Pattern selection in a predation model with self and cross diffusion

doi:10.1088/1674-1056/20/3/034702

Abstract In this paper, we present the amplitude equations for the excited modes in a cross-diffusive predator--prey model with zero-flux boundary conditions. From these equations, the stability of patterns towards uniform and inhomogenous perturbations is determined. Furthermore, we present novel numerical evidence of six typical turing patterns, and find that the model dynamics exhibits complex pattern replications: for $\mu_1<\mu\leq\mu_2$, the steady state is the only stable solution of the model; for $\mu_2<\mu\leq\mu_4$, by increasing the control parameter $\mu$, the sequence $H_{\pi}$-hexagons $\rightarrow$ $H_{\pi}$-hexagon-stripe mixtures $\rightarrow$ stripes $\rightarrow$ $H_{0}$-hexagon-stripe mixtures $\rightarrow$ $H_{0}$-hexagons is observed; for $\mu>\mu_4$, the stripe pattern emerges. This may enrich the pattern formation in the cross-diffusive predator--prey model.

Keywords: cross-diffusion turing instability pattern selection amplitude equations

Received: 29 July 2010
Revised: 19 November 2010
Accepted manuscript online:

PACS:	47.54.-r	(Pattern selection; pattern formation)
	87.23.Cc	(Population dynamics and ecological pattern formation)
	89.75.Kd	(Patterns)

Fund: Project supported by the Natural Science Foundation of Zhejiang Province, China (Grant No. Y7080041), and the Shanghai Postdoctoral Scientific Program, China (Grant No. 09R21410700).

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Wang Wei-Ming(王玮明), Wang Wen-Juan(王文娟), Lin Ye-Zhi(林晔智), and Tan Yong-Ji(谭永基) Pattern selection in a predation model with self and cross diffusion 2011 Chin. Phys. B 20 034702

[1]	Alonso D, Bartumeus F and Catalan J 2002 Ecology 83 28
[2]	Iida M, Mimura M and Ninomiya H 2006 J. Math. Biol. 53 617
[3]	Levin S 1992 Ecology 73 1943
[4]	Madzvamuse A and Maini P 2007 J. Comp. Phys. 225 100
[5]	McGehee E and Peacock-L'opez E 2005 Phys. Lett. A 342 90
[6]	Morozov A and Petrovskii S 2009 Bull. Math. Biol. 71 863
[7]	Murray J 2003 Mathematical Biology (3rd Edition) (New York: Springer)
[8]	Okubo A and Levin S 2001 Diffusion and Ecological Problems: Modern Perspectives (2nd Edition) (New York: Springer)
[9]	Cantrell R and Cosner C 2003 Spatial Ecology via Reaction-Diffusion Equations (Chichester, UK: Wiley)
[10]	Hoyle R 2006 Pattern Formation: An Introduction to Methods (Cambridge: Cambridge University Press)
[11]	Wang W, Liu Q and Jin Z 2007 Phys. Rev. E 75 051913
[12]	Sun G, Jin Z, Liu Q and Li L 2008 Chin. Phys. Lett. 25 2296
[13]	Li L, Jin Z and Sun G 2008 Chin. Phys. Lett. 25 3500
[14]	Baurmann M, Gross T and Feudel U 2007 J Theor. Biol. 245 220
[15]	Bartumeus F, Alonso D and Catalan J 2001 Phys. A 295 53
[16]	Bjornstad O N, Ims R A and Lambin X 1999 TREE 14 427
[17]	de Roos A M, McCauley E and Wilson W 1998 Theor. Popu. Biol. 53 108
[18]	Kawasaki K and Teramoto E 1979 J. Math. Biol. 8 33
[19]	Levin S and Segel L 1985 SIAM Rev. 27 45
[20]	Molofsky J 1994 Ecology 75 30
[21]	Neubert M, Kot M and Lewis M 1995 Theor. Popu. Biol. 48 7
[22]	Wang W, Zhang L, Wang H and Li Z 2010 Ecol. Model 221 131
[23]	Biktashev V, Brindley J, Holden A and Tsyganov M 2004 Chaos 14 988
[24]	Shukla J and Verma S 1981 Bull. Math. Biol. 43 593
[25]	Kerner E 1959 Bull. Math. Biol. 21 217
[26]	Shigesada N, Kawasaki K and Teramoto E 1979 J. Theor. Biol. 79 83
[27]	Chattopadhyay J and Tapaswi P 1997 Acta Appl. Math. 48 1
[28]	Chen L and J"ungel A 2006 J. Differential Equations 224 39
[29]	Dubey B, Das B and Hussain J 2001 Ecol. Model 141 67
[30]	Ko W and Ryu K 2008 J. Math. Anal. Appl. 341 1133
[31]	Ko W and Ryu K 2008 Appl. Math. Lett. 21 1177
[32]	Kuto K and Yamada Y 2004 J. Differential Equations 197 315
[33]	Pang R and Wang M 2004 J. Differential Equations 200 245
[34]	Pao C 2005 Nonl. Anal. 60 1197
[35]	Sun G, Jin Z, Liu Q and Li L 2008 Chin. Phys. B 17 3936
[36]	Vanag V and Epstein I 2009 Phys. Chem. Chem. Phys. 11 897
[37]	Dufiet V and Boissonade J 1992 J. Chem. Phys. 96 664
[38]	Dufiet V and Boissonade J 1992 Phys. A 188 158
[39]	Dufiet V and Boissonade J 1996 Phys. Rev. E 53 4883
[40]	Callahan T and Knobloch E 1999 Phys. D 132 339
[41]	Gunaratne G, Ouyang Q and Swinney H 1994 Phys. Rev. E 50 2802
[42]	Ipsen M, Hynne F and Sorensen P 1998 Chaos 8 834
[43]	Ipsen M, Hynne F and Sorensen P 2000 Phys. D 136 66
[44]	Newell A and Whitehead J 1969 J. Fluid. Mech. 38 279
[45]	Ouyang Q 2000 Pattern Formation in Reaction-Diffusion Systems (Shanghai: Shanghai Sci-Tech Education Publishing House)
[46]	Ouyang Q, Gunaratne G and Swinney H 1993 Chaos 3 707
[47]	Pe na B and P'erez-Garc'hia C 2000 Europhys. Lett. 51 300
[48]	Pe na B and P'erez-Garc'hia C 2001 Phys. Rev. E 64 56213
[49]	Segel L 1969 J. Fluid. Mech. 38 203
[50]	Tang Y and Zhang W 2005 J. Math. Biol. 50 699
[51]	Berezovskaya F, Karev G and Arditi R 2001 J. Math. Biol. 43 221
[52]	Li B and Kuang Y 2007 SIAM J. Appl. Math. 67 1453
[53]	Yang L Nonlinear Chemical Dynamics and Pattern Formation http://hopf.chem.brandeis.edu/ yanglingfa/pattern/.
[54]	Ciliberto S, Coullet P, Lega J, Pampaloni E and Perez-Garcia C 1990 Phys. Rev. Lett. 65 2370 endfootnotesize

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