Skip to main content Accessibility help
×
Home

Mathematics of Darwin’s Diagram

  • N. Bessonov (a1), N. Reinberg (a1) and V. Volpert (a2)

Abstract

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.

Copyright

Corresponding author

Corresponding author. E-mail: volpert@math.univ-lyon1.fr

References

Hide All
[1] Apreutesei, N., Bessonov, N., Volpert, V., Vougalter, V.. Spatial structures and generalized travelling waves for an integro-differential equation. DCDS B, 13 (2010), no. 3, 537-557.
[2] Apreutesei, N., Ducrot, A., Volpert, V.. Travelling waves for integro-differential equations in population dynamics. Discrete Contin. Dyn. Syst., Ser. B, 11 (2009), no. 3, 541561.
[3] Apreutesei, N., Ducrot, A., Volpert, V.. Competition of species with intra-specific competition. Math. Model. Nat. Phenom., 3 (2008), 127.
[4] Atamas, S.. Self-organization in computer simulated selective systems. Biosystems, 39 (1996), 143-151.
[5] Berestycki, H., Nadin, G., Perthame, B., Ryzhik, L.. The non-local Fisher-KPP equation: travelling waves and steady states. Nonlinearity, 22 (2009), no. 12, 28132844.
[6] Britton, N.F.. Spatial structures and periodic travelling waves in an integro-differential reaction-diffusion population model. SIAM J. Appl. Math., 6 (1990), 16631688.
[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] Demin, I., Volpert, V.. Existence of waves for a nonlocal reaction-diffusion equation. Math. Model. Nat. Phenom., 5 (2010), no. 5, 80101.
[10] Desvillettes, L., Jabin, P.E., Mischler, S., Raoul, G.. On selection dynamics for continuous structured populations. Commun. Math. Sci., 6 (2008), no. 3, 729747.
[11] Dieckmann, U., Doebeli, M.. On the origin of species by sympatric speciation. Nature, 400 (1999), 354357.
[12] Doebeli, M., Dieckmann, U.. Evolutionary branching and sympatric speciation caused by different types of ecological interactions. The American Naturalist, 156 (2000), S77S101.
[13] Ducrot, A., Marion, M., Volpert, V.. Spectrum of some integro-differential operators and stability of travelling waves. Nonlinear Analysis Series A: Theory, Methods and Applications, 74 (2011), no. 13, 4455-4473.
[14] Fisher, R.A.. The wave of advance of advantageous genes. Ann. Eugenics, 7 (1937), 355369.
[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] Genieys, S., Volpert, V., Auger, P.. Pattern and waves for a model in population dynamics with nonlocal consumption of resources. Math. Model. Nat. Phenom., 1 (2006), no. 1, 6582.
[18] Genieys, S., Volpert, V., Auger, P.. Adaptive dynamics: modelling Darwin’s divergence principle. Comptes Rendus Biologies, 329 (2006) no. 11, 876879.
[19] Gourley, S.A.. Travelling front solutions of a nonlocal Fisher equation. J. Math. Biol., 41 (2000), 272284.
[20] Gourley, S.A., Chaplain, M.A.J., Davidson, F.A.. Spatio-temporal pattern formation in a nonlocal reaction-diffusion equation. Dynamical systems, 16 (2001), no. 2, 173192.
[21] Iron, D., Ward, M.J.. A metastable spike solution for a nonlocal reaction-diffusion model. SIAM J. Appl. Math., 60 (2000), no. 3, 778802.
[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] J. Murray. Mathematical Biology. Second edition, 1993; Third edition, Volumes I and II, 2003. Springer, Heidelberg.
[26] Nadin, G., Rossi, L., Ryzhik, L., Perthame, B.. Wave-like solutions for nonlocal reaction-diffusion equations: a toy model. Math. Model. Nat.Phenom., 8 (2013), no. 3, 3341.
[27] Nec, Y., Ward, M. J.. The stability and slow dynamics of two-spike patterns for a class of reaction-diffusion system. Math. Model. Nat. Phenom., 8 (2013), no.5.
[28] Perthame, B., Genieys, S.. Concentration in the nonlocal Fisher equation: the Hamilton-Jacobi limit. Math. Model. Nat. Phenom., 4 (2007), 135151.
[29] A. Scheel. Radially symmetric patterns of reaction-diffusion systems. Memoirs of the AMS, 165 (2003), no. 3., 86 p.
[30] Tzou, J.C., Bayliss, A., Matkowsky, B.J., Volpert, V.A.. Stationary and slowly moving localised pulses in a singularly perturbed Brusselator model. Euro. Jnl. of Applied Mathematics, 22 (2011), 423453.
[31] Verhulst, P.-F.. Notice sur la loi que la population suit dans son accroissement. Correspondance mathématique et physique. 10 (1838), 113121.
[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] Volpert, V., Petrovskii, S.. Reaction-diffusion waves in biology. Physics of Life Reviews, 6 (2009), 267310.
[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] Wei, J., Winter, M.. Existence, classification and stability analysis of multiple-peaked solutions for the Gierer-Meinhardt system in R 1. Methods Appl. Anal., 14 (2007), no. 2, 119163.
[39] Zhang, F.. Coexistence of a pulse and multiple spikes and transition layers in the standing waves of a reaction-diffusion system. J. Differential Equations, 205 (2004), 77155.

Keywords

Mathematics of Darwin’s Diagram

  • N. Bessonov (a1), N. Reinberg (a1) and V. Volpert (a2)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.