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We investigate the connection between the existence of an explicit travelling wave solution and the travelling wave with minimal speed in a scalar monostable reaction-diffusion equation.

Compensated convex transforms have been introduced for extended real-valued functions defined over $\mathbb{R}^n$. In their application to image processing, interpolation, and shape interrogation, where one deals with functions defined over a bounded domain, one was making the implicit assumption that the function coincides with its transform at the boundary of the data domain. In this paper, we introduce local compensated convex transforms for functions defined in bounded open convex subsets $\Omega$ of $\mathbb{R}^n$ by making specific extensions of the function to the whole space, and establish their relations to globally defined compensated convex transforms via the mixed critical Moreau envelopes. We find that the compensated convex transforms of such extensions coincide with the local compensated convex transforms in the closure of $\Omega$. We also propose a numerical scheme for computing Moreau envelopes, establishing convergence of the scheme with the rate of convergence depending on the regularity of the original function. We give an estimate of the number of iterations needed for computing the discrete Moreau envelope. We then apply the local compensated convex transforms to image processing and shape interrogation. Our results are compared with those obtained by using schemes based on computing the convex envelope from the original definition of compensated convex transforms., Comment: 39 pages, 28 figures

We discuss minimality conditions for the speed of monotone travelling waves in a sample of smectic C* liquid crystal subject to a constant electric field, dealing with both isotropic and anisotropic cases. Such conditions are important in understanding the properties of domain wall switching across a smectic layer, and our focus here is on examining how the presence of anisotropy can affect the speed of this switching. We obtain an estimate of the influence of anisotropy on the minimal speed, sufficient conditions for linear and non-linear minimal speed selection mechanisms to hold in different parameter regimes, and a characterisation of the boundary separating the linear and non-linear regimes in parameter space.

We present a unified approach to characterising fast-reaction limits of systems of either two reaction–diffusion equations, or one reaction–diffusion equation and one ordinary differential equation, on unbounded domains, motivated by models of fast chemical reactions where either one or both reactant(s) is/are mobile. For appropriate initial data, solutions of four classes of problems each converge in the fast-reaction limit k→∞ to a self-similar limit profile that has one of four forms, depending on how many components diffuse and whether the spatial domain is a half or whole line. For fixed k, long-time convergence to these same self-similar profiles is also established, thanks to a scaling argument of Kamin. Our results generalise earlier work of Hilhorst, van der Hout and Peletier to a much wider class of problems, and provide a quantitative description of the penetration of one substance into another in both the fast-reaction and long-time regimes.

This paper is concerned with applications of the theory of approximation and interpolation based on compensated convex transforms developed in [K. Zhang, E. Crooks, A. Orlando, Compensated convexity methods for approximations and interpolations of sampled functions in Euclidean spaces: Theoretical Foundations. SIAM Journal on Mathematical Analysis 48 (2016) 4126-4154]. We apply our methods to $(i)$ surface reconstruction starting from the knowledge of finitely many level sets (or `contour lines'); $(ii)$ scattered data approximation; $(iii)$ image inpainting. For $(i)$ and $(ii)$ our methods give interpolations. For the case of finite sets (scattered data), in particular, our approximations provide a natural triangulation and piecewise affine interpolation. Prototype examples of explicitly calculated approximations and inpainting results are presented for both finite and compact sets. We also show numerical experiments for applications of our methods to high density salt and pepper noise reduction in image processing, for image inpainting and for approximation and interpolations of continuous functions sampled on finitely many level sets and on scattered points., Accepted to appear in SIAM Journal on Imaging Sciences

Energy harvesting by animals is important because it provides the power needed for all metabolic processes. Beyond this, efficient food finding enhances individual fitness [1] and population viability [2], although rates of energy accumulation are affected by the environment and food distribution. Typically, differences between individuals in the rate of food acquisition are attributed to varying competencies [3], even though food-encounter rates are known to be probabilistic [4]. We used animal-attached technology to quantify food intake in four disparate free-living vertebrates (condors, cheetahs, penguins, and sheep) and found that inter-individual variability depended critically on the probability of food encounter. We modeled this to reveal that animals taking rarer food, such as apex predators and scavengers, are particularly susceptible to breeding failure because this variability results in larger proportions of the population failing to accrue the necessary resources for their young before they starve and because even small changes in food abundance can affect this variability disproportionately. A test of our model on wild animals indicated why Magellanic penguins have a stable population while the congeneric African penguin population has declined for decades. We suggest that such models predicting probabilistic ruin can help predict the fortunes of species operating under globally changing conditions.

This paper is concerned with linear determinacy in monostable reaction- diffusion-convection equations and co-operative systems. We present sufficient conditions for minimal travelling-wave speeds (equivalent to spreading speeds) to equal values obtained from linearisations of the travelling-wave problem about the unstable equilibrium. These conditions involve both reaction and convection terms. We present separate conditions for non-increasing and non-decreasing travelling waves, called `right' and `left' conditions respectively, because of the asymmetry in propagation caused by the convection terms. We also give a necessary condition on the reaction term for the existence of convection terms such that both the right and left conditions can be satisfied simultaneously. Examples show that our sufficient conditions for linear determinacy are not necessary and compare these conditions in the scalar case with alternative conditions observed in Malaguti-Marcelli [15] and Benguria-Depassier-Mendez [3]. We also illustrate, for both an equation and a system, the existence of reaction and (non-trivial) convection terms for which the right and left linear determinacy conditions are simultaneously satisfied. An example is given of an equation which is right but not left linearly determinate.

