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A hunter-gatherer–farmer population model: Lie symmetries, exact solutions and their interpretation

Published online by Cambridge University Press:  27 February 2018

R. M. CHERNIHA  and
V. V. DAVYDOVYCH
R. M. CHERNIHA
Affiliation:
Institute of Mathematics, NAS of Ukraine, 3 Tereshchenkivs'ka Street, 01004 Kyiv, Ukraine emails: [email protected], [email protected]
V. V. DAVYDOVYCH
Affiliation:
Institute of Mathematics, NAS of Ukraine, 3 Tereshchenkivs'ka Street, 01004 Kyiv, Ukraine emails: [email protected], [email protected]
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Abstract

The Lie symmetry classification of the known three-component reaction–diffusion system modelling the spread of an initially localized population of farmers into a region occupied by hunter-gatherers is derived. The Lie symmetries obtained for reducing the system in question to systems of ordinary differential equations (ODEs) and constructing exact solutions are applied. Several exact solutions of travelling front type are also found, their properties are identified and biological interpretation is discussed.

Keywords

Reaction–diffusion systemdiffusive Lotka–Volterra systemLie symmetryexact solutiontravelling front
Type
Papers
Information
European Journal of Applied Mathematics , Volume 30 , Issue 2 , April 2019 , pp. 338 - 357
Copyright
Copyright © Cambridge University Press 2018 

