Travelling wave solutions to the KPP equation with branching noise arising from initial conditions with compact support

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.

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

European Journal of Applied Mathematics ◽

10.1017/s0956792519000159 ◽

2019 ◽

Vol 31 (3) ◽

pp. 470-510 ◽

Cited By ~ 1

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(∞).

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Shear instability of internal solitary waves in Euler fluids with thin pycnoclines

Journal of Fluid Mechanics ◽

10.1017/jfm.2012.366 ◽

2012 ◽

Vol 710 ◽

pp. 324-361 ◽

Cited By ~ 12

Author(s):

A. Almgren ◽

R. Camassa ◽

R. Tiron

Keyword(s):

Initial Conditions ◽

Nonlinear Models ◽

Maximum Amplitude ◽

Travelling Wave ◽

Stationary Solutions ◽

Stability Study ◽

Shear Instability ◽

Order Effects ◽

Travelling Wave Solutions ◽

Wave Solutions

AbstractThe stability with respect to initial condition perturbations of solitary travelling-wave solutions of the Euler equations for continuously, stably stratified, near two-layer fluids is examined numerically and analytically for a set of parameters of relevance for laboratory experiments. Numerical travelling-wave solutions of the Dubreil–Jacotin–Long equation are first obtained with a variant of Turkington, Eyland and Wang’s iterative code by testing convergence on the equation’s residual. In this way, stationary solutions with very thin pycnoclines (and small Richardson numbers) approaching the near two-layer configurations used in experiments can be obtained, allowing for a stability study free of non-stationary effects, introduced by lack of numerical resolution, which develop when these solutions are used as initial conditions in a time-dependent evolution code. The thin pycnoclines in this study permit analytical results to be derived from strongly nonlinear models and their predictions compared with carefully controlled numerical simulations. This brings forth shortcomings of simple criteria for shear instability manifestations based on parallel shear approximations due to subtle higher-order effects. In particular, evidence is provided that the fore–aft asymmetric growth observed in all simulations requires non-parallel shear analysis. Collectively, the results of this study reveal that while the wave-induced shear can locally reach unstable configurations and give rise to local convective instability, the global wave/self-generated shear system is in fact stable, even for extreme cases of thin pycnoclines and near-maximum-amplitude waves.

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

Applied Mathematics Letters ◽

10.1016/j.aml.2020.106918 ◽

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

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

10.1101/673202 ◽

2019 ◽

Cited By ~ 1

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.

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

Journal of Applied Probability ◽

10.1017/s0021900200016508 ◽

1998 ◽

Vol 35 (04) ◽

pp. 795-801 ◽

Cited By ~ 2

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.

Download Full-text