Doubly nonlocal reaction-diffusion equation and the emergence of species

The paper is devoted to a reaction-diffusion equation with doubly nonlocal nonlinearity arising in various applications in population dynamics. One of the integral terms corresponds to the nonlocal consumption of resources while another one describes reproduction with different phenotypes. Linear stability analysis of the homogeneous in space stationary solution is carried out. Existence of travelling waves is proved in the case of narrow kernels of the integrals. Periodic travelling waves are observed in numerical simulations. Existence of stationary solutions in the form of pulses is shown, and transition from periodic waves to pulses is studied. In the applications to the speciation theory, the results of this work signify that new species can emerge only if they do not have common offsprings. Thus, it is shown how Darwin's definition of species as groups of morphologically similar individuals is related to Mayr's definition as groups of individuals that can breed only among themselves.Comment: 15 pages, 4 figure

Solvability Conditions for a Linearized Cahn-Hilliard Equation of Sixth Order

We obtain solvability conditions in H6(ℝ3) for a sixth order partial differential equation which is the linearized Cahn-Hilliard problem using the results derived for a Schrödinger type operator without Fredholm property in our preceding articl

The enhancement of weakly exothermic polymerization fronts

Abstract The propagation of one-dimensional waves resulting from chemical reactions in a sandwich-type two-layer setting is considered. One layer, termed the polymerization layer, contains the monomer and initiator molecules needed for the initiation of a self-propagating polymer front. The other layer will be referred to as the enhancement layer, and it contains the necessary reactants to support a highly exothermic self-propagating reaction wave. Heat exchange occurs between the layers, and as a result, there is a net diffusion of heat away from the region undergoing the more exothermic reaction. As frontal polymerization (FP) reactions are known not to be very exothermic, an overall transfer of heat from the enhancement layer into the polymerization layer takes place. An analysis of the basic state of the system is carried out to investigate the effect of heat transfer on the polymerization reaction. An enhancement layer is shown to promote FP. This analysis is applicable to the manufacture of thin polymer films by FP

Mean Field Effects for Counterpropagating Traveling Wave Solutions of Reaction-Diffusion Systems

In many problems, e.g., in combustion or solidification, one observes traveling waves that propagate with constant velocity and shape in the x direction, say, are independent of y and z and describe transitions between two equilibrium states, e.g., the burned and the unburned reactants. As parameters of the system are varied, these traveling waves can become unstable and give rise to waves having additional structure, such as traveling waves in the y and z directions, which can themselves be subject to instabilities as parameters are further varied. To investigate this scenario we consider a system of reaction-diffusion equations with a traveling wave solution as a basic state. We determine solutions bifurcating from the basic state that describe counterpropagating traveling waves in directions orthogonal to the direction of propagation of the basic state and determine their stability. Specifically, we derive long wave modulation equations for the amplitudes of the counterpropagating traveling waves that are coupled to an equation for a mean field, generated by the translation of the basic state in the direction of its propagation. The modulation equations are then employed to determine stability boundaries to long wave perturbations for both unidirectional and counterpropagating traveling waves. The stability analysis is delicate because the results depend on the order in which transverse and longitudinal perturbation wavenumbers are taken to zero. For the unidirectional wave we demonstrate that it is sufficient to consider the cases of (i) purely transverse perturbations, (ii) purely longitudinal perturbations, and (iii) longitudinal perturbations with a small transverse component. These yield Eckhaus type, zigzag type, and skew type instabilities, respectively. The latter arise as a specific result of interaction with the mean field. We also consider the degenerate case of very small group velocity, as well as other degenerate cases, which yield several additional instability boundaries. The stability analysis is then extended to the case of counterpropagating traveling waves

PPARγ and Agonists against Cancer: Rational Design of Complementation Treatments

PPARγ is a member of the ligand-activated nuclear receptor superfamily: its ligands act as insulin sensitizers and some are approved for the treatment of metabolic disorders in humans. PPARγ has pleiotropic effects on survival and proliferation of multiple cell types, including cancer cells, and is now subject of intensive preclinical cancer research. Studies of the recent decade highlighted PPARγ role as a potential modulator of angiogenesis in vitro and in vivo. These observations provide an additional facet to the PPARγ image as potential anticancer drug. Currently PPARγ is regarded as an important target for the therapies against angiogenesis-dependent pathological states including cancer and vascular complications of diabetes. Some of the studies, however, identify pro-angiogenic and tumor-promoting effects of PPARγ and its ligands pointing out the need for further studies. Below, we summarize current knowledge of PPARγ regulatory mechanisms and molecular targets, and discuss ways to maximize the beneficial activity of the PPARγ agonists

Метод монотонных решений для уравнений реакции-диффузии

Existence of solutions of reaction-diffusion systems of equations in unbounded domains is studied by the Leray-Schauder (LS) method based on the topological degree for elliptic operators in unbounded domains and on a priori estimates of solutions in weighted spaces. We identify some reactiondiffusion systems for which there exist two subclasses of solutions separated in the function space, monotone and non-monotone solutions. A priori estimates and existence of solutions are obtained for monotone solutions allowing to prove their existence by the LS method. Various applications of this method are given.Методом Лере-Шаудера, основанном на топологической степени эллиптических операторов в неограниченных областях и на априорных оценках решений в весовых пространствах, изучается существование решений систем уравнений реакции-диффузии в неограниченных областях. Мы выделяем некоторые системы реакции-диффузии, для которых существуют два подкласса решений, отделенных друг от друга в функциональном пространстве: монотонные и немонотонные решения. Для монотонных решений получены априорные оценки, позволяющие доказать их существование методом Лере-Шаудера. Приводятся различные приложения этого метода

Non-homogeneous random walks, subdiffusive migration of cells and anomalous chemotaxis

This paper is concerned with a non-homogeneous in space and non-local in time random walk model for anomalous subdiffusive transport of cells. Starting with a Markov model involving a structured probability density function, we derive the non-local in time master equation and fractional equation for the probability of cell position. We show the structural instability of fractional subdiffusive equation with respect to the partial variations of anomalous exponent. We find the criteria under which the anomalous aggregation of cells takes place in the semi-infinite domain.Comment: 18 pages, accepted for publicatio

Stratégie multi-méthodes dans le domaine temporel

International audienceDans cet article nous présentons une stratégie multi-méthodes pour la simulation de problèmes de CEM. Dans cette approche, nous utilisons des méthodes d'ordre élevé permettant de tenir compte de la courbure des géométries et de limiter les erreurs de dispersions et/ou de dissipation. Ces méthodes sont basées sur des schémas Galerkin Discontinu et différences finies utilisant une approximation spatiale d'ordre élevé. Enfin pour tenir compte des câbles dans les structures, nous utilisons une équation de ligne de transmission, dans le domaine temporel que nous couplons aux méthodes de calcul de champs 3D

Preface. Bifurcations and Pattern Formation in Biological Applications

In the preface we present a short overview of articles included in the issue "Bifurcations and pattern formation in biological applications" of the journal Mathematical Modelling of Natural Phenomena