Abstract
Spatial component of ecological interactions has been identified as an important factor in how ecological communities are shaped. In this paper, we consider a Holling–Tanner model with spatial diffusion. Choosing appropriate parameter values in parameter spaces, we obtain rich patterns, including spotted, black-eye, and labyrinthine patterns. The numerical results show that predator–prey system can exhibit complicated behavior.
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References
Turing, A.M.: The chemical basis of morphogenesis. Philos. Trans. R. Soc. Lond. A 237, 37–72 (1952)
Li, L., Jin, Z.: Pattern dynamics of a spatial predator–prey model with noise. Nonlinear Dyn. (in press)
Medvinsky, A.B., Petrovskii, S.V., Tikhonova, I.A., Malchow, H., Li, B.L.: Spatio-temporal complexity of plankton and fish dynamics in simple model ecosystems. SIAM Rev. 44, 311–370 (2002)
Sun, G.-Q., Jin, Z., Liu, Q.-X., Li, L.: Pattern formation induced by cross-diffusion in a predator–prey system. Chinese Phys. B 17, 3936–3941 (2008)
Sun, G.-Q., Zhang, G., Jin, Z., Li, L.: Predator cannibalism can give rise to regular spatial pattern in a predator–prey system. Nonlinear Dyn. 58, 75–84 (2009)
Liu, Q.-X., Sun, G.-Q., Jin, Z., Li, B.-L.: Emergence of spatiotemporal chaos arising from far-field breakup of spiral waves in the plankton ecological systems. Chinese Phys. B 18, 506–515 (2009)
Liu, P.-P., Jin, Z.: Pattern formation of a predator–prey model. Nonlinear Anal. Hybrid Syst. 3, 177–183 (2009)
Liu, P.-P.: An analysis of a predator–prey model with both diffusion and migration. Math. Comput. Model. 51, 1064–1070 (2010)
Sun, G.-Q., Jin, Z., Li, L., Li, B.-L.: Self-organized wave pattern in a predator–prey model. Nonlinear Dyn. 60, 265–275 (2010)
Lou, Y., Ni, W.M.: Diffusion vs cross-diffusion: an elliptic approach. J. Differ. Equ. 154, 157–190 (1999)
Hsu, S.B., Hwang, T.W.: Global stability for a class of predator–prey systems. SIAM J. Appl. Math. 55, 763–783 (1995)
Peng, R., Wang, M.: Global stability of the equilibrium of a diffusive Holling–Tanner prey–predator model. Appl. Math. Lett. 20, 664–670 (2007)
Peng, R., Wang, M.: Positive steady-states of the Holling–Tanner prey–predator model with diffusion. Proc. R. Soc. Edinb. A 135, 149–164 (2005)
Hsu, S.B., Huang, T.W.: Hopf bifurcation analysis for a predator–prey system of Holling and leslie type. Taiwan. J. Math. 3, 35–53 (1999)
Braza, P.A.: The bifurcation structure of the Holling–Tanner model for predator–prey interactions using two-timing. SIAM J. Appl. Math. 63, 889–904 (2003)
Saez, E., Gonzalez-Olivares, E.: Dynamics of a predator–prey model. SIAM J. Appl. Math. 59, 1867–1878 (1999)
Hsu, S.B., Huang, T.W.: Uniqueness of limit cycles for a predator–prey system of Holling and Lesile type. Can. Appl. Math. Q. 6, 91–99 (1998)
Collings, J.B.: Bifurcation and stability analysis of a temperature dependent mite predator–prey interaction model incorporating a prey refuge. Bull. Math. Biol. 57, 63–76 (1995)
Wollkind, D.J., Collings, J.B., Logan, J.A.: Metastability in a temperature-dependent model system for predator–prey mite outbreak interactions on fruit trees. Bull. Math. Biol. 50, 379–409 (1988)
May, R.M.: Stability and Complexity in Model Ecosystems. Princeton University Press, Princeton (1973)
Shi, R., Chen, L.: The study of a ratio-dependent predator–prey model with stage structure in the prey. Nonlinear Dyn. 58, 443–451 (2009)
Sun, X.-K., Huo, H.-F., Xiang, H.: Bifurcation and stability analysis in predator–prey model with a stage-structure for predator. Nonlinear Dyn. 58, 497–513 (2009)
Wang, X., Tao, Y., Song, X.: A delayed HIV-1 infection model with Beddington–DeAngelis functional response. Nonlinear Dyn. 62, 67–72 (2010)
Pei, Y., Li, S., Li, C.: Effect of delay on a predator–prey model with parasitic infection. Nonlinear Dyn. 63, 311–321 (2011)
Murray, J.D.: Mathematical Biology. II. Spatial Models and Biomedical Applications. Springer, New York (2003)
Andresen, P., Bache, M., Mosekilde, E., Dewel, G., Borckmanns, P.: Stationary space-periodic structures with equal diffusion coefficients. Phys. Rev. E 60, 297–301 (1999)
Kuznetsov, S.P., Mosekilde, E., Dewel, G., Borckmans, P.: Absolute and convective instabilities in a one-dimensional brusselator flow model. J. Chem. Phys. 106, 7609–7616 (1997)
Callahan, T., Knobloch, E.: Pattern formation in three-dimensional reaction–diffusion systems. Physica D 132, 339–362 (1999)
Gunaratne, G., Ouyang, Q., Swinney, H.: Pattern formation in the presence of symmetries. Phys. Rev. E 50, 2802–2820 (1994)
Ipsen, M., Hynne, F., Soensen, P.: Amplitude equations for reaction–diffusion systems with a Hopf bifurcation and slow real modes. Physica D 136, 66–92 (2000)
Pena, B., Perez-Garcia, C.: Stability of Turing patterns in the Brusselator model. Phys. Rev. E 64, 056213 (2001)
Sun, G.-Q., Jin, Z., Liu, Q.-X., Li, L.: Dynamical complexity of a spatial predator–prey model with migration. Ecol. Model. 219, 248–255 (2008)
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Liu, PP., Xue, Y. Spatiotemporal dynamics of a predator–prey model. Nonlinear Dyn 69, 71–77 (2012). https://doi.org/10.1007/s11071-011-0246-5
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DOI: https://doi.org/10.1007/s11071-011-0246-5