Spreading properties of a three-component reaction-diffusion model for the population of farmers and hunter-gatherers
Annales de l'I.H.P. Analyse non linéaire, juillet – août 2021, Tome 38 (2021) no. 4, pp. 911-951.
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In this paper, we investigate the spreading properties of solutions of farmer and hunter-gatherer model which is a three-component reaction-diffusion system. Ecologically, the model describes the geographical spreading of an initially localized population of farmers into a region occupied by hunter-gatherers. This model was proposed by Aoki, Shida and Shigesada in 1996. By numerical simulations and some formal linearization arguments, they concluded that there are four different types of spreading behaviors depending on the parameter values. Despite such intriguing observations, no mathematically rigorous studies have been made to justify their claims. The main difficulty comes from the fact that the comparison principle does not hold for the entire system. In this paper, we give theoretical justification to all of the four types of spreading behaviors observed by Aoki et al. Furthermore, we show that a logarithmic phase drift of the front position occurs as in the scalar KPP equation. We also investigate the case where the motility of the hunter-gatherers is larger than that of the farmers, which is not discussed in the paper of Aoki et al.

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Révisé le :
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DOI : 10.1016/j.anihpc.2020.09.007
Mots-clés : Farmer and hunter-gatherer model, Long time behavior, Spreading speed, Logarithmic correction 
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     title = {Spreading properties of a three-component reaction-diffusion model for the population of farmers and hunter-gatherers},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {911--951},
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Xiao, Dongyuan; Mori, Ryunosuke. Spreading properties of a three-component reaction-diffusion model for the population of farmers and hunter-gatherers. Annales de l'I.H.P. Analyse non linéaire, juillet – août 2021, Tome 38 (2021) no. 4, pp. 911-951. doi : 10.1016/j.anihpc.2020.09.007. http://archive.numdam.org/articles/10.1016/j.anihpc.2020.09.007/

[1] Aoki, K.; Shida, M.; Shigesada, N. Travelling wave solutions for the spread of farmers into a region occupied by hunter-gatherers, Theor. Popul. Biol., Volume 50 (1996), pp. 1-17 | DOI | Zbl

[2] Aronson, D.G.; Weinberger, H.F. Multidimensional nonlinear diffusion arising in population genetics, Adv. Math., Volume 30 (1978), pp. 33-76 | DOI | MR | Zbl

[3] Bramson, M.D. Maximal displacement of branching Brownian motion, Commun. Pure Appl. Math., Volume 31 (1978), pp. 531-581 | DOI | MR | Zbl

[4] Bramson, M.D. Convergence of solutions of the Kolmogorov equation to travelling waves, Mem. Am. Math. Soc., Volume 44 (1983) | MR | Zbl

[5] Ducrot, A. On the large time behaviour of the multi-dimensional Fisher-KPP equation with compactly supported initial data, Nonlinearity, Volume 28 (2015), pp. 1043-1076 | DOI | MR | Zbl

[6] A. Ducrot, T. Giletti, H. Matano, Spreading speeds for multidimensional reaction-diffusion systems of the prey-predator type, preprint. | MR

[7] Fisher, R.A. The wave of advance of advantageous genes, Ann. Eugen., Volume 7 (1937), pp. 335-369 | DOI | JFM

[8] Girardin, L.; Lam, K. Invasion of an empty habitat by two competitors: spreading properties of monostable two-species competition-diffusion systems, 2018 | arXiv | MR | Zbl

[9] Kolmogorov, A.N.; Petrovskii, I.G.; Piskunov, N.S. A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem, Bull. Moscow State Univ. Ser. A: Math. Mech., Volume 1 (1937), pp. 1-25

[10] Hilhorst, D.; Mimura, M.; Weidenfeld, R. On a reaction-diffusion system for a population of hunters and farmers (Colli, P.; Verdi, C.; Visintin, A., eds.), Free Boundary Problems: Theory and Applications, 2014, pp. 189-196

[11] Hamel, F.; Nolen, J.; Roquejoffre, J.M.; Ryzhik, L. A short proof of the logarithmic Bramson correction in Fisher-KPP equations, Netw. Heterog. Media, Volume 8 (2013), pp. 261-289 | DOI | MR | Zbl

[12] Skellam, J.G. Random dispersal in theoretical populations, Biometrika, Volume 38 (1951), pp. 196-218 | DOI | MR | Zbl

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