Pattern formation of a predator-prey system with Ivlev-type functional response

Ecological Modelling

Volumn 221, Issue 2, 2010, Pages 131-140

  Wang, Weiming ^a,b   Zhang, Lei ^c   Wang, Hailing ^d   Li, Zhenqing ^e  

^a WENZHOU UNIVERSITY   (China)

^b FUDAN UNIVERSITY   (China)

^c EAST CHINA NORMAL UNIVERSITY   (China)

^d CITY UNIVERSITY OF HONG KONG   (Hong Kong)

^e INSTITUTE OF BOTANY   (China)

Author keywords

Hopf bifurcation; Spatiotemporal pattern; Spiral; Target; Turing instability

Indexed keywords

DEPENDENT TYPES; EVOLUTION PROCESS; FLUX BOUNDARY CONDITIONS; FUNCTIONAL RESPONSE; INITIAL CONDITIONS; NUMERICAL SIMULATION; PATTERN FORMATION; PREDATOR - PREY SYSTEM; RANDOM PERTURBATIONS; REACTION DIFFUSION; SMALL AMPLITUDE; SPATIAL PATTERN FORMATION; SPATIAL PATTERNS; SPATIOTEMPORAL PATTERN; SPATIOTEMPORAL PATTERNS; SPIRAL PATTERN; STEADY STATE; SUFFICIENT CONDITIONS; TARGET PATTERNS; TURING INSTABILITY;

ECOSYSTEMS; SIMULATORS; TARGETS;

HOPF BIFURCATION;

BIFURCATION; BOUNDARY CONDITION; CHAOTIC DYNAMICS; FUNCTIONAL RESPONSE; NUMERICAL MODEL; PREDATOR-PREY INTERACTION; SIMULATION; SPATIOTEMPORAL ANALYSIS;

EID: 70549086997     PISSN: 03043800     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.ecolmodel.2009.09.011     Document Type: Review

References (35)

