Search

We deal with the bistable reaction-diffusion equation on an unbounded metric graph consisting of half-lines and triple junctions. It is known that there exists an entire solution which asymptotically converges to the traveling front profile as t → − ∞ in the infinity of a half-line. Analyzing the behavior of the entire solution, we examine the condition that the front can propagate beyond the junctions of the graph having a tree-structure and a symmetry. As a result, the condition for blocking of the propagation depends not only a condition at the junctions but also the length between the neighboring junctions, specifically, in a certain case long segments between the two junctions can prevent blocking of the propagation. [ABSTRACT FROM AUTHOR]

We analyze the transition between pulled and pushed fronts both analytically and numerically from a model-independent perspective. Based on minimal conceptual assumptions, we show that pushed fronts bifurcate from a branch of pulled fronts with an effective speed correction that scales quadratically in the bifurcation parameter. Strikingly, we find that in this general context without assumptions on comparison principles, the pulled front loses stability and gives way to a pushed front when monotonicity in the leading edge is lost. Our methods rely on far-field core decompositions that identify explicitly asymptotics in the leading edge of the front. We show how the theoretical construction can be directly implemented to yield effective algorithms that determine spreading speeds and bifurcation points with exponentially small error in the domain size. Example applications considered here include an extended Fisher-KPP equation, a Fisher–Burgers equation, negative taxis in combination with logistic population growth, an autocatalytic reaction, and a Lotka-Volterra model. [ABSTRACT FROM AUTHOR]

The two-component Lotka-Volterra competition-diffusion system is well accepted as a model describing the invasion of a superior species into a new habitat. Under a bistable condition, we deal with the system in a domain of half-lines with a single junction and investigate the condition for the invasion from some of the half-lines beyond the junction or blocking the propagation of the superior species. We first give a sufficient condition for the invasion in the whole domain by a subsolution. Then, making use of sub- and supersolutions, we construct a standing front solution blocking the propagation if the number of half-lines occupied by the inferior species is sufficiently larger than that occupied by the superior species. [ABSTRACT FROM AUTHOR]

We study the effective fronts of first order front propagations in two dimensions (n = 2) in the periodic setting. Using PDE-based approaches, we show that for every a G (0,1), the class of centrally symmetric polygons with rational vertices (i.e., vectors in Uagr AZ2) and nonempty interior is admissible as effective fronts for front speeds in C1,a(T2, (0, oo)). This result can also be formulated in the language of stable norms corresponding to periodic metrics in T2. Similar results were known long ago when n > 3 for front speeds in Cco(Tn, (0, oo)). The two-dimensional case is much more subtle due to topological restrictions. In fact, for given C1,1(T2, (0, oo)) front speeds, the effective front is C1 and hence cannot be a polygon. Our regularity requirements on front speeds are hence optimal. To the best of our knowledge, this is the first time that polygonal effective fronts have been constructed in two dimensions. [ABSTRACT FROM AUTHOR]

Bacterial cells navigate their environment by directing their movement along chemical gradients. This process, known as chemotaxis, can promote the rapid expansion of bacterial populations into previously unoccupied territories. However, despite numerous experimental and theoretical studies on this classical topic, chemotaxis-driven population expansion is not understood in quantitative terms. Building on recent experimental progress, we here present a detailed analytical study that provides a quantitative understanding of how chemotaxis and cell growth lead to rapid and stable expansion of bacterial populations. We provide analytical relations that accurately describe the dependence of the expansion speed and density profile of the expanding population on important molecular, cellular, and environmental parameters. In particular, expansion speeds can be boosted by orders of magnitude when the environmental availability of chemicals relative to the cellular limits of chemical sensing is high. Analytical understanding of such complex spatiotemporal dynamic processes is rare. Our analytical results and the methods employed to attain them provide a mathematical framework for investigations of the roles of taxis in diverse ecological contexts across broad parameter regimes.

The current article is concerned with the traveling fronts for a class of double degenerate equations with delay. We first show that the traveling fronts decay algebraically at one end, while those may decay exponentially or algebraically at the other end, which depend on the wave speed of traveling fronts. Based on the asymptotical behavior, the uniqueness and stability of traveling fronts are then proved. Of particular interest is the effect of the lower order term and higher order term on the critical speed. We mention that, under the double degenerate case, the nonlinear reaction is less competitive due to the appearance of degeneracy. This yields that the critical speed depends on the lower order term and higher order term, which is different from the nondegenerate case.

