On the Form of Smooth-Front Travelling Waves in a Reaction-Diffusion Equation with Degenerate Nonlinear Diffusion

Sherratt, Jonathan A.

doi:10.1051/mmnp/20105505

Cited by 28 publications

(18 citation statements)

References 56 publications

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“…For specific parameter regimes, F (C) is degenerate at C = 0, that is, F (0) = R(0) = 0. This type of nonlinear diffusivity function, which we refer to as extinction-degenerate non-negative, leads to sharp-fronted travelling waves, provided that F (C) ≥ 0 for 0 ≤ C ≤ 1 [40,46,47]. For Equation (6), this corresponds to P i m = 0.…”

Section: Detailsmentioning

confidence: 99%

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Co-operation, competition and crowding: a discrete framework linking Allee kinetics, nonlinear diffusion, shocks and sharp-fronted travelling waves

Johnston¹,

Baker

McElwain³

et al. 2016

Preprint

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Abstract. Invasion processes are ubiquitous throughout cell biology and ecology. During invasion, individuals can become isolated from the bulk population and behave differently.We present a discrete, exclusion-based description of the birth, death and movement of individuals. The model distinguishes between individuals that are part of, or are isolated from, the bulk population by imposing different rates of birth, death and movement. This enables the simulation of various co-operative or competitive mechanisms, where there is either a positive or negative benefit associated with being part of the bulk population, respectively. The mean-field approximation of the discrete process gives rise to 22 different classes of partial differential equation, which can include Allee kinetics and nonlinear diffusion. Here we examine the ability of each class of partial differential equation to support travelling wave solutions and interpret the long time behaviour in terms of the individual-level parameters. For the first time we show that the strong Allee effect and nonlinear diffusion can result in shock-fronted travelling waves. We also demonstrate how differences in group and individual motility rates can influence the persistence of a population and provide conditions for the successful invasion of a population.

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Section: Detailsmentioning

confidence: 99%

“…x 0 x Fisher [42][43][44] Case 4: Sub-case 2 0 x Fisher [40,[45][46][47] Case 4: Sub-case 3 x 2 Fisher [36] Case 4: Sub-case 4 1 Fisher [38] Case 4: Sub-case 5 x 1 Fisher [38] Case 5 x x 0…”

Section: Introductionmentioning

confidence: 99%

Co-operation, competition and crowding: a discrete framework linking Allee kinetics, nonlinear diffusion, shocks and sharp-fronted travelling waves

Johnston¹,

Baker

McElwain³

et al. 2016

Preprint

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“…Models with degenerate diffusivities are also endowed with interesting mathematical properties. Among the new features one finds the emergence of traveling waves of "sharp" type (see, for example, [61,63,69]) and, notably, that solutions may exhibit finite speed of propagation of initial disturbances, in contrast with the strictly parabolic case (see, e.g., [19]). Clearly, not all models with degenerate diffusions are related to biology.…”

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confidence: 99%

Spectral Stability of Traveling Fronts for Reaction Diffusion-Degenerate Fisher-KPP Equations

