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

Recently 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 c_m(F₀)) of travelling wave solutions, and that c_m(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 c_m(0) > c_m(∞).