This paper deals with three fundamental issues associated with theoretical analyses of reactive infiltration instability (RII) problems in fluid‐saturated porous media. The first fundamental issue is to determine the spatial shapes of chemical dissolution‐fronts in the limit case of the mineral dissolution ratio approaching zero in both conventional time (daily life time) and unconventional time (abstract time) domains. The chemical dissolution‐front is commonly represented by the spatial shape of the porosity profile. The second fundamental issue is to conduct theoretical analyses of the RII problems associated with the mineral dissolution ratio approaching zero through directly solving dimensional mathematical governing equations in the conventional time domain. The third fundamental issue is to carry out theoretical analyses of the RII problems associated with the mineral dissolution ratio approaching zero through solving dimensionless mathematical governing equations in the unconventional time domain. Through purely mathematical deductions, it has been proven that: (1) the spatially sharp shape of a chemical dissolution‐front, which is theoretically predicted either at a much smaller timescale than the dissolution timescale associated with the daily life time or at a much larger timescale than the dissolution timescale associated with the daily life time, can be observed in the daily life time domain; (2) the theoretical instability criterion of an RII problem can be established directly at three different length‐scales in the daily life time domain; and (3) although the spatial shape of a chemical dissolution‐front at the dissolution timescale associated with the daily life time cannot be observed in the daily life time domain, the theoretical instability criterion of an RII problem at the dissolution timescale associated with the daily life time can be established when a physically consistent mathematical transform is used in the theoretical analysis. [ABSTRACT FROM AUTHOR]

Liquid–liquid phase separation has emerged as a fundamental mechanism underlying intracellular organization, with evidence for it being reported in numerous different systems. However, there is a growing concern regarding the lack of quantitative rigor in the techniques employed to study phase separation, and their ability to account for the complex nature of the cellular milieu, which affects key experimentally observable measures, such as the shape, size and transport dynamics of liquid droplets. Here, we bridge this gap by combining recent experimental data with theoretical predictions that capture the subtleties of nonlinear elasticity and fluid transport. We show that within a biologically accessible range of material parameters, phase separation is highly sensitive to elastic properties and can thus be used as a mechanical switch to rapidly transition between different states in cellular systems. Furthermore, we show that this active mechanically mediated mechanism can drive transport across cells at biologically relevant timescales and could play a crucial role in promoting spatial localization of condensates; whether cells exploit such mechanisms for transport of their constituents remains an open question. [ABSTRACT FROM AUTHOR]

The goal of this work is to understand and quantify how a line with nonlocal diffusion given by an integral enhances a reaction-diffusion process occurring in the surrounding plane. This is part of a long term programme where we aim at modelling, in a mathematically rigorous way, the effect of transportation networks on the speed of biological invasions or propagation of epidemics. We prove the existence of a global propagation speed and characterise in terms of the parameters of the system the situations where such a speed is boosted by the presence of the line. In the course of the study we also uncover unexpected regularity properties of the model. On the quantitative side, the two main parameters are the intensity of the diffusion kernel and the characteristic size of its support. One outcome of this work is that the propagation speed will significantly be enhanced even if only one of the two is large, thus broadening the picture that we have already drawn in our previous works on the subject, with local diffusion modelled by a standard Laplacian. We further investigate the role of the other parameters, enlightening some subtle effects due to the interplay between the diffusion in the half plane and that on the line. Lastly, in the context of propagation of epidemics, we also discuss the model where, instead of a diffusion, displacement on the line comes from a pure transport term., (© The Author(s) 2024. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.)

We consider here a model of accelerating fronts, consisting of one equation with nonlocal diffusion on a line, coupled via the boundary condition with a reaction-diffusion equation of the Fisher-KPP type in the upper half-plane. It was proposed in a previous work by H. Berestycki, L. Rossi and the authors, as a mechanism of front acceleration by a line of fast diffusion. In this latter work, it was indeed proved that the propagation in the direction of the line was exponentially fast in time. Inspired by numerical simulations of the first author, we make the estimate more precise by computing a time algebraic correction.

