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EMERGENCE OF OSCILLATORY TURING PATTERNS INDUCED BY CROSS DIFFUSION IN A PREDATOR–PREY SYSTEM

School of Electronic and Computer Science and Technology, North University of China, Taiyuan, Shan'xi 030051, P. R. China

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ZHEN JIN

Department of Mathematics, North University of China, Taiyuan, Shan'xi 030051, P. R. China

Corresponding author.

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LI LI

Department of Mathematics, North University of China, Taiyuan, Shan'xi 030051, P. R. China

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, and

JIAN-ZHONG WANG

School of Electronic and Computer Science and Technology, North University of China, Taiyuan, Shan'xi 030051, P. R. China

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https://doi.org/10.1142/S0217979212501937Cited by:6 (Source: Crossref)

Abstract

In this paper, we presented a predator–prey model with self diffusion as well as cross diffusion. By using theory on linear stability, we obtain the conditions on Turing instability. The results of numerical simulations reveal that oscillating Turing patterns with hexagons arise in the system. And the values of the parameters we choose for simulations are outside of the Turing domain of the no cross diffusion system. Moreover, we show that cross diffusion has an effect on the persistence of the population, i.e., it causes the population to run a risk of extinction. Particularly, our results show that, without interaction with either a Hopf or a wave instability, the Turing instability together with cross diffusion in a predator–prey model can give rise to spatiotemporally oscillating solutions, which well enrich the finding of pattern formation in ecology.

Keywords:

