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Sambath, M.; Balachandran, K.; Guin, L. N.

Spatiotemporal patterns in a predator-prey model with cross-diffusion effect. (English) Zbl 1388.35009

Int. J. Bifurcation Chaos Appl. Sci. Eng. 28, No. 2, Article ID 1830004, 12 p. (2018).
Summary: The present research deals with the emergence of spatiotemporal patterns of a two-dimensional (2D) continuous predator-prey system with cross-diffusion effect. First, we work out the critical lines of Hopf and Turing bifurcations of the current model system in a 2D spatial domain by means of bifurcation theory. More specifically, the exact Turing region is specified in a two-parameter space. In effect, by choosing the cross-diffusion coefficient as one of the momentous parameter, we demonstrate that the model system undergoes a sequence of spatiotemporal patterns in a homogeneous environment through diffusion-driven instability. Our results via numerical simulation authenticate that cross-diffusion be able to create stationary patterns which enrich the findings of pattern formation in an ecosystem.

Cited in 18 Documents

MSC:

35B36 Pattern formations in context of PDEs
35K57 Reaction-diffusion equations
35B32 Bifurcations in context of PDEs
35Q92 PDEs in connection with biology, chemistry and other natural sciences
92D25 Population dynamics (general)

Keywords:

cross-diffusion; Turing bifurcation; predator-prey interaction; pattern formation

Software:

PRED_PREY
Cite Review PDF
Full Text: DOI

References:

[1] Alonso, D., Bartumeus, F. & Catalan, J. [2002] “ Mutual interference between predators can give rise to Turing spatial patterns,” Ecology83, 28-34.
[2] Banerjee, M. [2010] “ Self-replication of spatial patterns in a ratio-dependent predator-prey model,” Math. Comput. Model.51, 44-52. · Zbl 1190.37085
[3] Bartumeus, F., Alonso, D. & Catalan, J. [2001] “ Self-organized spatial structures in a ratio-dependent predator-prey model,” Physica A295, 53-57. · Zbl 0978.35016
[4] Bazykin, A. D. [1998] Nonlinear Dynamics of Interacting Populations (World Scientific, Singapore).
[5] Benson, D. L., Sherratt, J. & Maini, P. K. [1993] “ Diffusion driven instability in an inhomogeneous domain,” Bull. Math. Biol.55, 365-384. · Zbl 0758.92003
[6] Choudhury, B. S. & Nasipuri, B. [2015] “ Self-organized spatial patterns due to diffusion in a Holling-Tanner predator-prey model,” Comput. Appl. Math.34, 177-195. · Zbl 1337.92172
[7] Chung, J. M. & Peacock-López, E. [2007] “ Bifurcation diagrams and Turing patterns in a chemical self-replicating reaction-diffusion system with cross diffusion,” J. Chem. Phys.127, 174903.
[8] DeAngelis, D. L., Goldstein, R. L. & O’Neill, R. V. [1975] “ A model for trophic interaction,” Ecology56, 881-892.
[9] Dubey, B., Das, B. & Hussain, J. [2001] “ A predator-prey interaction model with self and cross-diffusion,” Ecol. Model.141, 67-76.
[10] Flierl, G., Grunbaum, D., Levin, S. A. & Olson, D. [1999] “ Individual-based perspectives on grouping,” J. Theor. Biol.196, 397-454.
[11] Garvie, M. R. [2007] “ Finite-difference schemes for reaction-diffusion equations modeling predator-prey interactions in MATLAB,” Bull. Math. Biol.69, 931-956. · Zbl 1298.92081
[12] Guin, L. N. [2014] “ Existence of spatial patterns in a predator-prey model with self- and cross-diffusion,” Appl. Math. Comput.226, 320-335. · Zbl 1354.92062
[13] Guin, L. N. & Mandal, P. K. [2014] “ Spatiotemporal dynamics of reaction-diffusion models of interacting populations,” Appl. Math. Model.38, 4417-4427. · Zbl 1428.92086
[14] Levin, S. A., Powell, T. M. & Steele, J. H. [1992] “ The problem of pattern and scale in ecology: The Robert H. MacArthur Award Lecture,” Ecology73, 1943-1967.
[15] Li, L. & Wang, Z.-J. [2013] “ Dynamics in a predator-prey model with space and noise,” Appl. Math. Comput.219, 6542-6547. · Zbl 1283.92073
[16] Liu, P. P. & Jin, Z. [2009] “ Pattern formation of a predator-prey model,” Nonlin. Anal.: Hybrid Syst.3, 177-183. · Zbl 1184.92056
[17] McGehee, E. A. & Peacock-López, E. [2005] “ Turing patterns in a modified Lotka-Volterra model,” Phys. Lett. A342, 90-98. · Zbl 1222.92065
[18] Morozov, A., Petrovskii, S. & Li, B. L. [2004] “ Bifurcations and chaos in a predator-prey system with the Allee effect,” Proc. Roy. Soc. Lond. B271, 1407-1414.
[19] Murray, J. D. [1993] Mathematical Biology (Springer-Verlag, Berlin). · Zbl 0779.92001
[20] Pagea, K. M., Maini, P. K. & Monk, N. A. M. [2005] “ Complex pattern formation in reaction-diffusion systems with spatially varying parameters,” Physica D202, 95-115. · Zbl 1065.35145
[21] Rao, F. [2013] “ Spatiotemporal dynamics in a reaction-diffusion toxic-phytoplankton-zooplankton model,” J. Stat. Mech.: Th. Exper.2013, P08014. · Zbl 1456.92152
[22] Sambath, M. & Balachandran, K. [2012] “ Pattern formation for a ratio-dependent predator-prey model with cross diffusion,” J. Korean Soc. Ind. Appl. Math.16, 249-256. · Zbl 1316.92072
[23] Sambath, M. & Balachandran, K. [2013] “ Spatiotemporal dynamics of a predator-prey model incorporating a prey refuge,” J. Appl. Anal. Comput.3, 71-80. · Zbl 1300.92089
[24] Sambath, M., Gnanavel, S. & Balachandran, K. [2013] “ Stability and Hopf bifurcation of a diffusive predator-prey model with predator saturation and competition,” Appl. Anal.92, 2439-2456. · Zbl 1284.35229
[25] Sambath, M. & Balachandran, K. [2015] “ Bifurcations in a diffusive predator-prey model with predator saturation and competition response,” Math. Meth. Appl. Sci.38, 785-798. · Zbl 1310.35031
[26] Sun, X. K., Huo, H. F. & Xiang, H. [2009] “ Bifurcation and stability analysis in predator-prey model with a stage-structure for predator,” Nonlin. Dyn.58, 497-513. · Zbl 1183.92085
[27] Sun, G. Q., Zhang, G., Jin, Z. & Li, L. [2012a] “ Predator cannibalism can give rise to regular spatial pattern in a predator-prey system,” Nonlin. Dyn.58, 75-84. · Zbl 1183.92084
[28] Sun, G. Q., Jin, Z., Li, L., Haque, M. & Li, B. L. [2012b] “ Spatial patterns of a predator-prey model with cross diffusion,” Nonlin. Dyn.69, 1631-1638. · Zbl 1263.34062
[29] Sun, G. Q., Zhang, J., Song, L. P., Jin, Z. & Li, B. L. [2012c] “ Pattern formation of a spatial predator-prey system,” Appl. Comput. Math.218, 11151-11162. · Zbl 1278.92041
[30] Turing, A. M. [1952] “ The chemical basis of morphogenesis,” Philos. Trans. Roy. Soc. Lond. Ser. B: Biol. Sci.237, 37-72. · Zbl 1403.92034
[31] Wang, W., Zhang, L., Wang, H. & Li, Z. [2010] “ Pattern formation of a predator-prey system with Ivlev-type functional response,” Ecol. Model.221, 131-140.
[32] Wang, W., Liu, Q. X. & Jin, Z. [2011] “ Complex patterns in a predator-prey model with self and cross-diffusion,” Commun. Nonlin. Sci. Numer. Simul.16, 2006-2015. · Zbl 1221.35423
[33] Zhu, L. M., Wang, A. L., Liu, Y. J. & Wang, B. [2010] “ Stationary patterns of a predator-prey model with spatial effect,” Appl. Comput. Math.216, 3620-3626. · Zbl 1193.92084
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