scholarly journals Exact sharp-fronted travelling wave solutions of the Fisher–KPP equation

2021 ◽  
Vol 114 ◽  
pp. 106918
Author(s):  
Scott W. McCue ◽  
Maud El-Hachem ◽  
Matthew J. Simpson
Keyword(s):  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Kpp Equation ◽  
Wave Solutions

scholarly journals New travelling wave solutions for the Fisher–KPP equation with general exponents

2005 ◽  
Vol 18 (11) ◽  
pp. 1281-1285 ◽  
Author(s):  
Ariel Sánchez-Valdés ◽  
Benito Hernández-Bermejo
Keyword(s):  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Kpp Equation ◽  
Wave Solutions

Slow variation and uniqueness of solutions to the functional equation in the branching random walk

1998 ◽  
Vol 35 (4) ◽  
pp. 795-801 ◽  
Author(s):  
A. E. Kyprianou
Keyword(s):  
Functional Equation ◽  
Random Walk ◽  
Short Communication ◽  
Travelling Wave ◽  
Branching Random Walk ◽  
Discrete Version ◽  
Travelling Wave Solutions ◽  
Uniqueness Of Solutions ◽  
Kpp Equation ◽  
Wave Solutions

In this short communication, some of the recent results of Liu (1998) and Biggins and Kyprianou (1997), concerning solutions to a certain functional equation associated with the branching random walk, are strengthened. Their importance is emphasized in the context of travelling wave solutions to a discrete version of the KPP equation and the connection with the behaviour of the rightmost particle in the nth generation.

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(∞).

scholarly journals Revisiting the Fisher-KPP equation to interpret the spreading-extinction dichotomy

2019 ◽  
Author(s):  
Maud El-Hachem ◽  
Scott W. McCue ◽  
Wang Jin ◽  
Yihong Du ◽  
Matthew J. Simpson
Keyword(s):  
Moving Boundary ◽  
Travelling Wave ◽  
Phase Plane Analysis ◽  
Travelling Wave Solutions ◽  
Kpp Equation ◽  
Wave Speeds ◽  
Population Extinction ◽  
Plane Analysis ◽  
Wave Solutions ◽  
Stefan Condition

AbstractThe Fisher-KPP model supports travelling wave solutions that are successfully used to model numerous invasive phenomena with applications in biology, ecology, and combustion theory. However, there are certain phenomena that the Fisher-KPP model cannot replicate, such as the extinction of invasive populations. The Fisher-Stefan model is an adaptation of the Fisher-KPP model to include a moving boundary whose evolution is governed by a Stefan condition. The Fisher-Stefan model also supports travelling wave solutions; however, a key additional feature of the Fisher-Stefan model is that it is able to simulate population extinction, giving rise to aspreading-extinction dichotomy.In this work, we revisit travelling wave solutions of the Fisher-KPP model and show that these results provide new insight into travelling wave solutions of the Fisher-Stefan model and the spreading-extinction dichotomy. Using a combination of phase plane analysis, perturbation analysis and linearisation, we establish a concrete relationship between travelling wave solutions of the Fisher-Stefan model and often-neglected travelling wave solutions of the Fisher-KPP model. Furthermore, we give closed-form approximate expressions for the shape of the travelling wave solutions of the Fisher-Stefan model in the limit of slow travelling wave speeds,c≪ 1.

Slow variation and uniqueness of solutions to the functional equation in the branching random walk

1998 ◽  
Vol 35 (04) ◽  
pp. 795-801 ◽  
Author(s):  
A. E. Kyprianou
Keyword(s):  
Functional Equation ◽  
Random Walk ◽  
Short Communication ◽  
Travelling Wave ◽  
Branching Random Walk ◽  
Discrete Version ◽  
Travelling Wave Solutions ◽  
Uniqueness Of Solutions ◽  
Kpp Equation ◽  
Wave Solutions

In this short communication, some of the recent results of Liu (1998) and Biggins and Kyprianou (1997), concerning solutions to a certain functional equation associated with the branching random walk, are strengthened. Their importance is emphasized in the context of travelling wave solutions to a discrete version of the KPP equation and the connection with the behaviour of the rightmost particle in thenth generation.

Solitary and Travelling Wave Solutions of Certain Nonlinear Diffusion-Reaction Equations

2020 ◽  
Author(s):  
Miftachul Hadi
Keyword(s):  
Nonlinear Diffusion ◽  
Travelling Wave ◽  
Diffusion Reaction ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
Reaction Equations

We review the work of Ranjit Kumar, R S Kaushal, Awadhesh Prasad. The work is still in progress.

scholarly journals Singularity analysis and analytic solutions for the Benney–Gjevik equations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Andronikos Paliathanasis ◽  
Genly Leon ◽  
P. G. L. Leach
Keyword(s):  
Evolution Equations ◽  
Singularity Analysis ◽  
Analytic Solutions ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Painlevé Test ◽  
Wave Solutions ◽  
Algebraic Solutions

Abstract We apply the Painlevé test for the Benney and the Benney–Gjevik equations, which describe waves in falling liquids. We prove that these two nonlinear 1 + 1 evolution equations pass the singularity test for the travelling-wave solutions. The algebraic solutions in terms of Laurent expansions are presented.

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