Leyva

Plaza

2019

J Dyn Diff Equat

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This paper establishes the spectral stability of monotone traveling front solutions for reaction-diffusion equations where the reaction function is of Nagumo (or bistable) type and with diffusivities which are density dependent and degenerate at zero (one of the equilibrium points of the reaction). Spectral stability is understood as the property that the spectrum of the linearized operator around the wave, acting on an exponentially weighted space, is contained in the complex half plane with non-positive real part. Three different types of monotone waves are studied: (i) stationary diffusion-degenerate fronts, connecting the two stable equilibria of the reaction; (ii) traveling diffusiondegenerate fronts connecting zero with the unstable equilibrium; and, (iii) non-degenerate fronts. In the first two cases, the degeneracy is responsible of the loss of hyperbolicity of the asymptotic coefficient matrices of the spectral problem at one of the end points, precluding the application of standard techniques to locate the essential spectrum. This difficulty is overcome with a suitable partition of the spectrum, a generalized convergence of operators technique, the analysis of singular (or Weyl) sequences and the use of energy estimates. The monotonicity of the fronts, as well as detailed descriptions of the decay structure of eigenfunctions on a case by case basis, are key ingredients to show that all traveling fronts under consideration are spectrally stable in a suitably chosen exponentially weighted L 2 energy space.where u = u(x, t) ∈ R, x ∈ R, t > 0, the reaction function is of Nagumo (or bistable) type [52,45] and the diffusion coefficient is degenerate at u = 0, that is, D(0) = 0. More precisely, we assume that(1.2)As an example we have the quadratic functionfor some constant b > 0, proposed by Shigesada [70, 71] to model dispersive forces due to mutual interferences between individuals of an animal population.The reaction function f : R → R is supposed to be smooth enough and to have two stable equilibria at u = 0, u = 1, and one unstable equilibrium point at u = α ∈ (0, 1), that is,(1.4) for a certain α ∈ (0, 1). A well-known example is the widely used cubic polynomialwith α ∈ (0, 1). Scalar reaction-diffusion equations with a bistable reaction function appear in many different contexts and the nomenclature is not uniform. It is known as the bistable reaction-diffusion equation [16], the Nagumo equation in neurophysiological modeling [45,52], the real Ginzburg-Landau for the variational description of phase transitions [47], the parabolic Allen-Cahn equation [2], as well as the Chafee-Infante equation [9]. For simplicity, in this work we call it the Nagumo reaction diffusion equation. In terms of continuous models of the spread of biological populations, reaction functions of Nagumo type often describe kinetics exhibiting positive growth rate for population densities over a threshold value (u > α), and decay for densities below such value (u < α). The former is often described as the Allee effect, in which agg...

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“…Another property is the emergence of traveling waves of "sharp" type (cf. [53,56]). The cross-diffusion system (1.1) has been the subject of recent investigations.…”

Section: Introductionmentioning

confidence: 99%

Derivation of a bacterial nutrient-taxis system with doubly degenerate cross-diffusion as the parabolic limit of a velocity-jump process

Plaza

2019

J. Math. Biol.

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This paper is devoted to the justification of the macroscopic, mean-field nutrient taxis system with doubly degenerate cross-diffusion proposed by Leyva et al. [39] to model the complex spatio-temporal dynamics exhibited by the bacterium B. subtilis during experiments run in vitro. This justification is based on a microscopic description of the movement of individual cells whose changes in velocity (in both speed and orientation) obey a velocity jump process [47], governed by a transport equation of Boltzmann type. For that purpose, the asymptotic method introduced by Hillen and Othmer [27,48] is applied, which consists of the computation of the leading order term in a regular Hilbert expansion for the solution to the transport equation, under an appropriate parabolic scaling and a first order perturbation of the turning rate of Schnitzer type [54]. The resulting parabolic limit equation at leading order for the bacterial cell density recovers the degenerate nonlinear cross diffusion term and the associated chemotactic drift appearing in the original system of equations. Although the bacterium B. subtilis is used as a prototype, the method and results apply in more generality.2010 Mathematics Subject Classification. 35K65, 92C17, 60J75.

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On the Form of Smooth-Front Travelling Waves in a Reaction-Diffusion Equation with Degenerate Nonlinear Diffusion

Cited by 28 publications

References 56 publications

Co-operation, competition and crowding: a discrete framework linking Allee kinetics, nonlinear diffusion, shocks and sharp-fronted travelling waves

Co-operation, competition and crowding: a discrete framework linking Allee kinetics, nonlinear diffusion, shocks and sharp-fronted travelling waves

Spectral Stability of Traveling Fronts for Reaction Diffusion-Degenerate Fisher-KPP Equations

Derivation of a bacterial nutrient-taxis system with doubly degenerate cross-diffusion as the parabolic limit of a velocity-jump process

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