Nonlinear Dynamics

, Volume 67, Issue 3, pp 1737–1744 | Cite as

Pattern dynamics of a spatial predator–prey model with noise

  • Li Li
  • Zhen Jin
Original Paper


A spatial predator–prey model with colored noise is investigated in this paper. We find that the number of the spotted pattern is increased as the noise intensity is increased. When the noise intensity and temporal correlation are in appropriate levels, the model exhibits phase transition from spotted to stripe pattern. Moreover, we show the number of the spotted and stripe pattern, with respect to both noise intensity and temporal correlation. These studies raise important questions on the role of noise in the pattern formation of the populations, which may well explain some data obtained in the ecosystems.


Predator–prey Noise Phase transition 


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  1. 1.
    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) CrossRefMathSciNetGoogle Scholar
  2. 2.
    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) CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Sun, G.-Q., Jin, Z., Liu, Q.-X., Li, L.: Pattern formation induced by cross-diffusion in a predator–prey system. Chin. Phys. B 17, 3936–3941 (2008) CrossRefGoogle Scholar
  4. 4.
    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) CrossRefGoogle Scholar
  5. 5.
    Murray, J.D.: Mathematical Biology. Springer, Berlin (1989) zbMATHGoogle Scholar
  6. 6.
    Garvie, M.R.: Finite-difference schemes for reaction–diffusion equations modeling predator–prey interactions in Matlab. Bull. Math. Biol. 69, 931–956 (2007) CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Garvie, M.R., Trenchea, C.: Finite element approximations of spatially extended predator–prey interactions with the Holling type II functional response. Numer. Math. 107, 641–667 (2007) CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Holling, C.S.: Resilience and stability of ecological systems. Ann. Rev. Ecolog. Syst. 4, 1–23 (1973) CrossRefGoogle Scholar
  9. 9.
    Folke, C., Carpenter, S.R., Walker, B., Scheffer, M., Elmqvist, T., Gunderson, L.H., Holling, C.: Regime shifts resilience, and biodiversity in ecosystem management. Annu. Rev. Ecol. Evol. Syst. 35, 557–581 (2004) CrossRefGoogle Scholar
  10. 10.
    Richter, O.: Spatio-temporal patterns of gene flow and dispersal under temperature increase. Math. Biosci. 218, 15–23 (2009) CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Scheffer, M., Rinaldi, S., Kuznetsov, Y.A., van Nes, E.H.: Seasonal dynamics of daphnia and algae explained as a periodically forced predator–prey system. Oikos 80, 519–532 (1997) CrossRefGoogle Scholar
  12. 12.
    Scheffer, M., Rinaldi, S.: Minimal models of top-down control of phytoplankton. Freshw. Biol. 45, 265–283 (2000) CrossRefGoogle Scholar
  13. 13.
    Guttal, V., Jayaprakash, C.: Impact of noise on bistable ecological systems. Ecol. Model. 201, 420–428 (2007) CrossRefGoogle Scholar
  14. 14.
    García-Ojalvo, J., Sancho, J.M.: Noise in Spatially Extended Systems. Springer, New York (1999) CrossRefzbMATHGoogle Scholar
  15. 15.
    Horsthemke, W., Lefever, R.: Noise-Induced Transitions. Springer, Berlin (1984) zbMATHGoogle Scholar
  16. 16.
    Lesmes, F., Hochberg, D., Morán, F., Pérez-Mercader, J.: Noise-controlled self-replicating patterns. Phys. Rev. Lett. 91, 238301 (2003) CrossRefGoogle Scholar
  17. 17.
    Gammaitoni, L., Hanggi, P., Jung, P., Marchesoni, F.: Stochastic resonance. Rev. Mod. Phys. 70, 223–287 (1998) CrossRefGoogle Scholar
  18. 18.
    Vilar, J.M.G., Sole, R.V.: Effects of noise in symmetric two-species competition. Phys. Rev. Lett. 80, 4099 (1998) CrossRefGoogle Scholar
  19. 19.
    Scheffer, M., Carpenter, S., Foley, J., Folke, C., Walker, B.: Catastrophic shifts in ecosystems. Nature 413, 591–596 (2001) CrossRefGoogle Scholar
  20. 20.
    Giardina, I., Bouchaud, J.P., Mezard, M.: Proliferation assisted transport in a random environment. J. Phys. A, Math. Gen. 34, L245–252 (2001) CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Staliunas, K.: Spatial and temporal noise spectra of spatially extended systems with order–disorder phase transitions. Int. J. Bifurc. Chaos Appl. Sci. Eng. 11, 2845–2852 (2001) CrossRefGoogle Scholar
  22. 22.
    Bjornstad, O.N., Grenfell, B.T.: Noisy clockwork: time series analysis of population fluctuations in animals. Science 293, 638–643 (2001) CrossRefGoogle Scholar
  23. 23.