This study investigates transient wave dynamics in Turing pattern formation, focusing on waves emerging from localised disturbances. While the traditional focus of diffusion-driven instability has primarily centred on stationary solutions, considerable attention has also been directed towards understanding spatio-temporal behaviours, particularly the propagation of patterning from localised disturbances. We analyse these waves of patterning using both the well-established marginal stability criterion and weakly nonlinear analysis with envelope equations. Both methods provide estimates for the wave speed but the latter method, in addition, approximates the wave profile and amplitude. We then compare these two approaches analytically near a bifurcation point and reveal that the marginal stability criterion yields exactly the same estimate for the wave speed as the weakly nonlinear analysis. Furthermore, we evaluate these estimates against numerical results for Schnakenberg and CDIMA (chlorine dioxide–iodine–malonic acid) kinetics. In particular, our study emphasises the importance of the characteristic speed of pattern propagation, determined by diffusion dynamics and a complex relation with the reaction kinetics in Turing systems. This speed serves as a vital parameter for comparison with experimental observations, akin to observed pattern length scales. Furthermore, more generally, our findings provide systematic methodologies for analysing transient wave properties in Turing systems, generating insight into the dynamic evolution of pattern formation. • Near a bifurcation point, marginal stability criterion and envelope equations give exactly the same results. • There is a unique travelling wave depositing a pattern from a localised perturbation. • The travelling wave speed may also be used to discriminate models in applications. [ABSTRACT FROM AUTHOR]

The propagation of one and two-dimensional waves of populations are numerically investigated in the framework of the "predator-prey" model with the Arditi - Ginzburg trophic function. The propagation of prey and predator population waves and the propagation of co-existing populations' waves are considered. The simulations demonstrate that even in the case of an unstable quasi-equilibrium state of the system, which is established behind the front of a traveling wave, the propagation velocity of the joint population wave is a well-defined function. The calculated average propagation velocity of a cellular non-stationary wave front is determined uniquely for a given set of problem parameters. The estimations of the wave propagation velocity are obtained for both the case of a plane and cellular wave fronts of populations. The structure and velocity of outward propagating circular cellular wave are investigated to clarify the local curvature and scaling effects on the wave dynamics. [ABSTRACT FROM AUTHOR]

Although additive manufacturing (AM) allows for a large design freedom, there are some manufacturing limitations that have to be taken into consideration. One of the most restricting design rules is the minimum allowable overhang angle. To make topology optimization suitable for AM, several algorithms have been published to enforce a minimum overhang angle. In this work, the layer-by-layer overhang filter proposed by Langelaar (Struct Multidiscip Optim 55(3):871–883, 2017), and the continuous, front propagation-based, overhang filter proposed by van de Ven et al. (Struct Multidiscipl Optim 57(5):2075–2091, 2018) are compared in detail. First, it is shown that the discrete layer-by-layer filter can be formulated in a continuous setting using front propagation. Then, a comparison is made in which the advantages and disadvantages of both methods are highlighted. Finally, the continuous overhang filter is improved by incorporating complementary aspects of the layer-by-layer filter: continuation of the overhang filter and a parameter that had to be user-defined are no longer required. An implementation of the improved continuous overhang filter is provided. [ABSTRACT FROM AUTHOR]

This work considers an expansion of a dense monolayer of cells: a collective multicellular phenomenon, where cells divide, grow, and maintain contacts with their neighbors. During migration, cells display complex behavior, adjusting both their division rate and their growth after division to the local mechanical environment. Experimental observations show that cells near the edge of the expanding monolayer are larger and move faster than cells deep inside the colony. To explain these observations and describe cell migration patterns, we formulate a spatio-temporal theoretical model for multicellular dynamics in terms of the cell area distribution; the model includes cell growth after division and effective pressure. Numerical simulations of the model predict both the speed of invasion and the width of the outer proliferative rim; these predictions are in a good agreement with experimental observations. Theoretical analysis yields the equation for density of cells and reveals a novel type of propagating front with compact support. The velocity of front propagation (monolayer expansion) is derived analytically and its dependence on all the relevant parameters is determined. [ABSTRACT FROM AUTHOR]