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References

[1] Ablowitz, M. & Zeppetella, A. (1979) Explicit solutions of Fisher's equation for a special wave speed. Bull. Math. Biol. 41, 835–840.Google Scholar
[2] Ames, W.F. (1972) Nonlinear Partial Differential Equations in Engineering, Academic Press, New York.Google Scholar
[3] Aoki, K., Shida, M. & Shigesada, N. (1996) Travelling wave solutions for the spread of farmers into a region occupied by hunter-gatherers. Theor. Popul. Biol. 50, 1–17.Google Scholar
[4] Aris, R. (1975) The Mathematical Theory of the Diffusion and Reaction in Permeable Catalysts, Vol. 1–2, Oxford University Press, Oxford.Google Scholar
[5] Arrigo, D. J. (2015) Symmetry Analysis of Differential Equations, John Wiley & Sons, Inc., Hoboken, NJ.Google Scholar
[6] Bluman, G. W., Cheviakov, A. F. & Anco, S. C. (2010) Applications of Symmetry Methods to Partial Differential Equations, Springer, New York.Google Scholar
[7] Britton, N. F. (2003) Essential Mathematical Biology, Springer, Berlin.Google Scholar
[8] Chen, C.-C. & Hung, L.-C. (2012) Exact travelling wave solutions of three-species competition-diffusion systems. Discrete Contin. Dyn. Syst. Ser. B 17, 2653–2669.Google Scholar
[9] Cherniha, R. & Davydovych, V. (2013) Lie and conditional symmetries of the three-component diffusive Lotka–Volterra system. J. Phys. A: Math. Theor. 46, 185204 (14 pp).Google Scholar
[10] Cherniha, R. & Davydovych, V. (2017) Nonlinear Reaction-Diffusion Systems – Conditional Symmetry, Exact Solutions and Their Applications in Biology, Lecture Notes in Mathematics, Vol. 2196, Springer, Cham, Switzerland.Google Scholar
[11] Cherniha, R., Davydovych, V. & Muzyka, L. (2017) Lie symmetries of the Shigesada–Kawasaki–Teramoto system. Comm. Nonlinear Sci. Numer. Simul. 45, 81–92.Google Scholar
[12] Cherniha, R. & King, J. R. (2000) Lie symmetries of nonlinear multidimensional reaction-diffusion systems: I. J. Phys. A: Math. Gen. 33, 267–282.Google Scholar
[13] Cherniha, R. & King, J. R. (2000) Addendum: “Lie symmetries of nonlinear multidimensional reaction-diffusion systems: I''. J. Phys. A: Math. Gen. 33, 7839–7841.Google Scholar
[14] Cherniha, R. & King, J. R. (2003) Lie symmetries of nonlinear multidimensional reaction-diffusion systems: II. J. Phys. A: Math. Gen. 36, 405–425.Google Scholar
[15] Cherniha, R. & King, J. R. (2005) Nonlinear reaction-diffusion systems with variable diffusivities: Lie symmetries, ansätze and exact solutions. J. Math. Anal. Appl. 308, 11–35.Google Scholar
[16] Cherniha, R. & King, J. R. (2006) Lie symmetries and conservation laws of nonlinear multidimensional reaction-diffusion systems with variable diffusivities. IMA J. Appl. Math. 71, 391–408.Google Scholar
[17] Cherniha, R. M. & Wilhelmsson, H. (1996) Symmetry and exact solution of heat-mass transfer equations in thermonuclear plasma. Ukr. Math. J. 48, 1434–1449.Google Scholar
[18] Fisher, R. A. (1937) The wave of advance of advantageous genes. Ann. Eugen. 7, 353–369.Google Scholar
[19] Fushchych, W. I., Shtelen, W. M. & Serov, M. I. (1993) Symmetry Analysis and Exact Solutions of Equations of Nonlinear Mathematical Physics, Kluwer, Dordrecht.Google Scholar
[20] Gilding, B. H. & Kersner, R. (2004) Travelling Waves in Nonlinear Diffusion-Convection Reaction, Birkhauser Verlag, Basel.Google Scholar
[21] Hung, L.-C. (2011) Traveling wave solutions of competitive-cooperative Lotka–Volterra systems of three species. Nonlinear Anal. Real World Appl. 12, 3691–3700.Google Scholar
[22] Kamke, E. (1977) Differentialgleichungen. Lösungsmethoden und Lösungen. I: Gewöhnliche (in German), Teubner Stuttgart.Google Scholar
[23] Kandler, A., Unger, R. & Steele, J. (2010) Language shift, bilingualism and the future of Britain's Celtic languages. Philos. Trans. R. Soc. B 365, 3855–3864.Google Scholar
[24] Knyazeva, I. V. & Popov, M. D. (1994) A system of two diffusion equations. In: Ibragimov, N. H. (editor), CRC Handbook of Lie Group Analysis of Differential Equations, Vol. 1, CRC Press, Boca Raton, pp. 171–176.Google Scholar
[25] Kuang, Y., Nagy, J. D. & Eikenberry, S. E. (2016) Introduction to Mathematical Oncology, CRC Press, Boca Raton.Google Scholar
[26] Malfliet, W. (2004) The tanh method: A tool for solving certain classes of nonlinear evolution and wave equations. J. Comp. Appl. Math. 164, 529–541.Google Scholar
[27] Murray, J. D. (2003) An Introduction I: Models and Biomedical Applications, Springer, Berlin.Google Scholar
[28] Murray, J. D. (2003) Mathematical Biology II: Spatial Models and Biomedical Applications, Springer, Berlin.Google Scholar
[29] Nikitin, A. G (2005) Group classification of systems of nonlinear reaction-diffusion equations. Ukr. Math. Bull. 2, 153–204.Google Scholar
[30] Okubo, A. & Levin, S. A. (2001) Diffusion and Ecological Problems. Modern Perspectives, 2nd ed., Springer, Berlin.Google Scholar
[31] Olver, P. (1986) Applications of Lie Groups to Differential Equations, Springer, Berlin.Google Scholar
[32] Ovsiannikov, L. V. (1980) The Group Analysis of Differential Equations, Academic Press, New York.Google Scholar
[33] Polyanin, A. D. & Zaitsev, V. F. (2003) Handbook of Exact Solutions for Ordinary Differential Equations, CRC Press, London.Google Scholar
[34] Samarskii, A. A., Galaktionov, V. A., Kurdyumov, S. P. & Mikhailov, A. P. (1995) Blow-up in Quasilinear Parabolic Equations, Walter de Gruyter, Berlin.Google Scholar
[35] Straughan, B. (2014) Shocks and acceleration waves in modern continuum mechanics and in social systems. Evol. Equat. Contr. Theory 3, 541–555.Google Scholar
[36] Serov, M. & Omelian, O. (2008) Classification of the symmetry properties of a system of chemotaxis equations. Ukr. Math. Bull. 5, 529–557.Google Scholar
[37] Torrisi, M., Tracina, R. & Valenti, A. (1996) A group analysis approach for a non linear differential system arising in diffusion phenomena. J. Math. Phys. 37, 4758–4767.Google Scholar
[38] Turing, A. M. (1952) The chemical basis of morphogenesis. Philos. Trans. Roy. Soc. Lond. 237, 37–72.Google Scholar
[39] Wazwaz, A. M. (2008) The extended tanh method for the Zakharo–Kuznetsov (ZK) equation, the modified ZK equation, and its generalized forms. Commun. Nonlinear Sci. Numer. Simul. 13, 1039–1047.Google Scholar
[40] Zulehner, W. & Ames, W. F. (1983) Group analysis of a semilinear vector diffusion equation. Nonlinear Anal. 7, 945–969.Google Scholar