Minimal speed of fronts of reaction-convection-diffusion equations

We study the minimal speed of propagating fronts of convection reaction diffusion equations of the form $u_t + \mu \phi(u) u_x = u_{xx} +f(u)$ for positive reaction terms with $f'(0 >0$. The function $\phi(u)$ is continuous and vanishes at $u=0$. A variational principle for the minimal speed of the waves is constructed from which upper and lower bounds are obtained. This permits the a priori assesment of the effect of the convective term on the minimal speed of the traveling fronts. If the convective term is not strong enough, it produces no effect on the minimal speed of the fronts. We show that if $f''(u)/\sqrt{f'(0)} + \mu \phi'(u) < 0$, then the minimal speed is given by the linear value $2 \sqrt{f'(0)}$, and the convective term has no effect on the minimal speed. The results are illustrated by applying them to the exactly solvable case $u_t + \mu u u_x = u_{xx} + u (1 -u)$. Results are also given for the density dependent diffusion case $u_t + \mu \phi(u) u_x = (D(u)u_x)_x +f(u)$.Comment: revised, new results adde

A nonhomogeneous boundary value problem in mass transfer theory

We prove a uniqueness result of solutions for a system of PDEs of Monge-Kantorovich type arising in problems of mass transfer theory. The results are obtained under very mild regularity assumptions both on the reference set $\Omega\subset\mathbf{R}^n$, and on the (possibly asymmetric) norm defined in $\Omega$. In the special case when $\Omega$ is endowed with the Euclidean metric, our results provide a complete description of the stationary solutions to the tray table problem in granular matter theory.Comment: 22 pages, 2 figure

Three-points interfacial quadrature for geometrical source terms on nonuniform grids

International audienceThis paper deals with numerical (finite volume) approximations, on nonuniform meshes, for ordinary differential equations with parameter-dependent fields. Appropriate discretizations are constructed over the space of parameters, in order to guarantee the consistency in presence of variable cells' size, for which $L^p$-error estimates, $1\le p < +\infty$, are proven. Besides, a suitable notion of (weak) regularity for nonuniform meshes is introduced in the most general case, to compensate possibly reduced consistency conditions, and the optimality of the convergence rates with respect to the regularity assumptions on the problem's data is precisely discussed. This analysis attempts to provide a basic theoretical framework for the numerical simulation on unstructured grids (also generated by adaptive algorithms) of a wide class of mathematical models for real systems (geophysical flows, biological and chemical processes, population dynamics)

Can a Species Keep Pace with a Shifting Climate?

Consider a patch of favorable habitat surrounded by unfavorable habitat and assume that due to a shifting climate, the patch moves with a fixed speed in a one-dimensional universe. Let the patch be inhabited by a population of individuals that reproduce, disperse, and die. Will the population persist? How does the answer depend on the length of the patch, the speed of movement of the patch, the net population growth rate under constant conditions, and the mobility of the individuals? We will answer these questions in the context of a simple dynamic profile model that incorporates climate shift, population dynamics, and migration. The model takes the form of a growth-diffusion equation. We first consider a special case and derive an explicit condition by glueing phase portraits. Then we establish a strict qualitative dichotomy for a large class of models by way of rigorous PDE methods, in particular the maximum principle. The results show that mobility can both reduce and enhance the ability to track climate change that a narrow range can severely reduce this ability and that population range and total population size can both increase and decrease under a moving climate. It is also shown that range shift may be easier to detect at the expanding front, simply because it is considerably steeper than the retreating back

Evolutionary dynamics of Lewis signaling games: signaling systems vs. partial pooling

Transfer of information between senders and receivers, of one kind or another, is essential to all life. David Lewis introduced a game theoretic model of the simplest case, where one sender and one receiver have pure common interest. How hard or easy is it for evolution to achieve information transfer in Lewis signaling?. The answers involve surprising subtleties. We discuss some if these in terms of evolutionary dynamics in both finite and infinite populations, with and without mutation

Ein inverses Eigenwertproblem

AbstractThe paper ios concerned with the problem of finding a real, diagonal matrix M such that A + M has prescribed eigenvalues where A is any given symmetric matrix. This problem represents a discrete analog of the inverse eigenvalue problem in which we seek to determine a “potential” g(x) such that the operator in Hilbert space, l(y) = − y″ + g(x)y with appropriate boundary conditions, possesses a prescribed spectrum.Section 2 defines the “inverse problem” and its generalization.In Section 3, upper and lower bounds are given for the eigenvalues of a symmetric matrix, obtained by the iterative application of Temple's theorem. Also, an estimate is given for the changes in the eigenvalues produced by perturbing a given symmetric matrix by a diagonal matrix.Section 4 contains a complete discussion of the case of 2 × 2 matrices.In Section 5, from simple considerations concerning rotations in an n(n + 1)2-dimensional space, necessary conditions are given for the inverse problem to have a solution.In Section 6, the Brouwer fixed point theorem is applied to an operator equation which is equivalent to the “inverse problem.” A sufficient condition is given for the problem to have a solution: let k denote a certain seminorm on the set of all symmetric matrices and let d be the minimal mutual distance of the given eigenvalues. Then the inverse problem has a solution provided k(A)d ⩽ 0.288.Under only slightly stronger conditions (k(A)d ⩽ 0.25), it is shown in Section 7, using the Banach contraction theorem, that the iteration converges and the solution is locally unique.Section 8 contains a numerical example