We model the growth, dispersal and mutation of two phenotypes of a species using reaction&ndash, diffusion equations, focusing on the biologically realistic case of small mutation rates. Having verified that the addition of a small linear mutation term to a Lotka&ndash, Volterra system limits it to only two steady states in the case of weak competition, an unstable extinction state and a stable coexistence state, we exploit the fact that the spreading speed of the system is known to be linearly determinate to show that the spreading speed is a nonincreasing function of the mutation rate, so that greater mixing between phenotypes leads to slower propagation. We also find the ratio at which the phenotypes occur at the leading edge in the limit of vanishing mutation.

We prove the existence of additional unstable equilibria E ± for a class of “monostable” reaction–diffusion systems u t = A u x x + f ( u ) , u ( x , t ) ∈ R N , having known equilibria S T with S “stable” and T “unstable” such that the derivative d f [ T ] has an even number of unstable eigenvalues. The proofs use positive-operator theory and degree theory. Systems in the class considered are known to admit monotone increasing travelling-wave solutions w ( x − c t ) connecting ordered stable and unstable equilibria for all c greater than or equal to a positive critical value and the existence of E ± > S yields two one-parameter families of travelling waves connecting S to E ± in addition to the family connecting S to T .

We apply compensated convex transforms to define a multiscale Hausdorff stable method to extract intersections between smooth compact manifolds represented by their characteristic functions or as point clouds embedded in Rn. We prove extraction results on intersections of smooth compact manifolds and for points of high curvature. As a result of the Hausdorff?Lipschitz continuity of our transforms, we show that our method is stable against dense sampling of smooth manifolds with noise. Examples of explicitly calculated prototype models for some simple cases are presented, which are also used in the proofs of our main results. Numerical experiments in two- and three-dimensional space, and applications to geometric objects are also shown. Fil: Zhang, Kewei. The University of Nottingham; Reino Unido Fil: Orlando, Antonio. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de Tucuman. Facultad de Ciencias Exactas y Tecnologia. Instituto de Estructuras ; Argentina Fil: Crooks, Elaine. Swansea University; Reino Unido

We develop and apply the theory of lower and upper compensated convex transforms introduced in [K. Zhang, Compensated convexity and its applications, Ann. Inst. H. Poincaré Anal. Non Linéaire 25 (2008) 743?771] to define multiscale, parametrized, geometric singularity extraction transforms of ridges, valleys and edges of function graphs and sets in Rn. These transforms can be interpreted as "tight" opening and closing operators, respectively, with quadratic structuring functions. We show that these geometric morphological operators are invariant with respect to translation, and stable under curvature perturbations, and establish precise locality and tight approximation properties for compensated convex transforms applied to bounded functions and continuous functions. Furthermore, we establish multiscale and Hausdorff stable versions of such transforms. Specifically, the stable ridge transforms can be used to extract exterior corners of domains defined by their characteristic functions. Examples of explicitly calculated prototype mathematical models are given, as well as some numerical experiments illustrating the application of these transforms to 2d and 3d objects. Fil: Kewei Zhang. The University of Nottingham; Reino Unido Fil: Orlando, Antonio. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de Tucuman. Facultad de Ciencias Exactas y Tecnologia. Instituto de Estructuras "Ing. Arturo M. Guzman"; Argentina Fil: Elaine Crooks. Swansea University; Reino Unido

This article is concerned with the effect of different types of competitive interaction terms on the large-interaction limit of nonlinear elliptic systems modelling the steady states of populations that compete in some region. As the competition rate tends to infinity, we show that non-negative solutions of quite simple-looking systems converge to the positive and negative parts of a solution of a scalar limit problem which may be much more strongly nonlinear than the original system, possibly with quadratic growth in the gradient of the limit function.

We are concerned with the asymptotic behaviour of classical solutions of systems of the form $$ \cases u_{t} = A u_{xx} + f(u, u_{x}) &\text{for } x \in {\mathbb R},\ t> 0,\ u(x,t) \in {\mathbb R}^N,\\ u(x,0) = \varphi (x),& \endcases\tag{1} $$ where $A$ is a positive-definite diagonal matrix and $f$ is a "bistable" nonlinearity satisfying conditions which guarantee the existence of a comparison principle for (1). Suppose that (1) has a travelling-front solution $w$ with velocity $c$, that connects two stable equilibria of $f$. (There are hypotheses on $f$ under which such a front is known to exist [E. C. M. Crooks and J. F. Toland, Travelling waves for reaction-diffusion-convection systems , Topol. Methods Nonlinear Anal. 11 (1998), 19–43].) We show that if $\varphi$ is bounded, uniformly continuously differentiable and such that $\Vert w(x) - \varphi (x) \Vert $ is small when $|x|$ is large, then there exists $\chi \in {\mathbb R}$ such that $$ \Vert u(\cdot, t) - w(\cdot + \chi - ct) \Vert _{BUC^{1}} \rightarrow 0 \quad\text{as } t \rightarrow \infty.\tag{2} $$ Our approach extends an idea developed by Roquejoffre, Terman and Volpert in the convectionless case, where $f$ is independent of $u_{x}$. First $\varphi$ is assumed to be increasing in $x$, and (2) proved via a homotopy argument. Then we deduce the result for arbitrary $\varphi$ by showing that there is an increasing function in the $\omega$-limit set of $\varphi$.

