Spatiotemporal complexity of a predator-prey system with constant harvest rate. (English) Zbl 1198.35127

Summary: We investigate the emergence of a predator-prey system with Michaelis-Menten-type predator-prey systems with reaction-diffusion and constant harvest rate. We derive the conditions for Hopf and Turing bifurcation on the spatial domain. The results of spatial pattern analysis, via numerical simulations, typical spatial pattern formation is isolated groups, i.e., stripe-like, patch-like and so on. Our results show that modeling by reaction-diffusion equations is an appropriate tool for investigating fundamental mechanisms of complex spatiotemporal dynamics. It will be useful for studying the dynamic complexity of ecosystems.
Editorial remark: There are doubts about a proper peer-reviewing procedure of this journal. The editor-in-chief has retired, but, according to a statement of the publisher, articles accepted under his guidance are published without additional control.


35K57 Reaction-diffusion equations
92D25 Population dynamics (general)
Full Text: DOI


[1] Alonso, D.; Bartumeus, F.; Catalan, J., Mutual interference between predators can give rise to Turing spatial patterns, Ecology, 83, 28-34 (2002)
[2] Baurmann, M.; Gross, T.; Feudel, U., Instabilities in spatially extended predator-prey systems: spatio-temporal patterns in the neighborhood of Turing-Hopf bifurcations, J Theor Biol, 245, 220-229 (2007) · Zbl 1451.92248
[3] Callahan, T. K.; Knobloch, E., Pattern formation in the three-dimensional reaction-diffusion systems, Phys D, 132, 339-362 (1999) · Zbl 0935.35065
[4] Garvie, M. R., Finite-difference schemes for reaction-diffusion equations modelling predator-prey interactions in matlab, Bull Math Biol, 69, 931-956 (2007) · Zbl 1298.92081
[5] Grevenstettea, M.; Linz, S. J., Model for pattern formation of granular matter on vibratory conveyors, Chaos Solitons & Fractals, 39, 1698-1714 (2007)
[6] Griffith, D. A.; Peres-Netob, P. R., Spatial modeling in ecology: the flexibility of eigenfunction spatial analyses, Ecology, 87, 2603-2613 (2006)
[7] Hutt, A.; Atay, F. M., Spontaneous and evoked activity in extended neural populations with gamma-distributed spatial interactions and transmission delay, Chaos Solitons & Fractals, 32, 547-560 (2007) · Zbl 1133.92005
[8] Kuang, Y.; Beretta, E., Global qualitative analysis of a ratio-dependent predator-prey system, J Math Biol, 36, 389-406 (1998) · Zbl 0895.92032
[9] Leppänen T. Computational studies of pattern formation in Turing systems, PhD thesis, Finland: Helsinki University of Technology; 2004.; Leppänen T. Computational studies of pattern formation in Turing systems, PhD thesis, Finland: Helsinki University of Technology; 2004.
[10] Leppänen, T.; Karttunen, M.; Kaski, K.; Barrio, R. A., Dimensionality effects in Turing pattern formation, Int J Mod Phys B, 17, 5541-5553 (2003)
[11] Levin, S. A.; Grenfell, B.; Hastings, A.; Perelson, A. S., Mathematical and computational challenges in population biology and ecosystems science, Science, 275, 334-343 (1997) · Zbl 1225.92058
[12] Li, P.; Li, Z.; Halang, W. A.; Chen, G., Li-yorke chaos in a spatiotemporal chaotic system, Chaos Solitons & Fractals, 33, 335-341 (2007) · Zbl 1133.37311
[13] Li, W.; Wu, S., Traveling waves in a diffusive predator-prey model with holling type-iii functional response, Chaos Solitons & Fractals, 37, 476-486 (2006) · Zbl 1155.37046
[14] Liu, Q.; Jin, Z., Formation of spatial patterns in epidemic model with constant removal rate of the infectives, J Stat Mech, P05002 (2007)
[15] Maini, P. K., Using mathematical models to help understand biological pattern formation, Comp Rend Biol, 327, 225-234 (2004)
[16] Maini, P. K.; Baker, R. E.; Chuong, C., The Turing model comes of molecular age, Science, 314, 1397-1398 (2006)
[17] Medvinsky, A. B.; Petrovskii, S. V.; Tikhonova, I. A.; Malchow, H.; Li, B., Spatiotemporal complexity of plankton and fish dynamics, SIAM Rev, 44, 311-370 (2002) · Zbl 1001.92050
[18] Murray, J. D., (Mathematical biology ii: spatial models and biomedical applications. Mathematical biology ii: spatial models and biomedical applications, Biomathematics, vol. 18 (2003), Springer: Springer New York) · Zbl 1006.92002
[19] Neuhauser, C., Mathematical challenges in spatial ecology, Not Amer Math Soc, 47, 1304-1314 (2001) · Zbl 1128.92328
[20] Palaniyandi, P.; Muruganandam, P.; Lakshmanan, M., Coexistence of synchronized and desynchronized patterns in coupled chaotic dynamical systems, Chaos Solitons & Fractals, 36, 991-1018 (2006)
[21] Sanduloviciu, M.; Lozneanu, E.; Popescu, S., On the physical basis of pattern formation in nonlinear systems, Chaos Solitons & Fractals, 17, 183-188 (2003)
[22] Sprott, J. C.; Wildenberg, J. C.; Azizi, Y., A simple spatiotemporal chaotic lotkacvolterra model, Chaos Solitons & Fractals, 26, 1035-1043 (2006) · Zbl 1081.37057
[23] Turing, A. M., The chemical basis of morphogenesis, Phil Trans R Soc London B, 237, 37-72 (1952) · Zbl 1403.92034
[24] Wang, W.; Liu, Q.; Jin, Z., Spatiotemporal complexity of a ratio-dependent predator-prey system, Phys Rev E, 75, 051913 (2007)
[25] Xiao, D.; Jennings, L. S., Bifurcations of a ratio-dependent predator-prey system with constant rate harvesting, SIAM J Appl Math, 65, 737-753 (2005) · Zbl 1094.34024
[26] Xiao, D.; Li, W.; Han, M., Dynamics in a ratio-dependent predator-prey model with predator harvesting, J Math Anal Appl, 324, 14-29 (2006) · Zbl 1122.34035
[27] Yang, L.; Dolnik, M.; Zhabotinsky, A. M.; Epstein, I. R., Pattern formation arising from interactions between Turing and wave instabilities, J Chem Phys, 117, 7259-7265 (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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.