PACS: 87.23.-n, 87.23.Cc, 89.75.Kd, 89.75.Fb, 47.54.-r

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References

V. Voltera , Opere Matematiche: Mmemorie e Note V ( Acc. Naz. dei Lincei , Roma, Cremona , 1962 ) . Google Scholar
A. J. Lotka , Elements of Mathematical Biology ( Dover , New York , 1962 ) . Google Scholar
C. Jost, Ph.D. thesis, Institute National Agronomique, Paris-Grignon (1998) . Google Scholar
S.-B. Hsu and T.-W. Huang, SIAM J. Appl. Math. 55, 763 (1995), DOI: 10.1137/S0036139993253201. Web of Science, Google Scholar
K. S. Cheng, S. B. Hsu and S. S. Lin, J. Math. Biol. 12, 115 (1981), DOI: 10.1007/BF00275207. Web of Science, Google Scholar
Y. Kuang, J. Math. Biol. 28, 463 (1990), DOI: 10.1007/BF00178329. Web of Science, Google Scholar
S. Ruan and D. Xiao, SIAM J. Appl. Math. 61, 1445 (2001). Web of Science, Google Scholar
A. B. Medvinskyet al., SIAM Rev. 44, 311 (2002), DOI: 10.1137/S0036144502404442. Web of Science, ADS, Google Scholar
G.-Q. Sunet al., Chin. Phys. B 17, 3936 (2008). Web of Science, ADS, Google Scholar
P. H. Crowley, Am. Nat. 118, 673 (1981), DOI: 10.1086/283861. Web of Science, Google Scholar
W. S. C. Gurneyet al., Ecology 79, 2516 (1998). Web of Science, Google Scholar
G.-Q. Sunet al., Nonlin. Dyn. 58, 75 (2009), DOI: 10.1007/s11071-008-9462-z. Web of Science, Google Scholar
G.-Q. Sunet al., Int. J. Mod. Phys. C 20, 71 (2009), DOI: 10.1142/S0129183109013467. Link, Web of Science, ADS, Google Scholar
Z. Fei, B. J. Green and J. L. Hudson, J. Phys. Chem. B 103, 2178 (1999), DOI: 10.1021/jp983252j. Web of Science, Google Scholar
J.-J. Perraudet al., Phys. Rev. Lett. 71, 1272 (1993), DOI: 10.1103/PhysRevLett.71.1272. Web of Science, ADS, Google Scholar
M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65, 851 (1993), DOI: 10.1103/RevModPhys.65.851. Web of Science, ADS, Google Scholar
A. De. Witet al., Phys. Rev. E 54, 261 (1996). Web of Science, ADS, Google Scholar
L. Yang, A. M. Zhabotinsky and I. R. Epstein, Phys. Rev. Lett. 92, 198303 (2004), DOI: 10.1103/PhysRevLett.92.198303. Web of Science, Google Scholar
L. Yang and I. R. Epstein, Phys. Rev. Lett. 90, 178303 (2003), DOI: 10.1103/PhysRevLett.90.178303. Web of Science, ADS, Google Scholar
A. Kaminaga, V. K. Vanag and I. R. Epstein, J. Phys. Chem. 122, 174706 (2005). Web of Science, Google Scholar
L. Yanget al., J. Chem. Phys. 117, 7259 (2002), DOI: 10.1063/1.1507110. Web of Science, ADS, Google Scholar
R.-T. Liu, S.-S. Liaw and P. K. Maini, J. Kor. Phys. Soc. 50, 234 (2007), DOI: 10.3938/jkps.50.1827. Web of Science, Google Scholar
V. K. Vanag and I. R. Epstein, Phys. Rev. E 71, 066212 (2005), DOI: 10.1103/PhysRevE.71.066212. Web of Science, Google Scholar
E. P. Zemskovet al., Nonlinearity 13, 2063 (2000), DOI: 10.1088/0951-7715/13/6/309. Web of Science, Google Scholar
E. P. Zemskovet al., Physica D 183, 117 (2003), DOI: 10.1016/S0167-2789(03)00138-6. Web of Science, ADS, Google Scholar
V. N. Biktashev and M. A. Tsyganov, Proc. R. Soc. Lond. Ser. A 461, 3711 (2005). Web of Science, ADS, Google Scholar
M. A. Tsyganovet al., Phys. Rev. Lett. 91, 218102 (2003), DOI: 10.1103/PhysRevLett.91.218102. Web of Science, Google Scholar
M. A. Tsyganov and V. N. Biktashev, Phys. Rev. E 70, 031901 (2004), DOI: 10.1103/PhysRevE.70.031901. Web of Science, Google Scholar
B. Dubey, B. Das and J. Hussain, Ecol. Model. 141, 67 (2001), DOI: 10.1016/S0304-3800(01)00255-1. Web of Science, Google Scholar
E. P. Zemskov, K. Kassner and M. J. B. Hauser, Phys. Rev. E 77, 036219 (2008), DOI: 10.1103/PhysRevE.77.036219. Web of Science, Google Scholar
Y. Kuang and E. Beretta, J. Math. Biol. 36, 389 (1998), DOI: 10.1007/s002850050105. Web of Science, Google Scholar
D. M. Xiao and L. S. Jennings, SIAM J. Appl. Math. 65, 737 (2005), DOI: 10.1137/S0036139903428719. Web of Science, Google Scholar
G.-Q. Sunet al., Ecol. Model. 219, 248 (2008), DOI: 10.1016/j.ecolmodel.2008.08.009. Web of Science, Google Scholar
A. M. Turing, Philos. Trans. R. Soc. Lond. B 237, 37 (1952), DOI: 10.1098/rstb.1952.0012. Web of Science, ADS, Google Scholar
V. Castetset al., Phys. Rev. Lett. 64, 2953 (1990), DOI: 10.1103/PhysRevLett.64.2953. Web of Science, Google Scholar
J. M. Chung and E. Peacock-López, J. Chem. Phys. 127, 174903 (2007), DOI: 10.1063/1.2784554. Web of Science, ADS, Google Scholar
J. M. Chung and E. Peacock-López, Phys. Lett. A 371, 41 (2007), DOI: 10.1016/j.physleta.2007.04.114. Web of Science, ADS, Google Scholar
L. F. Yang, A. M. Zhabotinsky and I. R. Epstein, Phys. Rev. Lett. 92, 19 (2004). Google Scholar
A. J. Koch and H. Meinhart, Rev. Mod. Phys. 66, 1481 (1994), DOI: 10.1103/RevModPhys.66.1481. Web of Science, ADS, Google Scholar