Finite Travelling Waves for a Semilinear Degenerate Reaction-Diffusion System

2001 ◽  
Vol 17 (4) ◽  
pp. 603-612
Author(s):  
Shu Wang ◽  
Cheng Fu Wang ◽  
Dang Luo
Keyword(s):  
Travelling Waves ◽  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System

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(∞).

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