Strategy and stationary pattern in a three-species predator-prey model

Journal of Differential Equations

Volumn 200, Issue 2, 2004, Pages 245-273

  Pang, Peter Y H ^a   Wang, Mingxin ^b  

^a NATIONAL UNIVERSITY OF SINGAPORE   (Singapore)

^b SOUTHEAST UNIVERSITY   (China)

Author keywords

Cross diffusion; Predator prey model; Stationary pattern; Strongly coupled elliptic system

Indexed keywords

EID: 2442652435     PISSN: 00220396     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.jde.2004.01.004     Document Type: Article

References (41)

1. H. Amann, Dynamic theory of quasilinear parabolic equations II: reaction-diffusion systems, Differential Integral Equations 3 (1990) 13-75.
2. K.J. Brown, P.C. Dunne, R.A. Gardner, A semilinear parabolic system arising in the theory of superconductivity, J. Differential Equations 40 (1981) 232-252.
3. V. Capasso, O. Diekmann (Eds), Mathematics Inspired by Biology, Lecture Notes in Mathematics Vol. 1714, Springer, Berlin, CIME, Florence, 1999.
4. V. Castets, E. Dulos, J. Boissonade, P. DeKepper, Experimental evidence of a sustained Turing-type equilibrium chemical pattern, Phys. Rev. Lett. 64 (1990) 2953-2956.
5. J. Chattopadhyay, P.K. Tapaswi, Effect of cross-diffusion on pattern formation: a nonlinear analysis, Acta Appl. Math. 48 (1997) 1-12.
6. W.Y. Chen, R. Peng, Stationary patterns created by cross-diffusion for the competitor-mutualist model, J. Math. Anal. Appl. 291 (2) (2004) 550-564.
7. X.F. Chen, W.M. Ni, Y.W. Qi, M.X. Wang, A strongly coupled predator-prey system with non-monotonic functional response, preprint.
8. X.F. Chen, W.M. Ni, Y.W. Qi, M.X. Wang, Steady states of a strongly coupled prey-predator model, preprint.
9. M.C. Cross, P.S. Hohenberg, Pattern formation outside of equilibrium, Rev. Modern Phys. 65 (1993) 851-1112.
10. E.N. Dancer, Z.T. Zhang, Dynamics of Lotka-Volterra competition systems with large interaction, J. Differential Equations 182 (2) (2002) 470-489.
11. F.A. Davidson, B.P. Rynne, A priori bounds and global existence of solutions of the steady-state Sel'kov model, Proc. Roy. Soc. Edinburgh Sect. A 130 (2000) 507-516.
12. Y.H. Du, Y. Lou, Qualitative behaviour of positive solutions of a predator-prey model: effects of saturation, Proc. Roy. Soc. Edinburgh Sect. A 131 (2001) 321-349.
13. B. Ermentrout, Strips or spots? Nonlinear effects in bifurcation of reaction diffusion equation on the square, Proc. Roy. Soc. London 434 (1991) 413-417.
14. A. Gierer, H. Meinhardt, A theory of biological pattern formation, Kybernetik 12 (1972) 30-39.
15. J.K. Hall, Ordinary Differential Equations, Krieger, Malabar FL, 1980.
16. D. Henry, in: Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, Vol. 840, Springer, Berlin, New York, 1993.
17. T. Ikeda, M. Mimura, An interfacial approach to regional segregation of two competing species mediated by a predator, J. Math. Biol. 31 (1993) 215-240.
18. D. Iron, M.J. Ward, J.C. Wei, The stability of spike solutions to the one-dimensional Gierer-Meinhardt model, Phys. D 150 (1-2) (2001) 25-62.
19. Y. Kan-on, Existence and instability of Neumann layer solutions for a 3-component Lotka-Volterra model with diffusion, J. Math. Anal. Appl. 243 (2000) 357-372.
20. Y. Kan-on, M. Mimura, Singular perturbation approach to a 3-component reaction-diffusion system arising in population dynamics, SIAM J. Math. Anal. 29 (1998) 1519-1536.
21. C.S. Lin, W.M. Ni, I. Takagi, Large amplitude stationary solutions to a chemotaxis systems, J. Differential Equations 72 (1988) 1-27.
22. Y. Lou, S. Martinez, W.M. Ni, On 3 × 3 Lotka-Volterra competition systems with cross-diffusion, Discrete Contin. Dynamic Systems 6 (1) (2000) 175-190.
23. Y. Lou, W.M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations 131 (1996) 79-131.
24. Y. Lou, W.M. Ni, Diffusion vs cross-diffusion: an elliptic approach, J. Differential Equations 154 (1999) 157-190.
25. W.M. Ni, Diffusion, cross-diffusion and their spike-layer steady states, Notices Amer. Math. Soc. 45 (1) (1998) 9-18.
26. L. Nirenberg, Topics in Nonlinear Functional Analysis, American Mathematical Society, Providence, RI, 2001.
27. A. Okubo, Diffusion and Ecological Problems: Mathematical Models, Springer, Berlin, New York, 1980.
28. Q. Ouyang, R. Li, G. Li, H.L. Swinney, Dependence of Turing pattern wavelength on diffusion rate, Notices J. Chem. Phys. 102 (1) (1995) 2551-2555.
29. P.Y.H. Pang, M.X. Wang, Qualitative analysis of a ratio-dependent predator-prey system with diffusion, Proc. Roy. Soc. Edinburgh Sect. A 133 (2003) 919-942.
30. P.Y.H. Pang, M.X. Wang, Non-constant positive steady states of a predator-prey system with non-monotonic functional response and diffusion, Proc. London Math. Soc. 88 (1) (2004) 135-157.
31. R. Peng, M.X. Wang, Positive steady-state solutions of the Noyes-Field model for Belousov-Zhabotinskii reaction, Nonlinear Anal. TMA 56 (3) (2004) 451-464.
32. M. Saleem, A.K. Tripathi, A.H. Sadiyal, Coexistence of species in a defensive switching model, Math. Biosci. 181 (2003) 145-164.
33. S. Takahashi, M. Hori, Unstable evolutionarily stable strategy and oscillation: a model of lateral asymmetry in scale-eating cichlids, Amer. Nat. 144 (1994) 1001-1020.
34. A. Turing, The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. Ser. B 237 (1952) 37-72.
35. M.X. Wang, Nonlinear Partial Differential Equations of Parabolic Type, Science Press, Beijing, 1993 (in Chinese).
36. M.X. Wang, Non-constant positive steady states of the Sel'kov model, J. Differential Equations 190 (2) (2003) 600-620.
37. M.X. Wang, Stationary patterns of strongly coupled predator-prey models, J. Math. Anal. Appl., in press.
38. X.F. Wang, Qualitative behavior of solutions of chemotactic diffusion systems: effects of motility and chemotaxis and dynamics, SIAM J. Math. Anal. 31 (3) (2000) 535-560.
39. M.J. Ward, J.C. Wei, Asymmetric spike patterns for the one-dimensional Gierer-Meinhardt model: equilibria and stability, European J. Appl. Math. 13 (3) (2002) 283-320.
40. J.C. Wei, M. Winter, Spikes for the Gierer-Meinhardt system in two dimensions: the strong coupling case, J. Differential Equations 178 (2) (2002) 478-518.
41. E. Yanagida, Mini-maximizer for reaction-diffusion systems with skew-gradient structure, J. Differential Equations 179 (1) (2002) 311-335.