Travelling wave phenomena in non-linear diffusion degenerate Nagumo equations

 In this paper we study the existence of one-dimensional travelling wave solutions u(x, t)=φ(x−ct) for the non-linear degenerate (at u=0) reaction-diffusion equation u _t =[D(u)u _x ] _x +g(u) where g is a generalisation of the Nagumo equation arising in nerve conduction theory, as well as describing the Allee effect. We use a dynamical systems approach to prove: 1. the global bifurcation of a heteroclinic cycle (two monotone stationary front solutions), for c=0, 2. The existence of a unique value c ^*>0 of c for which φ(x−c ^* t) is a travelling wave solution of sharp type and 3. A continuum of monotone and oscillatory fronts for c≠c ^*. We present some numerical simulations of the phase portrait in travelling wave coordinates and on the full partial differential equation. Received 15 December 1995; received in revised form 14 May 1996     

Existence and stability of periodic travelling wave solutions to Nagumo's nerve equation

Summary Nagumo's nerve conduction equation has a one-parameter family of spatially periodic travelling wave solutions. First, we prove the existence of these solutions by using a topological method. (There are some exceptional cases in which this method cannot be applied in showing the existence.) A periodic travelling wave solution corresponds to a closed orbit of a third-order dynamical system. The Poincaré index of the closed orbit is determined as a direct consequence of the proof of the existence. Second, we prove that the periodic travelling wave solution is unstable if the Poincaré index of the corresponding closed orbit is + 1. By using this result, together with the result of the author's previous paper, it is concluded that the slow periodic travelling wave solutions are always unstable. Third, we consider the stability of the fast periodic travelling wave solutions. We denote by L(c) the spatial period of the travelling wave solution with the propagation speed c. It is shown that the fast solution is unstable if its period is close to L_min, the minimum of L(c).     

Segregation and mixing in degenerate diffusion in population dynamics

 We study the qualitative properties of degenerate diffusion equations used to describe dispersal processes in population dynamics. For systems of interacting populations, the forms of the diffusion models used determine if the population will intermix or remain disjoint (segregated). The dynamics and stability of segregation boundaries between competing populations is analyzed. General population models with segregation and mixing interactions are derived and connections to behavior in fluid mechanical systems are addressed. Received 19 January 1996; received in revised form 4 April 1996     

From periodic travelling waves to travelling fronts in the spike-diffuse-spike model of dendritic waves

In the vertebrate brain excitatory synaptic contacts typically occur on the tiny evaginations of neuron dendritic surface known as dendritic spines. There is clear evidence that spine heads are endowed with voltage-dependent excitable channels and that action potentials invade spines. Computational models are being increasingly used to gain insight into the functional significance of a spine with an excitable membrane. The spike-diffuse-spike (SDS) model is one such model that admits to a relatively straightforward mathematical analysis. In this paper we demonstrate that not only can the SDS model support solitary travelling pulses, already observed numerically in more detailed biophysical models, but that it has periodic travelling wave solutions. The exact mathematical treatment of periodic travelling waves in the SDS model is used, within a kinematic framework, to predict the existence of connections between two periodic spike trains of different interspike interval. The associated wave front in the sequence of interspike intervals travels with a constant velocity without degradation of shape, and might therefore be used for the robust encoding of information.     

一类具有分布时滞的扩散种群模型行波解的存在性

研究了一类具有分布时滞的扩散种群模型行波解的存在性,证明了当平均时滞充分小时,方程具有连接两个平衡点的单调行波解.     

Activation waves in a model of platelet aggregation: existence of solutions and stability of travelling fronts

Platelets cohere to one another to form platelet aggregates as part of the blood's clotting response. The ability of a platelet to participate in this process depends on its prior activation by chemicals released into the blood plasma by other activated platelets. We study the piecewise-linear system of reaction-diffusion equations which, in one spatial dimension, describe the chemically-mediated spread of platelet activation. We establish the existence of classical solutions to this system of equations, and show that these solutions do not blow up in finite time. We also explicitly construct travelling front solutions and discuss their stability. Finally, we present numerical evidence which suggests that for a broad range of initial data with the correct limiting values at ± , the solution to the initial value problem rapidly evolves into the travelling front solution provided the front is linearly stable.     

One way traffic of pulses in a neuron

In this paper we investigate the effect of a change in geometry of a nerve axon on the propagation of potential waves along the axon. In particular we show that potential waves are stopped at a sudden large increase of cross-section area such as increase of diameter or branching. Some special examples are treated. The results do also apply to problems in population genetics and chemical reaction theory.     

