Geodesic
Mathematics of Darwin’s Diagram
N. Bessonov1 ; N. Reinberg1 ; V. Volpert2
1 Institute of Problems of Mechanical Engineering, Russian Academy of Sciences 199178 Saint Petersburg, Russia
2 Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, 69622 Villeurbanne, France
Mathematical modelling of natural phenomena, Tome 9 (2014) no. 3, pp. 5-25.

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Darwin illustrated his theory about emergence and evolution of biological species with a diagram. It shows how species exist, evolve, appear and disappear. The goal of this work is to give a mathematical interpretation of this diagram and to show how it can be reproduced in mathematical models. It appears that conventional models in population dynamics are not sufficient, and we introduce a number of new models which take into account local, nonlocal and global consumption of resources, and models with space and time dependent coefficients.
DOI : 10.1051/mmnp/20149302

N. Bessonov 1 ; N. Reinberg 1 ; V. Volpert 2

1 Institute of Problems of Mechanical Engineering, Russian Academy of Sciences 199178 Saint Petersburg, Russia
2 Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, 69622 Villeurbanne, France 
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N. Bessonov; N. Reinberg; V. Volpert. Mathematics of Darwin’s Diagram. Mathematical modelling of natural phenomena, Tome 9 (2014) no. 3, pp. 5-25. doi : 10.1051/mmnp/20149302. http://geodesic.mathdoc.fr/articles/10.1051/mmnp/20149302/

[1] N. Apreutesei, N. Bessonov, V. Volpert, V. Vougalter DCDS B 2010 537 557

[2] N. Apreutesei, A. Ducrot, V. Volpert Discrete Contin. Dyn. Syst., Ser. B 2009 541 561

[3] N. Apreutesei, A. Ducrot, V. Volpert Math. Model. Nat. Phenom. 2008 1 27

[4] S. Atamas Biosystems 1996 143 151

[5] H. Berestycki, G. Nadin, B. Perthame, L. Ryzhik Nonlinearity 2009 2813 2844

[6] N.F. Britton SIAM J. Appl. Math. 1990 1663 1688

[7] J.A. Coyne, H.A. Orr. Speciation. Sinauer Associates, Sunderland, 2004.

[8] C. Darwin. The origin of species by means of natural selection. Barnes Noble Books, New York, 2004. Publication prepared on the basis of the first edition appeared in 1859.

[9] I. Demin, V. Volpert Math. Model. Nat. Phenom. 2010 80 101

[10] L. Desvillettes, P.E. Jabin, S. Mischler, G. Raoul Commun. Math. Sci. 2008 729 747

[11] U. Dieckmann, M. Doebeli Nature 1999 354 357

[12] M. Doebeli, U. Dieckmann The American Naturalist 2000 S77 S101

[13] A. Ducrot, M. Marion, V. Volpert Nonlinear Analysis Series A: Theory, Methods and Applications 2011 4455 4473

[14] R.A. Fisher Ann. Eugenics 1937 355 369

[15] S. Gavrilets. Fitness Landscape and the Origin of Species. Princeton University Press, Princeton, 2004.

[16] S. Genieys, N. Bessonov, V. Volpert. Mathematical model of evolutionary branching. Mathematical and computer modelling, 2008, doi: 10/1016/j.mcm.2008.07.023

[17] S. Genieys, V. Volpert, P. Auger Math. Model. Nat. Phenom. 2006 65 82

[18] S. Genieys, V. Volpert, P. Auger Comptes Rendus Biologies 2006 876 879

[19] S.A. Gourley J. Math. Biol. 2000 272 284

[20] S.A. Gourley, M.A.J. Chaplain, F.A. Davidson Dynamical systems 2001 173 192

[21] D. Iron, M.J. Ward SIAM J. Appl. Math. 2000 778 802

[22] A.N. Kolmogorov, I.G. Petrovsky, N.S. Piskunov. A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem. Bull. Moscow Univ., Math. Mech., 1:6 (1937), 1-26. In: Selected Works of A.N. Kolmogorov, Vol. 1, V.M. Tikhomirov, Editor, Kluwer, London, 1991.

[23] A. Lotka. Elements of Physical Biology. Williams Wilkins, Baltimore, 1925.

[24] T.R. Malthus. Essay on the Principle of Population. Printed for J. Johnson, in St. Paul’s Church-Yard, 1798.

[25]

[26] G. Nadin, L. Rossi, L. Ryzhik, B. Perthame Math. Model. Nat.Phenom. 2013 33 41

[27] Y. Nec, M. J. Ward Math. Model. Nat. Phenom. 2013

[28] B. Perthame, S. Genieys Math. Model. Nat. Phenom. 2007 135 151

[29] A. Scheel. Radially symmetric patterns of reaction-diffusion systems. Memoirs of the AMS, 165 (2003), no. 3., 86 p.

[30] J.C. Tzou, A. Bayliss, B.J. Matkowsky, V.A. Volpert Euro. Jnl. of Applied Mathematics 2011 423 453

[31] P.-F. Verhulst. Notice sur la loi que la population suit dans son accroissement. Correspondance mathématique et physique. 10 (1838), 113–121.

[32] A.I. Volpert, V. Volpert, V.A. Volpert. Traveling Wave Solutions of Parabolic Systems. Translation of Mathematical Monographs, Vol. 140, AMS, Providence, 1994.

[33] V. Volpert. Elliptic Partial Differential Equations. Volume 1. Fredholm Theory of Elliptic Problems in Unbounded Domains. Birkhäuser, 2011.

[34] V. Volpert. Elliptic Partial Differential Equations. Volume 2. Reaction-diffusion Equations. Birkhäuser, 2014.

[35] V. Volpert, S. Petrovskii Physics of Life Reviews 2009 267 310

[36] V. Volpert, V. Vougalter. Emergence and propagation of patterns in nonlocal reaction-diffusion equations arising in the theory of speciation. In: Dispersal, individual movement and spatial ecology. M. Lewis, Ph. Maini, S. Petrovskii. Editors. Springer Applied Interdisciplinary Mathematics Series. Lecture Notes in Mathematics, Volume 2071, 2013, 331-353.

[37] V. Volterra. Leçons sur la théorie mathématique de la lutte pour la vie. Paris, 1931.

[38] J. Wei, M. Winter Existence, classification and stability analysis of multiple-peaked solutions for the Gierer-Meinhardt system in R1 Methods Appl. Anal. 2007 119 163

[39] F. Zhang J. Differential Equations 2004 77 155

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