on the mathematical foundations of buckling theory. Since the critical buckling load can usually be characterized as the largest (or smallest) eigenvalue of a variational problem, the approximation of this load is a special case of the approximation of all the eigenvalues of a variational problem. A rather detailed analysis of this problem is to be found in the author’s book Variational Methods for Eigenvalue Approximation, CBMS Regional Conference Series in Applied Mathematics 15, SIAM, Philadelphia, 1974. We refer the interested reader to this book for more details and for references.

The asymptotic behavior as t----infinity of the solutions with values in the interval (0, 1) of a reaction-diffusion equation of the form (Formula: see text) is studied. Conditions on m which are satisfied when m is nonincreasing in mu and which imply that every solution converges to some periodic limit function are found. Except in some very special and well-defined circumstances, the limit is the same for all solutions, so that it is a global attractor. This global attractor may be one of the trivial solutions 0 or 1, or it may be a spatial-temporal cline. The linear stability properties of the trivial states serve to distinguish between these cases.

The purpose of this paper is to derive some inequalities of the form $${u _k} + R < {ambda _k} for k = 1, 2, ...$$ (1.1) between the eigenvalues λ 1 < λ 2 ≦ ... of the Dirichlet problem $$egin{gathered}{elta _u} + {ambda _u} = 0 in D ubset {R^N}, fill u = 0 on artial D fill nd{gathered} $$ (1.2) and the eigenvalues 0 = µ 1 < u 2 < ... of the Neumann problem $$ elta psilon + u psilon = 0 in D, fill rac{{artial psilon }}{{artial v}} = 0 on artial D fill nd{gathered} $$ (1.3) for some classes of N-dimensional domains D. Here ∂/∂v denotes the outward normal derivative.

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.

The 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?

The problem is, given A, B, C, to find D such that $\left\| ( {\begin{array}{*{20}c} A & C \\ B & D \\ \end{array} )} \right\| \leqq \mu $; here we deal with Hilbert-space operators, A, B, and C ar...

Let S be a bounded linear transformation from a. Hilbert space B to a Hilbert space Σ. Then Su can be thought of as the solution of a linear differential equation with right-hand side, initial data, or boundary data u . Given the incomplete information Nu = v , ∥ u ∥ B ⩽ 1 about the data, where N is a linear operator from B to a Euclidean space E n , and a linear interpolation M from E m to Σ, one defines the optimal approximation to Su to be the point Mâ(v) in the range of M which is the center of the smallest ball containing all points of the form Su with Nu = v and ∥ u ∥ B ⩽ 1 and centered in M . A characterization is given for the optimal approximation Mâ(v) . It is shown to be unique and, in general, nonlinear. Simpler approximations and relations with other concepts of optimality are investigated.

We study stationary solutions of a parabolic system arising in population dynamics. An uncountable infinity of such solutions exists. Solutions belonging to a certain class are shown to be stable in a suitable sense.

It is shown that for a given body and a given orientationgthere is always a position of the centre of mass which produces a stable falling motion in a very viscous fluid withgvertical and, in general, with a spin about the vertical axis. The corresponding terminal settling speed is bounded by means of several variational principles.Relations between the terminal speeds for falls with different downward directions and between the terminal speed and the geometry of the body are deduced. In particular, it is proved that for a large class of slender bodies the first approximation to the drag obtained from the slender-body theory of Burgers (1938) is correct. It follows that the ratio of the terminal speeds for falls with the long axis vertical and horizontal is near two.

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