zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Pattern formation in a diffusive ratio-dependent Holling-Tanner predator-prey model with Smith growth. (English) Zbl 1264.34169
Summary: The spatiotemporal dynamics of a diffusive ratio-dependent Holling-Tanner predator-prey model with Smith growth subject to zero-flux boundary condition are investigated analytically and numerically. The asymptotic stability of the positive equilibrium and the existence of Hopf bifurcation around the positive equilibrium are shown; the conditions of Turing instability are obtained. And with the help of numerical simulations, it is found that the model exhibits complex pattern replication: stripes, spots-stripes mixtures, and spots Turing patterns.

34K60Qualitative investigation and simulation of models
34K18Bifurcation theory of functional differential equations
34K20Stability theory of functional-differential equations
92D25Population dynamics (general)
Full Text: DOI
[1] S. V. Petrovskii and H. Malchow, “A minimal model of pattern formation in a prey-predator system,” Mathematical and Computer Modelling, vol. 29, no. 8, pp. 49-63, 1999. · Zbl 0990.92040 · doi:10.1016/S0895-7177(99)00070-9
[2] D. Alonso, F. Bartumeus, and J. Catalan, “Mutual interference between predators can give rise to turing spatial patterns,” Ecology, vol. 83, no. 1, pp. 28-34, 2002.
[3] A. M. Turing, “The chemical basis of morphogenesis,” Philosactions Transactions of the Royal Society of London. Series B, vol. 237, no. 641, pp. 37-72, 1952.
[4] L. A. Segel and J. L. Jackson, “Dissipative structure: an explanation and an ecological example,” Journal of Theoretical Biology, vol. 37, no. 3, pp. 545-559, 1972.
[5] L. S. Luckinbill, “The effects of space and enrichment on a predator-prey system,” Ecology, vol. 55, no. 5, pp. 1142-1147, 1974.
[6] S. A. Levin, “The problem of pattern and scale in ecology: the Robert H. MacArthur award lecture,” Ecology, vol. 73, no. 6, pp. 1943-1967, 1992.
[7] M. Pascual, “Diffusion-induced chaos in a spatial predator-prey system,” Proceedings of the Royal Society B, vol. 251, no. 1330, pp. 1-7, 1993.
[8] A. B. Medvinsky, S. V. Petrovskii, I. A. Tikhonova, H. Malchow, and B.-L. Li, “Spatiotemporal complexity of plankton and fish dynamics,” SIAM Review, vol. 44, no. 3, pp. 311-370, 2002. · Zbl 1001.92050 · doi:10.1137/S0036144502404442
[9] J. D. Murray, Mathematical Biology. II: Spatial Models and Biomedical Applications, vol. 18 of Interdisciplinary Applied Mathematics, Springer, New York, NY, USA, 3rd edition, 2003.
[10] A. B. Peet, P. A. Deutsch, and E. Peacock-López, “Complex dynamics in a three-level trophic system with intraspecies interaction,” Journal of Theoretical Biology, vol. 232, no. 4, pp. 491-503, 2005. · doi:10.1016/j.jtbi.2004.08.028
[11] D. A. Griffith and P. R. Peres-Neto, “Spatial modeling in ecology: the flexibility of eigenfunction spatial analyses,” Ecology, vol. 87, no. 10, pp. 2603-2613, 2006. · doi:10.1890/0012-9658(2006)87[2603:SMIETF]2.0.CO;2
[12] I. G. Pearce, M. A. J. Chaplain, P. G. Schofield, A. R. A. Anderson, and S. F. Hubbard, “Modelling the spatio-temporal dynamics of multi-species host-parasitoid interactions: heterogeneous patterns and ecological implications,” Journal of Theoretical Biology, vol. 241, no. 4, pp. 876-886, 2006. · doi:10.1016/j.jtbi.2006.01.026
[13] W. Wang, Q.-X. Liu, and Z. Jin, “Spatiotemporal complexity of a ratio-dependent predator-prey system,” Physical Review E, vol. 75, no. 5, Article ID 051913, 9 pages, 2007. · doi:10.1103/PhysRevE.75.051913
[14] W. Wang, L. Zhang, H. Wang, and Z. Li, “Pattern formation of a predator-prey system with Ivlev-type functional response,” Ecological Modelling, vol. 221, no. 2, pp. 131-140, 2010. · doi:10.1016/j.ecolmodel.2009.09.011
[15] W. Wang, Y. Lin, F. Rao, L. Zhang, and Y. Tan, “Pattern selection in a ratio-dependent predator-prey model,” Journal of Statistical Mechanics: Theory and Experiment, vol. 2010, Article ID P11036, 2010.