Large time behavior in a nonlinear age-dependent population dynamics problem with spatial diffusion

In this work we analyze the large time behavior in a nonlinear model of population dynamics with age-dependence and spatial diffusion. We show that when t+ either the solution of our problem goes to 0 or it stabilizes to a nontrivial stationary solution. We give two typical examples where the stationary solutions can be evaluated upon solving very simple partial differential equations. As a by-product of the extinction case we find a necessary condition for a nontrivial periodic solution to exist. Numerical computations not described below show a rapid stabilization.This work was partially supported by the Centre National de la Recherche Scientifique through ATP 95939900     

Biological Control Through Intraguild Predation: Case Studies in Pest Control,Invasive Species and Range Expansion

Intraguild predation (IGP), the interaction between species that eat each other and compete for shared resources, is ubiquitous in nature. We document its occurrence across a wide range of taxonomic groups and ecosystems with particular reference to non-indigenous species and agricultural pests. The consequences of IGP are complex and difficult to interpret. The purpose of this paper is to provide a modelling framework for the analysis of IGP in a spatial context. We start by considering a spatially homogeneous system and find the conditions for predator and prey to exclude each other, to coexist and for alternative stable states. Management alternatives for the control of invasive or pest species through IGP are presented for the spatially homogeneous system. We extend the model to include movement of predator and prey. In this spatial context, it is possible to switch between alternative stable steady states through local perturbations that give rise to travelling waves of extinction or control. The direction of the travelling wave depends on the details of the nonlinear intraguild interactions, but can be calculated explicitly. This spatial phenomenon suggests means by which invasions succeed or fail, and yields new methods for spatial biological control. Freshwater case studies are used to illustrate the outcomes.     

Some remarks about the wave speed and travelling wave solutions of a nonlinear integral operator

In this paper, we extend the existence of travelling wave solutions of a nonlinear operator in the inferior case to higher dimensions and establish the uniqueness of that solution. In addition we determine the sign of the wave speed in a special case.     

A model of herd grazing as a travelling wave,chemotaxis and stability

Possible constitutive models are examined for the formation of a herd, under the assumption that a herd forms a travelling wave while grazing. Under quite general conditions, it is found that the only possibility for a travelling wave is a balance between food seeking and natural dispersion, such as in chemotaxis. The stability of the travelling wave previously conjectured, is shown both for one- and two-dimensional perturbations.     

Effects of macromolecular crowding on intracellular diffusion from a single particle perspective

Compared to biochemical reactions taking place in relatively well-defined aqueous solutions in vitro, the corresponding reactions happening in vivo occur in extremely complex environments containing only 60-70% water by volume, with the remainder consisting of an undefined array of bio-molecules. In a biological setting, such extremely complex and volume-occupied solution environments are termed 'crowded'. Through a range of intermolecular forces and pseudo-forces, this complex background environment may cause biochemical reactions to behave differently to their in vitro counterparts. In this review, we seek to highlight how the complex background environment of the cell can affect the diffusion of substances within it. Engaging the subject from the perspective of a single particle's motion, we place the focus of our review on two areas: (1) experimental procedures for conducting single particle tracking experiments within cells along with methods for extracting information from these experiments; (2) theoretical factors affecting the translational diffusion of single molecules within crowded two-dimensional membrane and three-dimensional solution environments. We conclude by discussing a number of recent publications relating to intracellular diffusion in light of the reviewed material.     

Dynamics of a transmural scroll wave in ground squirrel myocardium

A study was made of tachyarrhythmia evoked by a premature stimulus (4 ms at 4–5 diastolic thresholds) following a train of rectangular pulses (4 ms at 2 diastolic thresholds, repetition rate of 0.5 or 2 s^?1). The spatiotemporal distribution of the potential over the endo- and epicardial surfaces of a thin (～1 mm) specimen of ground squirrel ventricular myocardium was monitored with two arrays of 32 unipolar electrodes each. The electrographic data were processed into isochrone maps reflecting the spread of activation over the surfaces. These maps were further analyzed to infer the 3D structure and dynamics of the vortex (scroll) wave. During the evolution of a transmural scroll, (i) the filament could be normal to the myocardial surfaces as well as oblique at varying angles (down to 12°); (ii) the scroll could drift as a whole, whereby the filament remained self-parallel or changed its inclination; in other cases, one (endocardial) core was anchored while the other changed its position (precession of the filament); (iii) the vortex cores on both or only one surface changed in size and shape; (iv) the filament could be repeatedly twisted through various angles and untwisted. Scroll rotation was attended with excitation breakthroughs that might have originated from filament bending as well as from focal sources.     