(1) ut = Auxx + f(u), u ∈ R , x ∈ R, t ∈ [0,∞), where A is a real, positive-definite, N × N matrix and f : R → R is a continuously differentiable nonlinear function. The vector u may represent, for example, the concentrations of chemicals or the population densities of interacting species, the interactions between components of u being modelled by the reaction term f(u) and their diffusion by Auxx. Travelling waves are solutions u of (1) in the form

We construct families of front-like entire solutions for problems with convection, both for bistable and monostable reaction–diffusion–convection equations, and, via vanishing-viscosity arguments, for bistable and monostable balance laws. The unified approach employed is inspired by ideas of Chen and Guo and based on a robust method using front-dependent sub and supersolutions. Unlike convectionless problems, the equations studied here lack symmetry between increasing and decreasing travelling waves, which affects the choice of sub and supersolutions used. Our entire solutions include both those that behave like two fronts coming together and annihilating as time increases, and, for bistable equations, those that behave like two fronts merging to propagate like a single front. We also characterise the long-time behaviour of each family of entire solutions, which in the case of solutions constructed from a monostable front merging with a bistable front answers a question that was open even for reaction–diffusion equations without convection.

We introduce Lipschitz continuous and C1;1 geometric approximation and interpolation methods for sampled bounded uniformly continuous functions over compact sets and over complements of bounded open sets in Rn by using compensated convex transforms. Error estimates are provided for the approximations of bounded uniformly continuous functions, of Lipschitz functions, and of C1;1 functions. We also prove that our approximation methods, which are differentiation and integration free and not sensitive to sample type, are stable with respect to the Hausdorff distance between samples. Fil: Zhang, Kewei. The University of Nottingham; Reino Unido Fil: Crooks, Elaine. Swansea University; Reino Unido Fil: Orlando, Antonio. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Tucumán; Argentina. Universidad Nacional de Tucumán. Facultad de Ciencias Exactas y Tecnología. Instituto de Estructuras ; Argentina

We consider a two-component competition-diffusion system with equal diffusion coefficients and inhomogeneous Dirichlet boundary conditions. Provided certain conditions on a limit problem hold and provided that the competition rate is sufficiently large, all non-negative solutions of the system converge to a stationary solution of the system as $ t \rightarrow \infty$.

We apply upper and lower compensated convex transforms, which are `tight' one-sided approximations of a given function, to the extraction of fine geometric singularities from semiconvex/semiconcave functions and DC-functions in $\mathbb{R}^n$ (difference of convex functions). Well-known examples of (locally) semiconcave functions include the Euclidean distance and squared distance functions. For a locally semiconvex function $f$ with general modulus, we show that `locally' a point is a singular (non-differentiable) point if and only if it is a scale $1$-valley point, and if $x$ is a singular point, then locally the limit of the scaled valley transform exists at every point $x$ and $ \lim_{\lambda\to \infty}\lambda V_\lambda (f)(x)=r_x^2/4$, where $r_x$ is the radius of the minimal bounding sphere of the (Fr\'echet) subdifferential $\partial_- f(x)$ and $V_\lambda (f)(x)$ is the valley transform at $x$. Thus the limit function $\mathcal{V}_\infty(f)(x):=\lim_{\lambda\to+\infty}\lambda V_\lambda (f)(x)=r_x^2/4$ gives a `scale $1$-valley landscape function' of the singular set for a locally semiconvex function $f$, and also provides an asymptotic expansion of the upper transform $C^u_\lambda(f)(x)$ when $\lambda \to \infty$. For a locally semiconvex function $f$ with linear modulus we show that the limit of the gradient of the upper compensated convex transform $\lim_{\lambda\to+\infty}\nabla C^u_\lambda(f)(x)$ exists and equals the centre of the minimal bounding sphere of $\partial_- f(x$, and that for a DC-function $f=g-h$, the scale $1$-edge transform satisfies $\liminf_{\lambda\to+\infty}\lambda E_\lambda (f)(x)\geq (r_{g,x}-r_{h,x})^2/4$, where $r_{g,x}$ and $r_{h,x}$ are the radii of the minimal bounding spheres of the subdifferentials $\partial_- g$ and $\partial_- h$ of the convex functions $g$ and $h$ at $x$ respectively., Comment: A Chinese version of the material in this manuscript has been published in Zhang, Kewei, Crooks, Elaine and Orlando, Antonio, Compensated convex transforms and geometric singularity extraction from semiconvex functions (in Chinese), Sci. Sin. Math., 46 (2016) 747-768, doi: 10.1360/N012015-00339