American Institute of Mathematical Sciences

Advanced
doi: 10.3934/dcdsb.2020082

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

Ján Eliaš 1, , Masayasu Mimura 2, and  Ryunosuke Mori 3,,

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.

Keywords: Neolithic transition, farmers, hunter-gatherers, reaction-diffusion equations, asymptotic behavior of solutions.
Mathematics Subject Classification: Primary: 35K57, 35B40; Secondary: 35Q92.
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

show all references

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

[1]

Lin Shi, Xuemin Wang, Dingshi Li. Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5367-5386. doi: 10.3934/cpaa.2020242
[2]

José Luiz Boldrini, Jonathan Bravo-Olivares, Eduardo Notte-Cuello, Marko A. Rojas-Medar. Asymptotic behavior of weak and strong solutions of the magnetohydrodynamic equations. Electronic Research Archive, 2021, 29 (1) : 1783-1801. doi: 10.3934/era.2020091
[3]

Shin-Ichiro Ei, Shyuh-Yaur Tzeng. Spike solutions for a mass conservation reaction-diffusion system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3357-3374. doi: 10.3934/dcds.2020049
[4]

Masaharu Taniguchi. Axisymmetric traveling fronts in balanced bistable reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3981-3995. doi: 10.3934/dcds.2020126
[5]

Hideki Murakawa. Fast reaction limit of reaction-diffusion systems. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1047-1062. doi: 10.3934/dcdss.2020405
[6]

Junyong Eom, Kazuhiro Ishige. Large time behavior of ODE type solutions to nonlinear diffusion equations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3395-3409. doi: 10.3934/dcds.2019229
[7]

Nabahats Dib-Baghdadli, Rabah Labbas, Tewfik Mahdjoub, Ahmed Medeghri. On some reaction-diffusion equations generated by non-domiciliated triatominae, vectors of Chagas disease. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021004
[8]

Chungang Shi, Wei Wang, Dafeng Chen. Weak time discretization for slow-fast stochastic reaction-diffusion equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021019
[9]

Hoang The Tuan. On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1749-1762. doi: 10.3934/dcdsb.2020318
[10]

Kaixuan Zhu, Ji Li, Yongqin Xie, Mingji Zhang. Dynamics of non-autonomous fractional reaction-diffusion equations on $ \mathbb{R}^{N} $ driven by multiplicative noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020376
[11]

Weiwei Liu, Jinliang Wang, Yuming Chen. Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020316
[12]

Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020321
[13]

Maho Endo, Yuki Kaneko, Yoshio Yamada. Free boundary problem for a reaction-diffusion equation with positive bistable nonlinearity. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3375-3394. doi: 10.3934/dcds.2020033
[14]

Chihiro Aida, Chao-Nien Chen, Kousuke Kuto, Hirokazu Ninomiya. Bifurcation from infinity with applications to reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3031-3055. doi: 10.3934/dcds.2020053
[15]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319
[16]

H. M. Srivastava, H. I. Abdel-Gawad, Khaled Mohammed Saad. Oscillatory states and patterns formation in a two-cell cubic autocatalytic reaction-diffusion model subjected to the Dirichlet conditions. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020433
[17]

Guillaume Cantin, M. A. Aziz-Alaoui. Dimension estimate of attractors for complex networks of reaction-diffusion systems applied to an ecological model. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020283
[18]

Shin-Ichiro Ei, Hiroshi Ishii. The motion of weakly interacting localized patterns for reaction-diffusion systems with nonlocal effect. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 173-190. doi: 10.3934/dcdsb.2020329
[19]

El Haj Laamri, Michel Pierre. Stationary reaction-diffusion systems in $ L^1 $ revisited. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 455-464. doi: 10.3934/dcdss.2020355
[20]

Klemens Fellner, Jeff Morgan, Bao Quoc Tang. Uniform-in-time bounds for quadratic reaction-diffusion systems with mass dissipation in higher dimensions. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 635-651. doi: 10.3934/dcdss.2020334

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (155)
  • HTML views (437)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences