# American Institute of Mathematical Sciences

## Asymptotic behavior of solutions of Aoki-Shida-Shigesada model in bounded domains

 1 Institute of Mathematics and Scientific Computing, University of Graz, Heinrichstrasse 36, 8010 Graz, Austria 2 Graduate School of Integrated Sciences for Life, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima City, Hiroshima 739-8526, Japan 3 Meiji Institute for Advanced Study of Mathematical Sciences, Meiji University, 4–21–1 Nakano, Nakano ku, Tokyo, 164–8525, Japan

* Corresponding author: Ryunosuke Mori

Received  July 2019 Published  January 2020

The beginning of the transition from a hunter-gatherer way of life to a more settled, farming-based one in Europe is dated to the Neolithic period. The spread of farming culture from the Middle East is associated, among other things, with the transformation of landscape, cultivation of domesticated plants, domestication of animals, as well as it is identified with the distribution of certain human genetic lineages. Ecological models attribute the Neolithic transition either to the spread of the initial farming populations or to the dispersal of farming knowledge and ideas with the simultaneous conversion of hunter-gatherers to farmers. A reaction-diffusion model proposed by Aoki, Shida and Shigesada in 1996 is the first model that includes the populations of initial farmers and converted farmers from hunter-gatherers. Both populations compete for the same resources in this model, however, otherwise they evolve independently of each other from a genetic point of view. We study the large time behaviour of solutions to this model in bounded domains and we explain which farmers under what conditions dominate over the other and eventually occupy the whole habitat.

Citation: Ján Eliaš, Masayasu Mimura, Ryunosuke Mori. Asymptotic behavior of solutions of Aoki-Shida-Shigesada model in bounded domains. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020082
##### References:
 [1] A. J. Ammerman and L. L. Cavalli-Sforza, Measuring the rate of spread of early farming in Europe, Man, 6 (1971), 674-688.   Google Scholar [2] K. Aoki, M. Shida and N. Shigesada, Travelling wave solutions for the spread of farmers into a region occupied by hunter-gatherers, Theoretical Population Biology, 50 (1996), 1-17.   Google Scholar [3] B. Bramanti, M. G. Thomas, W. Haak, M. Unterlaender, P. Jores, K. Tambets, I. Antanaitis-Jacobs, M. N. Haidle, R. Jankauskas, C.-J. Kind, F. Lueth, T. Terberger, J. Hiller, S. Matsumura, P. Forster and J. Burger, Genetic discontinuity between local hunter-gatherers and Central Europe's first farmers, Science, 326 (2009), 137-140.   Google Scholar [4] J. Eliaš, D. Hilhorst and M. Mimura, Large time behaviour of the solution of a nonlinear diffusion problem in anthropology, Journal of Mathematical Study, 51 (2018), 309-336.  doi: 10.4208/jms.v51n3.18.04.  Google Scholar [5] J. Fort, Synthesis between demic and cultural diffusion in the Neolithic transition in Europe, Proceedings of the National Academy of Sciences, 109 (2012), 18669-18673.   Google Scholar [6] J. Fort, Demic and cultural diffusion propagated the Neolithic transition across different regions of Europe, Journal of The Royal Society Interface, 12 (2015), 20150166. Google Scholar [7] J. Fort, E. R. Crema and M. Madella, Modeling demic and cultural diffusion: An introduction, Human Biology, 87 (2015), 141-149.   Google Scholar [8] M. Gkiasta, T. Russell, S. Shennan and J. Steele, Neolithic transition in Europe: The radiocarbon record revisited, Antiquity, 77 (2003), 45-62.   Google Scholar [9] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, New York/Berlin, 1981.  Google Scholar [10] D. Hilhorst, M. Mimura and R. Weidenfeld, On a reaction-diffusion system for a population of hunters and farmers, in Free Boundary Problems: Theory and Applications (eds. P. Colli, C. Verdi and A. Visintin), Birkhäuser, Basel, 147 (2004), 189–196.  Google Scholar [11] R. Mori and D. Xiao, Spreading properties of a three-component reaction-diffusion model for the population of farmers and hunter-gatherers, preprint, arXiv: 1812.04440. Google Scholar [12] J. Steele, Human dispersals: Mathematical models and the archaeological record, Human Biology, 81 (2009), 121-140.   Google Scholar

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##### References:
 [1] A. J. Ammerman and L. L. Cavalli-Sforza, Measuring the rate of spread of early farming in Europe, Man, 6 (1971), 674-688.   Google Scholar [2] K. Aoki, M. Shida and N. Shigesada, Travelling wave solutions for the spread of farmers into a region occupied by hunter-gatherers, Theoretical Population Biology, 50 (1996), 1-17.   Google Scholar [3] B. Bramanti, M. G. Thomas, W. Haak, M. Unterlaender, P. Jores, K. Tambets, I. Antanaitis-Jacobs, M. N. Haidle, R. Jankauskas, C.-J. Kind, F. Lueth, T. Terberger, J. Hiller, S. Matsumura, P. Forster and J. Burger, Genetic discontinuity between local hunter-gatherers and Central Europe's first farmers, Science, 326 (2009), 137-140.   Google Scholar [4] J. Eliaš, D. Hilhorst and M. Mimura, Large time behaviour of the solution of a nonlinear diffusion problem in anthropology, Journal of Mathematical Study, 51 (2018), 309-336.  doi: 10.4208/jms.v51n3.18.04.  Google Scholar [5] J. Fort, Synthesis between demic and cultural diffusion in the Neolithic transition in Europe, Proceedings of the National Academy of Sciences, 109 (2012), 18669-18673.   Google Scholar [6] J. Fort, Demic and cultural diffusion propagated the Neolithic transition across different regions of Europe, Journal of The Royal Society Interface, 12 (2015), 20150166. Google Scholar [7] J. Fort, E. R. Crema and M. Madella, Modeling demic and cultural diffusion: An introduction, Human Biology, 87 (2015), 141-149.   Google Scholar [8] M. Gkiasta, T. Russell, S. Shennan and J. Steele, Neolithic transition in Europe: The radiocarbon record revisited, Antiquity, 77 (2003), 45-62.   Google Scholar [9] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, New York/Berlin, 1981.  Google Scholar [10] D. Hilhorst, M. Mimura and R. Weidenfeld, On a reaction-diffusion system for a population of hunters and farmers, in Free Boundary Problems: Theory and Applications (eds. P. Colli, C. Verdi and A. Visintin), Birkhäuser, Basel, 147 (2004), 189–196.  Google Scholar [11] R. Mori and D. Xiao, Spreading properties of a three-component reaction-diffusion model for the population of farmers and hunter-gatherers, preprint, arXiv: 1812.04440. Google Scholar [12] J. Steele, Human dispersals: Mathematical models and the archaeological record, Human Biology, 81 (2009), 121-140.   Google Scholar
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