1. Alonso D., Bartumeus F., and Catalan J. Mutual interference between predators can give rise to Turing spatial patterns. Ecology 83 (2002) 28-34
2. Baurmann M., Gross T., and Feudel U. Instabilities in spatially extended predator-prey systems: spatio-temporal patterns in the neighborhood of Turing-Hopf bifurcations. J. Theor. Biol. 245 (2007) 220-229
3. Bhattacharyay A. Spirals and targets in reaction-diffusion systems. Phys. Rev. E 64 (2001) 016113(4)
4. Callahan T., and Knobloch E. Pattern formation in the three-dimensional reaction-diffusion systems. Physica D 132 (1999) 339-362
5. Garvie M. Finite-difference schemes for reaction-diffusion equations modelling predator-prey interactions in matlab. Bull. Math. Biol. 69 (2007) 931-956
6. Griffith D., and Peres-Netob P. Spatial modeling in ecology: the flexibility of eigenfunction spatial analyses. Ecology 87 (2006) 2603-2613
7. Hoyle R. Pattern formation: an introduction to methods (2006), Cambridge University Press
8. Hawick, K., James, H., Scogings, C., 2006. A zoology of emergent life patterns in a predator-prey simulation model. Technical Note CSTN-015 and in Proc. IASTED International Conference on Modelling, Simulation and Optimization, September 2006. Gabarone, Botswana, 507-115.
9. Ivlev V. Experimental Ecology of the Feeding Fishes (1961), Yale University Press, New Haven
10. Kay A., and Sherratt J. Spatial noise stabilizes periodic wave patterns in oscillatory systems on finite domains. SIAM J. Appl. Math. 61 (2000) 1013-1041
11. Kooij R. A predator-prey model with Ivlev's functional response. J. Math. Anal. Appl. 198 (1996) 473-489
12. Leppänen, T., 2004. Computational studies of pattern formation in Turing systems. PhD Thesis, Helsinki University of Technology, Finland.
13. Leppänen T., Karttunen M., Kaski K., and Barrio R. Dimensionality effects in Turing pattern formation. Int. J. Mod. Phys. B 17 (2003) 5541-5553
14. Levin S. The problem of pattern and scale in ecology. Ecology 73 (1992) 1943-1967
15. Liu Q.-X., and Jin Z. Formation of spatial patterns in epidemic model with constant removal rate of the infectives. J. Stat. Mech. (2007) P05002
16. Malchow H., Radtke B., Kallache M., Medvinsky A., Tikhonov D., and Petrovskii S. Spatio-temporal pattern formation in coupled models of plankton dynamics and fish school motion. Nonlinear Anal.: Real World Appl. 1 (2000) 53-67
17. May R.M. Stability and Complexity in Model Ecosystems (1981), Princeton University Press, Princeton, New Jersey
18. Medvinsky A., Petrovskii S., Tikhonova I., Malchow H., and Li B.-L. Spatiotemporal complexity of plankton and fish dynamics. SIAM Rev. 44 (2002) 311-370
19. Metz J., and Diekmann O. A gentle introduction to structured population models: three worked examples. In: Metz J.A.J., and Diekmann O. (Eds). The Dynamics of Physiologically Structured Populations (Lecture Notes in Biomathematics 68) (1986), Springer, Berlin 3-45
20. Murray J. Mathematical Biology II: Spatial Models and Biomedical Applications (2003), Springer, Berlin
21. Neuhauser C. Mathematical challenges in spatial ecology. Notices Am. Math. Soc. 47 (2001) 1304-1314
22. Okubo A., and Levin S.A. Diffusion and Ecological Problems: Modern Perspectives (2001), Springer Verlag
23. Pearce I., Chaplain M., Schofield P., Anderson A., and Hubbard S. Modelling the spatio-temporal dynamics of multi-species host-parasitoid interactions: heterogeneous patterns and ecological implications. J. Theor. Biol. 241 (2006) 876-886
24. Preedy K., Schofield P., Chaplain M., and Hubbard S. Disease induced dynamics in host-parasitoid systems: chaos and coexistence. J. R. Soc. Interface 4 (2006) 463-471
25. Savill N., and Hogeweg P. Competition and dispersal in predator-prey waves. Theor. Popul. Biol. 56 (1999) 243-263
26. Sherratt J., Lewis M., and Fowler A. Ecological chaos in the wake of invasion. Proc. Natl. Acad. Sci. 92 (1995) 2524-2528
27. Sherratt J., Eagan B., and Lewis M. Oscillations and chaos behind predator-prey invasion: mathematical artifact or ecological reality?. Phil. Trans. Roy. Soc. Lond. B 352 (1997) 21-38
28. Shoji H., Yamada K., Ueyama D., and Ohta T. Turing patterns in three dimensions. Phys. Rev. E 75 (2007) 046212(13)
29. Sugie J. Two-parameter bifurcation in a predator-prey system of Ivlev type. J. Math. Anal. Appl. 217 (1998) 349-371
30. Tian R. Toward standard parameterization in marine biological modeling. Ecol. Model. 193 (2006) 363-386
31. Turing A. The chemical basis of morphogenesis. Phil. Trans. Roy. Soc. Lond. B 237 B641 (1952) 37-72
32. Uriu K., and Iwasa Y. Turing pattern formation with two kinds of cells and a diffusive chemical. Bull. Math. Biol. 69 (2007) 2515-2536
33. Wang H., and Wang W. The dynamical complexity of a Ivlev-type prey-predator system with impulsive effect. Chaos Solitons Fractals 38 (2007) 1168-1176
34. Wang W., Liu Q.-X., and Jin Z. Spatiotemporal complexity of a ratio-dependent predator-prey system. Phys. Rev. E 75 (2007) 051913
35. Yang L., Dolnik M., Zhabotinsky A., and Epstein I. Pattern formation arising from interactions between Turing and wave instabilities. J. Chem. Phys. 117 (2002) 7259-7265