Non-constant positive steady states of a prey-predator system with cross-diffusions

Journal of Mathematical Analysis and Applications

Volumn 332, Issue 2, 2007, Pages 989-1009

  Zeng, Xianzhong ^a,b  

^a CENTRAL SOUTH UNIVERSITY   (China)

^b HUNAN UNIVERSITY OF SCIENCE AND TECHNOLOGY   (China)

Author keywords

Cross diffusion; Degree theory; Non constant positive steady states; Prey predator system

Indexed keywords

EID: 34247391161     PISSN: 0022247X     EISSN: 10960813     Source Type: Journal    
DOI: 10.1016/j.jmaa.2006.10.075     Document Type: Article

References (19)

1. Brown K.J., and Davidson F.A. Global bifurcation in the brusselator system. Nonlinear Anal. 24 12 (1995) 1713-1725
2. Blat J., and Brown K.J. Global bifurcation of positive solutions in some systems of elliptic equations. SIAM J. Math. Anal. 17 6 (1986) 1339-1353
3. Chen W.Y., and Peng R. Stationary patterns created by cross-diffusion for the competitor-competitor-mutualist model. J. Math. Anal. Appl. 291 (2004) 550-564
4. Crandall M.G., and Rabinowitz P.H. Bifurcation from simple eigenvalue. J. Funct. Anal. 8 (1971) 321-340
5. Du Y.H., and Lou Y. S-shaped global bifurcation curve and Hopf bifurcation of positive solutions to a predator-prey model. J. Differential Equations 144 (1998) 390-440
6. Gilbarg D., and Trudinger N.S. Elliptic Partial Differential Equations of Second Order (1977), Springer, New York
7. Gu Y.G., and Wang M.X. Existence of positive stationary solutions and threshold results for a reaction-diffusion system. J. Differential Equations 130 2 (1996) 277-291
8. Kuto K. Stability of steady-state solutions to a prey-predator system with cross-diffusion. J. Differential Equations 197 (2004) 293-314
9. Kuto K., and Yamada Y. Multiple coexistence states for a prey-predator system with cross-diffusion. J. Differential Equations 197 (2004) 315-348
10. Lou Y., and Ni W.N. Diffusion vs cross-diffusion: An elliptic approach. J. Differential Equations 154 (1999) 157-190
11. Lin C.S., Ni W.M., and Takagi I. Large amplitude stationary solutions to a chemotaxis system. J. Differential Equations 72 (1988) 1-27
12. Nirenberg L. Topics in Nonlinear Functional Analysis (2001), Amer. Math. Soc.
13. Pang P.Y.H., and Wang M.X. Strategy and stationary pattern in a three-species predator-prey model. J. Differential Equations 200 (2004) 245-273
14. Schreiber S.J. Coexistence for species sharing a predator. J. Differential Equations 196 (2004) 209-225
15. Sander E., and Wanner T. Pattern formation in a nonlinear model for animal coats. J. Differential Equations 191 (2003) 143-174
16. Wang M.X. Non-constant positive steady states of the Sel'kov model. J. Differential Equations 190 (2003) 600-620
17. Wang M.X. Stationary patterns of strongly coupled prey-predator models. J. Math. Anal. Appl. 292 (2004) 484-505
18. Y.G. Gu, X.Z. Zeng, Existence of positive stationary solutions for a prey-predator model with the third boundary value to the prey, Acta Math. Sci., in press (in Chinese)
19. Zeng X.Z. A ratio-dependent predator-prey model with diffusion, Nonlinear Anal. Real World Appl., in press. doi:10.1016/j.nonrwa.2006.06.006