We study propagation over R d of the solution to a doubly nonlocal reaction–diffusion equation of the Fisher–KPP-type with anisotropic kernels. We present both necessary and sufficient conditions which ensure linear in time propagation of the solution in a direction. For kernels with a finite exponential moment over R d we prove front propagation in all directions for a general class of initial conditions decaying in all directions faster than any exponential function (that includes, for the first time in the literature about the considered type of equations, compactly supported initial conditions). [ABSTRACT FROM AUTHOR]

We present a shape-oriented data assimilation strategy suitable for front-tracking tumor growth problems. A general hyperbolic/elliptic tumor growth model is presented as well as the available observations corresponding to the location of the tumor front over time extracted from medical imaging as MRI or CT scans. We provide sufficient conditions allowing one to design a state observer by proving the convergence of the observer model to the target solution for exact parameters. In particular, the similarity measure chosen to compare observations and simulation of tumor contour is presented. A specific joint state-parameter correction with a Luenberger observer correcting the state and a reduced-order Kalman filter correcting the parameters is introduced and studied. We then illustrate and assess our proposed observer method with synthetic problems. Our numerical trials show that state estimation is very effective with the proposed Luenberger observer, but specific strategies are needed to accurately perform parameter estimation in a clinical context. We then propose strategies to deal with the fact that data is very sparse in time and that the initial distribution of the proliferation rate is unknown. The results on synthetic data are very promising, and work is ongoing to apply our strategy on clinical cases. [ABSTRACT FROM AUTHOR]

An axisymmetric problem on propagation of the phase transformation front in the solid porous medium in conditions of the boundary heat transfer in the presence of heatproof layer is considered. The parametric analysis that permits to determine the influence of the heat flow rate on the boundary, and the thermophysical properties of the insulating layer on the parameters of the phase transformation wave is made.

We study the effect of small random advection in two models in turbulent combustion. Assuming that the velocity field decorrelates sufficiently fast, we (i) identify the order of the fluctuations of the front with respect to the size of the advection; and (ii) characterize them by the solution of a Hamilton–Jacobi equation forced by white noise. In the simplest case, the result yields, for both models, a front with Brownian fluctuations of the same scale as the size of the advection. That the fluctuations are the same for both models is somewhat surprising, in view of known differences between the two models. [ABSTRACT FROM AUTHOR]

We propose an alternative approach to the direct polynomial chaos expansion in order to approximate one-dimensional uncertain field exhibiting steep fronts. The principle of our non-intrusive approach is to decompose the level points of the quantity of interest in order to avoid the spurious oscillations encountered in the direct approach. This method is more accurate and less expensive than the direct approach since the regularity of the level points with respect to the input parameters allows achieving the convergence with low-order polynomial series. The additional computational cost induced in the post-processing phase is largely offset by the use of low-level sparse grids that require a weak number of direct model evaluations in comparison with high-level sparse grids. We apply the method to subsurface flows problem with uncertain hydraulic conductivity. Infiltration test cases having different levels of complexity are presented. [ABSTRACT FROM AUTHOR]

We study the one-dimensional Fisher–KPP equation, with an initial condition u 0 (x) that coincides with the step function except on a compact set. A well-known result of Bramson in [Maximal displacement of branching Brownian motion, Comm. Pure Appl. Math. 31 (1978) 531–581; Convergence of Solutions of the Kolmogorov Equation to Travelling Waves (American Mathematical Society, Providence, RI, 1983)] states that, as t → + ∞ , the solution converges to a traveling wave located at the position X (t) = 2 t − (3 / 2) log t + x 0 + o (1) , with the shift x 0 that depends on u 0 . Ebert and Van Saarloos have formally derived in [Front propagation into unstable states: Universal algebraic convergence towards uniformly translating pulled fronts, Phys. D 146 (2000) 1–99; Front propagation into unstable states, Phys. Rep. 386 (2003) 29–222] a correction to the Bramson shift, arguing that  X (t) = 2 t − (3 / 2) log t + x 0 − 3 π / t + O (1 / t). Here, we prove that this result does hold, with an error term of the size O (1 / t 1 − γ) , for any γ > 0. The interesting aspect of this asymptotics is that the coefficient in front of the 1 / t -term does not depend on u 0 . [ABSTRACT FROM AUTHOR]

