Turing patterns in a modified Lotka-Volterra model. (English) Zbl 1222.92065

Summary: We consider a modified Lotka-Volterra model, widely known as the Bazykin model, which is the MacArthur-Rosenzweig (MR) model that includes a prey-dependent response function and is modified with the inclusion of intraspecies interactions. We show that a quadratic intra-prey interaction term, which is the most realistic nonlinearity, yields sufficient conditions for Turing patterns. For the Bazykin model we find the Turing region the in parameter space and Turing patterns in one dimension.
Editorial remark: Note that the content of the article actually argues the intra-predator interaction term as the crucial factor instead of yielding sufficient conditions.


92D40 Ecology
92C15 Developmental biology, pattern formation


Full Text: DOI


[1] Turing, A. M., Philos. Trans. R. Soc. London B, 237, 37 (1952)
[2] Bartumeus, F.; Alonso, D.; Catalan, J., Physica A, 295, 53 (2001) · Zbl 0978.35016
[3] Alonso, D.; Bartumeus, F.; Catalan, J., Ecology, 83, 28 (2002)
[4] DeAngelis, D. L.; Goldstein, R. L.; O’Neill, R. V., Ecology, 56, 881 (1975)
[5] Turchin, P., Complex Population Dynamics (2003), Princeton Univ. Press: Princeton Univ. Press Princeton · Zbl 1062.92077
[6] Kot, M., Elements of Mathematical Ecology (2001), Cambridge Univ. Press: Cambridge Univ. Press Cambridge
[7] Beddington, J. R., J. Anim. Ecol., 44, 331 (1975)
[8] Bazykin, A. D., (Voprosy Matematicheskoy Genetiki (1974), Nauka: Nauka Novosibirsk), 103
[9] Rosenzweig, M. L.; MacArthur, R. H., Am. Naturalist, 97, 209 (1963)
[10] Peacock-López, E., WSEAS Trans. Biol. Med., 1, 76 (2004)
[11] Peet, A. B.; Deutsch, P. A.; Peacock-López, E., J. Theor. Biol., 232, 491 (2005)
[12] Holling, C. S., Mem. Entomol. Soc. Can., 45, 1 (1965)
[13] Okubo, A.; Levin, S. A., Diffusion and Ecological Problems (2001), Springer: Springer New York
[14] Murray, J. D., Mathematical Biology II (2003), Springer: Springer Berlin
[15] Neubert, M. G.; Klanjscek, T.; Caswell, H., Ecol. Modelling, 179, 29 (2004)
[16] Segel, L.; Jackson, J., J. Theor. Biol., 37, 545 (1972)
[17] Mimura, M.; Murray, J. D., J. Theor. Biol., 75, 249 (1978)
[18] Neubert, M. G.; Caswell, H.; Murray, J. D., Math. Biosci., 175, 1 (2002)
[19] Wolfram, S., Mathematica, Ver. 5.1 (2004), Springer-Verlag: Springer-Verlag New York
[20] Ermentrout, B., Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students (2002), SIAM: SIAM Philadelphia · Zbl 1003.68738
[21] Castets, V.; Dulos, E.; Boissonade, J.; De Kepper, P., Phys. Rev. Lett., 64, 2953 (1990)
[22] Rovinsky, A. B.; Menzinger, M., Phys. Rev. Lett., 70, 778 (1993)
[23] Buceta, J.; Lindenberg, K., Phys. Rev. E, 66, 046202 (2002)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.