    Mantegna, R.N., Spagnolo, B.: Stochastic resonance in a tunnel diode. Phys. Rev. E 49, R1792–R1795 (1994) CrossRefGoogle Scholar
  24. 24.
    Mantegna, R.N., Spagnolo, B.: Noise enhanced stability in an unstable system. Phys. Rev. Lett. 76, 563–566 (1996) CrossRefGoogle Scholar
  25. 25.
    Spagnolo, B., Fiasconaro, A., Valenti, D.: Noise induced phenomena in Lotka–Volterra systems. Fluct. Noise Lett. 3, L177–L185 (2003) CrossRefGoogle Scholar
  26. 26.
    Valenti, D., Fiasconaro, A., Spagnolo, B.: Stochastic resonance and noise delayed extinction in a model of two competing species. Physica A 331, 477–486 (2004) CrossRefMathSciNetGoogle Scholar
  27. 27.
    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) CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    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) zbMATHGoogle Scholar
  29. 29.
    Hsu, S.B., Huang, T.W.: Global stability for a class of predator–prey systems. SIAM J. Appl. Math. 55, 763–783 (1995) CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Wollkind, D.J., Collings, J.B., Logan, J.A.: Metastability in a temperature-dependent model system for predator–prey mite outbreak interactions on fruit flies. Bull. Math. Biol. 50, 379–409 (1988) zbMATHMathSciNetGoogle Scholar
  31. 31.
    May, R.M.: Stability and Complexity in Model Ecosystems. Princeton University Press, Princeton (1973) Google Scholar
  32. 32.
    Reichenbach, T., Mobilia, M., Frey, E.: Noise and correlations in a spatial population model with cyclic competition. Phys. Rev. Lett. 99, 238105 (2007) CrossRefGoogle Scholar
  33. 33.
    Reichenbach, T., Mobilia, M., Frey, E.: Mobility promotes and jeopardizes biodiversity in rock–paper–scissors games. Nature 448, 1046–1049 (2007) CrossRefGoogle Scholar
  34. 34.
    Liu, Q.-X., Li, B.-L., Jin, Z.: Resonance and frequency-locking phenomena in a spatially extended phytoplankton–zooplankton system with additive noise and periodic forces. J. Stat. Mech. Theory Exp. 5, P05011 (2008) CrossRefGoogle Scholar
  35. 35.
    Blasius, B., Huppert, A., Stone, L.: Complex dynamics and phase synchronization in spatially extended ecological systems. Nature 399, 354–359 (1999) CrossRefGoogle Scholar
  36. 36.
    Mankin, R., Ainsaar, A., Haljas, A., Reiter, E.: Trichotomous-noise-induced catastrophic shifts in symbiotic ecosystems. Phys. Rev. E 65, 051108 (2002) CrossRefGoogle Scholar
  37. 37.
    Mankin, R., Sauga, A., Ainsaar, A., Haljas, A., Paunel, K.: Colored-noise-induced discontinuous transitions in symbiotic ecosystems. Phys. Rev. E 69, 061106 (2004) CrossRefGoogle Scholar
  38. 38.
    Sun, G.-Q., Li, L., Jin, Z., Li, B.-L.: Effect of noise on the pattern formation in an epidemic model. Num. Methods Partial Differ. Equ. 26, 1168–1179 (2010) zbMATHMathSciNetGoogle Scholar
  39. 39.
    Higham, D.J.: An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev. 43, 525–546 (2001) CrossRefzbMATHMathSciNetGoogle Scholar
  40. 40.
    Petrovskii, S., Li, B.L., Malchow, H.: Transition to spatiotemporal chaos can resolve the paradox of enrichment. Ecol. Complex. 1, 37–47 (2004) CrossRefGoogle Scholar
  41. 41.
    Malchow, H., Petrovskii, S.V., Venturino, E.: Spatiotemporal Patterns in Ecology and Epidemiology: Theory, Models, and Simulations. Chapman & Hall/CRC, London (2008) zbMATHGoogle Scholar
  42. 42.
    Sun, G.-Q., Jin, Z., Li, L., Liu, Q.-X.: The role of noise in a predator–prey model with allee effect. J. Biol. Phys. 35, 185–196 (2009) CrossRefGoogle Scholar
  43. 43.
    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) CrossRefzbMATHGoogle Scholar
  44. 44.
    Freund, J.A., Schimansky-Geier, L., Beisner, B., Neiman, A., Russell, D.F., Yakusheva, T., Moss, F.: Behavioral stochastic resonance: how the noise from a daphnia swarm enhances individual prey capture by juvenile paddlefish. J. Theor. Biol. 214, 71–83 (2002) CrossRefGoogle Scholar
  45. 45.
    Sun, G.-Q., Liu, Q.-X., Jin, Z., Chakraborty, A., Li, B.-L.: Influence of infection rate and migration on extinction of disease in spatial epidemics. J. Theor. Biol. 264, 95–103 (2010) CrossRefGoogle Scholar
  46. 46.
    Tanner, J.T.: The stability and the intrinsic growth rates of prey and predator populations. Ecology 56, 855–867 (1975) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Department of MathematicsNorth University of ChinaTaiyuanPeople’s Republic of China
  2. 2.Department of MathematicsTaiyuan Institute of TechnologyTaiyuanPeople’s Republic of China

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