We study two-dimensional (2D) fronts propagating up a comoving reaction rate gradient in finite number reaction-diffusion systems. We show that in a 2D rectangular channel, planar solutions to the deterministic mean-field equation are stable with respect to deviations from planarity. We argue that planar fronts in the corresponding stochastic system, on the other hand, are unstable if the channel width exceeds a critical value. Furthermore, the velocity of the stochastic fronts is shown to depend on the channel width in a simple and interesting way, in contrast to fronts in the deterministic mean-field equation. Thus fluctuations alter the behavior of these fronts in an essential way. These effects are shown to be partially captured by introducing a density cutoff in the reaction rate. Moreover, some of the predictions of the cutoff mean-field approach are shown to be in quantitative accord with the stochastic results.

Coupled oscillators can exhibit complex spatiotemporal dynamics. Here, we study the propagation of nonlinear waves into an unstable state in dissipative coupled oscillators. To this, we consider the dissipative Frenkel–Kontorova model, which accounts for a chain of coupled pendulums or Josephson junctions and coupling superconducting quantum interference devices. As a function of the dissipation parameter, the front that links the stable and unstable state is characterized by having a transition from monotonous to non-monotonous profile. In the conservative limit, these traveling nonlinear waves are unstable as a consequence of the energy released in the propagation. Traveling waves into unstable states are peculiar of dissipative coupling systems. When the coupling and the dissipation parameter are increased, the average front speed decreases. Based on an averaging method, we analytically determine the front speed. Numerical simulations show a quite fair agreement with the theoretical predictions. To show that our results are generic, we analyze a chain of coupled logistic equations. This model presents the predicted dynamics, opening the door to investigate a wider class of systems. [ABSTRACT FROM AUTHOR]

Analyses have disagreed on whether the velocity u T of bulk advancement of a Huygens front in turbulence vanishes or remains finite in the limit of vanishing local front propagation speed u 0 . Here, a connection to the large-deviation statistics of log-correlated random processes enables a definitive determination of the correct small- u 0 asymptotics. This result reconciles several theoretical and phenomenological perspectives with the conclusion that u T remains finite for vanishing u 0 , which implies a propagation anomaly akin to the energy-dissipation anomaly in the limit of vanishing viscosity. Various leading-order structural properties such as a novel u 0 dependence of a bulk length scale associated with front geometry are predicted in this limit. The analysis involves a formal analogy to random advection of diffusive scalars. [ABSTRACT FROM AUTHOR]

Symmetric random walks in Rd and Zd are considered. It is assumed that the jump distribution density has moderate tails, i.e., several density moments are finite, including the second one. The global (for all x and t) asymptotic behavior at infinity of the transition probability (i.e., the fundamental solution of the corresponding parabolic convolution operator) is found. Front propagation of ecological waves in the corresponding population dynamics models is described. [ABSTRACT FROM AUTHOR]

We consider the accelerated propagation of solutions to equations with a nonlocal linear dispersion on the real line and monostable nonlinearities (both local or nonlocal, however, not degenerated at 0), in the case when either of the dispersion kernel or the initial condition has regularly heavy tails at both , perhaps different. We show that, in such case, the propagation to the right direction is fully determined by the right tails of either the kernel or the initial condition. We describe both cases of integrable and monotone initial conditions which may give different orders of the acceleration. Our approach is based, in particular, on the extension of the theory of sub-exponential distributions, which we introduced early. [ABSTRACT FROM AUTHOR]

