scholarly journals Travelling-wave analysis of a model describing tissue degradation by bacteria

2007 ◽  
Vol 18 (5) ◽  
pp. 583-605 ◽  
Author(s):  
D. HILHORST ◽  
J. R. KING ◽  
M. RÖGER
Keyword(s):  
Reaction Diffusion ◽  
Travelling Wave ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Travelling Wave Solutions ◽  
Large Wave ◽  
Minimal Speed ◽  
Wave Solutions ◽  
Degradation Rates ◽  
Tissue Degradation

We study travelling-wave solutions for a reaction-diffusion system arising as a model for host-tissue degradation by bacteria. This system consists of a parabolic equation coupled with an ordinary differential equation. For large values of the ‘degradation-rate parameter’ solutions are well approximated by solutions of a Stefan-like free boundary problem, for which travelling-wave solutions can be found explicitly. Our aim is to prove the existence of travelling waves for all sufficiently large wave speeds for the original reaction-diffusion system and to determine the minimal speed. We prove that for all sufficiently large degradation rates, the minimal speed is identical to the minimal speed of the limit problem. In particular, in this parameter range,non-linearselection of the minimal speed occurs.

Travelling waves in a reaction-diffusion system modelling farmer and hunter-gatherer interaction in the Neolithic transition in Europe

2019 ◽  
Vol 31 (3) ◽  
pp. 470-510 ◽  
Author(s):  
JE-CHIANG TSAI ◽  
M. HUMAYUN KABIR ◽  
MASAYASU MIMURA
Keyword(s):  
Travelling Waves ◽  
Reaction Diffusion ◽  
Travelling Wave ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
System Modelling ◽  
Travelling Wave Solutions ◽  
Kpp Equation ◽  
Neolithic Transition ◽  
Wave Solutions

AbstractRecently we have proposed a monostable reaction-diffusion system to explain the Neolithic transition from hunter-gatherer life to farmer life in Europe. The system is described by a three-component system for the populations of hunter-gatherer (H), sedentary farmer (F1) and migratory one (F2). The conversion between F1 and F2 is specified by such a way that if the total farmers F1 + F2 are overcrowded, F1 actively changes to F2, while if it is less crowded, the situation is vice versa. In order to include this property in the system, the system incorporates a critical parameter (say F0) depending on the development of farming technology in a monotonically increasing way. It determines whether the total farmers are either over crowded (F1 + F2 >F0) or less crowded (F1 + F2 <F0) ( [9, 20]). Previous numerical studies indicate that the structure of travelling wave solutions of the system is qualitatively similar to the one of the Fisher-KPP equation, that the asymptotically expanding velocity of farmers is equal to the minimal velocity (say cm(F0)) of travelling wave solutions, and that cm(F0) is monotonically decreasing as F0 increases. The latter result suggests that the development of farming technology suppresses the expanding velocity of farmers. As a partial analytical result to this property, the purpose of this paper is to consider the two limiting cases where F0 = 0 and F0 → ∞, and to prove cm(0)>cm(∞).

Periodic travelling wave solutions for a reaction-diffusion system on landscape fitted domains

2020 ◽  
Vol 139 ◽  
pp. 110300
Author(s):  
Sangkwon Kim ◽  
Jintae Park ◽  
Chaeyoung Lee ◽  
Darae Jeong ◽  
Yongho Choi ◽  
...  
Keyword(s):  
Reaction Diffusion ◽  
Travelling Wave ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
Periodic Travelling Wave

scholarly journals FKPP equation with impulses on unbounded domain

2017 ◽  
Vol 1 ◽  
pp. 1 ◽  
Author(s):  
Valaire Yatat ◽  
Yves Dumont
Keyword(s):  
Travelling Waves ◽  
Reaction Diffusion ◽  
Bare Soil ◽  
Travelling Wave ◽  
Reaction Diffusion Equations ◽  
Spreading Speed ◽  
Systems Dynamics ◽  
Travelling Wave Solutions ◽  
Minimal Speed ◽  
Wave Solutions

This paper deals with the problem of travelling wave solutions in a scalar impulsive FKPP-like equation. It is a first step of a more general study that aims to address existence of travelling wave solutions for systems of impulsive reaction-diffusion equations that model ecological systems dynamics such as fire-prone savannas. Using results on scalar recursion equations, we show existence of populated vs. extinction travelling waves invasion and compute an explicit expression of their spreading speed (characterized as the minimal speed of such travelling waves). In particular, we find that the spreading speed explicitly depends on the time between two successive impulses. In addition, we carry out a comparison with the case of time-continuous events. We also show that depending on the time between two successive impulses, the spreading speed with pulse events could be lower, equal or greater than the spreading speed in the case of time-continuous events. Finally, we apply our results to a model of fire-prone grasslands and show that pulse fires event may slow down the grassland vs. bare soil invasion speed.

Travelling waves for a reaction–diffusion system in population dynamics and epidemiology

2005 ◽  
Vol 135 (4) ◽  
pp. 663-675 ◽  
Author(s):  
Shangbing Ai ◽  
Wenzhang Huang
Keyword(s):  
Population Dynamics ◽  
Travelling Waves ◽  
Three Dimensional ◽  
Diffusion Constant ◽  
Reaction Diffusion ◽  
Travelling Wave ◽  
Reaction Diffusion Equations ◽  
Dimensional System ◽  
Travelling Wave Solutions ◽  
Wave Solutions

The existence and uniqueness of travelling-wave solutions is investigated for a system of two reaction–diffusion equations where one diffusion constant vanishes. The system arises in population dynamics and epidemiology. Travelling-wave solutions satisfy a three-dimensional system about (u, u′, ν), whose equilibria lie on the u-axis. Our main result shows that, given any wave speed c > 0, the unstable manifold at any point (a, 0, 0) on the u-axis, where a ∈ (0, γ) and γ is a positive number, provides a travelling-wave solution connecting another point (b, 0, 0) on the u-axis, where b:= b(a) ∈ (γ, ∞), and furthermore, b(·): (0, γ) → (γ, ∞) is continuous and bijective

Travelling wave solutions of a reaction—infiltration problem and a related free boundary problem

1994 ◽  
Vol 5 (3) ◽  
pp. 255-265 ◽  
Author(s):  
John Chadam ◽  
Xinfu Chen ◽  
Elena Comparini ◽  
Riccardo Ricci
Keyword(s):  
Free Boundary ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
Reaction Diffusion ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
Suitable Norm ◽  
Formal Limit ◽  
A Free Boundary Problem

We consider travelling wave solutions of a reaction–diffusion system arising in a model for infiltration with changing porosity due to reaction. We show that the travelling wave solution exists, and is unique modulo translations. A small parameter ε appears in this problem. The formal limit as ε → 0 is a free boundary problem. We show that the solution for ε > 0 tends, in a suitable norm, to the solution of the formal limit.

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