Development of a gas diffusion probe for the determination of methane concentrations and diffusion characteristics in flooded paddy soil

Abstract A probe for the measurement of dissolved CH₄ in anoxic methanogenic environments was developed. The probe was based on the diffusion of dissolved CH₄ through a silicone membrane into a gas space at the end of the probe. This gas space was flushed with N₂ and analyzed gas-chromatographically for CH₄. The probe had a spatial resolution of < 1.3 mm, the detection limit was about 20 μM CH₄, the precision of the measurement was 9%, and consecutive measurements could be made every 4 min. Memory effects after analysis of high CH₄ concentrations could be avoided by flushing the probe with N₂ between each measurement. The probe was sensitive for water movement and, therefore, was calibrated in an artificial sediment of glass beads (100 μm diam.) immersed by aqueous solutions of known CH₄ concentrations. Sensitivity of the probe for changes in the sediment's porosity could not presently be excluded. The probe was used to measure vertical profiles of dissolved CH₄ in microcosms of anoxic paddy soil. The vertical CH₄ profiles measured with the probe compared fairly well with those measured after an extraction procedure. The profiles clearly showed that CH₄ was produced in deeper layers and diffused upwards to be consumed in the oxic top 2 mm soil layers. The probe was also used to determine the diffusion coefficient of CH₄ in an inactivated paddy soil microcosm using a set-up which allowed modelling of a measured CH₄ concentration profile using Fick's 2nd law.     

Kinetics and thermodynamics of gas diffusion in a NiFe hydrogenase

We have investigated O₂ and H₂ transport across a NiFe hydrogenase at the atomic scale by means of computational methods. The Wild Type protein has been compared with the V74Q mutant. Two distinct methodologies have been applied to study the gas access to the active site. Temperature locally enhanced sampling simulations have emphasized the importance of protein dynamics on gas diffusion. The O₂ diffusion free energy profiles, obtained by umbrella sampling, are in agreement with the known kinetic data and show that in the V74Q mutant, the inhibition process is lowered from both a kinetic and a thermodynamic point of view. Proteins 2011. © 2012 Wiley Periodicals, Inc.     

三种群互惠模型抛物系统解的整体存在与爆破

研究了三种群互惠模型的抛物系统,用上下解方法研究解的整体存在性与爆破问题,并给出了相应的条件.结果表明当种群自身的竞争强时解整体存在,反之则有可能爆破.     

Statistical aspects of design and validation of microtitre-plate-based linear and non-linear parallel in vitro bioassays

Assay validation was performed using four consecutive experiments with the related statistical evaluation. A cell-based assay on microtitre plates measured repeatedly within 1 day and on consecutive days was chosen as the model. The following problems were addressed: (i) choosing an appropriate design on a plate to avoid heterogeneities, (ii) quantification of all sources of variability and (iii) selecting between linear and non-linear parallel line assays. A mixed model was used with the random factors: rows, columns and plates and fixed effect factors with either linear or non-linear parallel line models.     

Craniometric data support a mosaic model of demic and cultural Neolithic diffusion to outlying regions of Europe

The extent to which the transition to agriculture in Europe was the result of biological (demic) diffusion from the Near East or the adoption of farming practices by indigenous hunter-gatherers is subject to continuing debate. Thus far, archaeological study and the analysis of modern and ancient European DNA have yielded inconclusive results regarding these hypotheses. Here we test these ideas using an extensive craniometric dataset representing 30 hunter-gatherer and farming populations. Pairwise population craniometric distance was compared with temporally controlled geographical models representing evolutionary hypotheses of biological and cultural transmission. The results show that, following the physical dispersal of Near Eastern/Anatolian farmers into central Europe, two biological lineages were established with limited gene flow between them. Farming communities spread across Europe, while hunter-gatherer communities located in outlying geographical regions adopted some cultural elements from the farmers. Therefore, the transition to farming in Europe did not involve the complete replacement of indigenous hunter-gatherer populations despite significant gene flow from the Southwest Asia. This study suggests that a mosaic process of dispersal of farmers and their ideas was operating in outlying regions of Europe, thereby reconciling previously conflicting results obtained from genetic and archaeological studies.     

一类扩散循环系统正解的存在性与稳定性

利用上下解方法及全藕合线性互惠系统的最大值原理,研究了一类非线性椭圆系统,给出了其正解存在的充分必要条件,同时也得到了其正解局部稳定的某些结果．