We consider a curvature flow $V=\unicode[STIX]{x1D705}+A$ in a two-dimensional undulating cylinder $\unicode[STIX]{x1D6FA}$ described by $\unicode[STIX]{x1D6FA}:=\{(x,y)\in \mathbb{R}^{2}\mid -g_{1}(y) , where $V$ is the normal velocity of a moving curve contacting the boundaries of $\unicode[STIX]{x1D6FA}$ perpendicularly, $\unicode[STIX]{x1D705}$ is its curvature, $A>0$ is a constant and $g_{1}(y),g_{2}(y)$ are positive smooth functions. If $g_{1}$ and $g_{2}$ are periodic functions and there are no stationary curves, Matano et al. ['Periodic traveling waves in a two-dimensional cylinder with saw-toothed boundary and their homogenization limit', Netw. Heterog. Media 1 (2006), 537–568] proved the existence of a periodic travelling wave. We consider the case where $g_{1},g_{2}$ are general nonperiodic positive functions and the problem has some stationary curves. For each stationary curve $\unicode[STIX]{x1D6E4}$ unstable from above/below, we construct an entire solution growing out of it, that is, a solution curve $\unicode[STIX]{x1D6E4}_{t}$ which increases/decreases monotonically, converging to $\unicode[STIX]{x1D6E4}$ as $t\rightarrow -\infty$ and converging to another stationary curve or to $+\infty /-\infty$ as $t\rightarrow \infty$. [ABSTRACT FROM AUTHOR]

We investigate the influence of steady periodic flows on the propagation of chemical fronts in an infinite channel domain. We focus on the sharp front arising in Fisher--Kolmogorov--Petrovskii--Piskunov (FKPP) type models in the limit of small molecular diffusivity and fast reaction (large P\'eclet and Damk\"ohler numbers, Pe and Da) and on its heuristic approximation by the G equation. We introduce a variational formulation that expresses the two front speeds in terms of periodic trajectories minimizing the time of travel across the period of the flow, under a constraint that differs between the FKPP and G equations. This formulation makes it plain that the FKPP front speed is greater than or equal to the G equation front speed. We study the two front speeds for a class of cellular vortex flows used in experiments. Using a numerical implementation of the variational formulation, we show that the differences between the two front speeds are modest for a broad range of parameters. However, large differences appear when a strong mean flow opposes front propagation; in particular, we identify a range of parameters for which FKPP fronts can propagate against the flow while G fronts cannot. We verify our computations against closed-form expressions derived for Da \ll Pe and for Da \gg Pe. [ABSTRACT FROM AUTHOR]

We consider the mean curvature flow V = H in a cylinder Ω ≔ B × R , where B is the unit ball in R N , V is the normal velocity of a moving surface Γ t : x N + 1 = u ( x , t ) ( x ∈ B ) and H is the mean curvature of Γ t . Assume that the surface is radially symmetric and it contacts the boundary of Ω with a prescribed angle θ = θ ( t , u ) . In case θ tends to two (spatially or temporally) periodic functions as u → ± ∞ , we show that the problem has a unique entire solution U , which propagates from − ∞ to ∞ and connects two (spatially or temporally) periodic traveling waves at t = − ∞ and at t = ∞ . [ABSTRACT FROM AUTHOR]

Order-disorder phase transitions driven by temperature or light in soft matter materials exhibit complex dissipative structures. Here, we investigate the spatio-temporal phenomena induced by light in a dye-doped nematic liquid crystal layer. Experimentally, for planar anchoring of the nematic layer and high enough input power, photoisomerization processes induce a nematic- isotropic phase transition mediated by interface propagation between the two phases. In the case of a twisted nematic layer and for intermediate input power, the light induces a spatially modulated phase, which exhibits stripe patterns. The pattern originates as an instability mediated by interface propagation between the modulated and the homogeneous nematic states. Theoretically, the phase transition, emergence of stripe patterns and front dynamics are described on the basis of a proposed model for the dopant concentration coupled with the nematic order parameter. Numerical simulations show quite a fair agreement with the experimental observations. This article is part of the theme issue 'Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 2)'. [ABSTRACT FROM AUTHOR]

We study the averaging of fronts moving with positive oscillatory normal velocity, which is periodic in space and stationary ergodic in time. The problem can be formulated as the homogenization of coercive level set Hamilton-Jacobi equations with spatio-temporal oscillations. To overcome the difficulties due to the oscillations in time and the linear growth of the Hamiltonian, we first study the long time averaged behavior of the associated reachable sets using geometric arguments. The results are new for higher than one dimensions even in the space-time periodic setting. [ABSTRACT FROM AUTHOR]