[16] W. Wang, Y. Lin, L. Zhang, F. Rao, and Y. Tan, “Complex patterns in a predator-prey model with self and cross-diffusion,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 4, pp. 2006-2015, 2011. · Zbl 1221.35423 · doi:10.1016/j.cnsns.2010.08.035
[17] X. Guan, W. Wang, and Y. Cai, “Spatiotemporal dynamics of a Leslie-Gower predator-prey model incorporating a prey refuge,” Nonlinear Analysis: Real World Applications, vol. 12, no. 4, pp. 2385-2395, 2011. · Zbl 1225.49038 · doi:10.1016/j.nonrwa.2011.02.011
[18] Y. Zhu, Y. Cai, S. Yan, and W. Wang, “Dynamical analysis of a delayed reaction-diffusion predator-prey system,” Abstract and Applied Analysis, vol. 2012, Article ID 323186, 23 pages, 2012. · Zbl 1256.35183 · doi:10.1155/2012/323186
[19] Y. Cai, W. Wang, and J. Wang, “Dynamics of a diffusive predator-prey model with additive Allee effect,” International Journal of Biomathematics, vol. 5, no. 2, Article ID 1250023, 11 pages, 2012. · Zbl 1297.92060 · doi:10.1142/S1793524511001659
[20] W. Wang, Z. Guo, R. K. Upadhyay, and Y. Lin, “Pattern formation in a cross-diffusive Holling-Tanner model,” Discrete Dynamics in Nature and Society, vol. 2012, Article ID 828219, 12 pages, 2012. · Zbl 1257.92012 · doi:10.1155/2012/828219
[21] R. Arditi, L. R. Ginzburg, and H. R. Akcakaya, “Variation in plankton densities among lakes: a case for ratio-dependent models,” The American Naturalist, vol. 138, no. 5, pp. 287-1296, 1991.
[22] R. Arditi and H. Saiah, “Empirical evidence of the role of heterogeneity in ratio-dependent consumption,” Ecology, vol. 73, no. 5, pp. 1544-1551, 1992.
[23] A. P. Gutierrez, “Physiological basis of ratio-dependent predator-prey theory: the metabolic pool model as a paradigm,” Ecology, vol. 73, no. 5, pp. 1552-1563, 1992.
[24] T. Saha and C. Chakrabarti, “Dynamical analysis of a delayed ratio-dependent Holling-Tanner predator-prey model,” Journal of Mathematical Analysis and Applications, vol. 358, no. 2, pp. 389-402, 2009. · Zbl 1177.34103 · doi:10.1016/j.jmaa.2009.03.072
[25] R. Arditi and L. R. Ginzburg, “Coupling in predator-prey dynamics: ratio-dependence,” Journal of Theoretical Biology, vol. 139, no. 3, pp. 311-326, 1989.
[26] R. Arditi, N. Perrin, and H. Saiah, “Functional responses and heterogeneities: an experimental test with cladocerans,” Oikos, vol. 60, no. 1, pp. 69-75, 1991.
[27] Z. Liang and H. Pan, “Qualitative analysis of a ratio-dependent Holling-Tanner model,” Journal of Mathematical Analysis and Applications, vol. 334, no. 2, pp. 954-964, 2007. · Zbl 1124.34030 · doi:10.1016/j.jmaa.2006.12.079
[28] M. Banerjee and S. Banerjee, “Turing instabilities and spatio-temporal chaos in ratio-dependent Holling-Tanner model,” Mathematical Biosciences, vol. 236, no. 1, pp. 64-76, 2012. · Zbl 06036392 · doi:10.1016/j.mbs.2011.12.005
[29] F. E. Smith, “Population dynamics in Daphnia Magna and a new model for population growth,” Ecology, vol. 44, pp. 651-663, 1963.
[30] E. C. Pielou, An Introduction to Mathematical Ecology, John Wiley & Sons, New York, NY, USA, 1969. · Zbl 0259.92001
[31] T. G. Hallam and J. T. Deluna, “Effects of toxicants on populations: a qualitative approach III,” Journal of Theoretical Biology, vol. 109, no. 3, pp. 411-429, 1984.
[32] K. Gopalsamy, M. R. S. Kulenović, and G. Ladas, “Environmental periodicity and time delays in a “food-limited” population model,” Journal of Mathematical Analysis and Applications, vol. 147, no. 2, pp. 545-555, 1990. · Zbl 0701.92021 · doi:10.1016/0022-247X(90)90369-Q
[33] W. Feng and X. Lu, “On diffusive population models with toxicants and time delays,” Journal of Mathematical Analysis and Applications, vol. 233, no. 1, pp. 373-386, 1999. · Zbl 0927.35049 · doi:10.1006/jmaa.1999.6332
[34] M. Fan and K. Wang, “Periodicity in a “food-limited” population model with toxicants and time delays,” Acta Mathematicae Applicatae Sinica, vol. 18, no. 2, pp. 309-314, 2002. · Zbl 1025.34070 · doi:10.1007/s102550200030
[35] M. R. Garvie, “Finite-difference schemes for reaction-diffusion equations modeling predator-prey interactions in MATLAB,” Bulletin of Mathematical Biology, vol. 69, no. 3, pp. 931-956, 2007. · Zbl 1298.92081 · doi:10.1007/s11538-006-9062-3
[36] A. Munteanu and R. V. Solé, “Pattern formation in noisy self-replicating spots,” International Journal of Bifurcation and Chaos, vol. 16, no. 12, pp. 3679-3685, 2006. · Zbl 1113.92007 · doi:10.1142/S0218127406017063