Multiple coexistence states for a prey-predator system with cross-diffusion

Journal of Differential Equations

Volumn 197, Issue 2, 2004, Pages 315-348

  Kuto, Kousuke ^a   Yamada, Yoshio ^a  

^a WASEDA UNIVERSITY   (Japan)

Author keywords

Bifurcation; Cross diffusion; Lyapunov Schmidt reduction; Steady state

Indexed keywords

EID: 1342286051     PISSN: 00220396     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.jde.2003.08.003     Document Type: Article

References (21)

1. J. Blat, K.J. Brown, Bifurcation of steady-state solutions in predator-prey and competition systems, Proc. Royal. Soc. Edinburgh 97A (1984) 21-34.
2. M.G. Crandall, P.H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal. 8 (1971) 321-340.
3. M.G. Crandall, P.H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues, and linearized stability, Arch. Rational Mech. Anal. 52 (1973) 161-180.
4. E.N. Dancer, On positive solutions of some pairs of differential equations, Trans. Amer. Math. Soc. 284 (1984) 729-743.
5. E.N. Dancer, On positive solutions of some pairs of differential equations, II, J. Differential Equations 60 (1985) 236-258.
6. E.N. Dancer, On uniqueness and stability for solutions of singularly perturbed predator-prey type equations with diffusion, J. Differential Equations 102 (1993) 1-32.
7. Y. Du, Y. Lou, S-shaped global bifurcation curve and Hopf bifurcation of positive solutions to a predator-prey model, J. Differential Equations 144 (1998) 390-440.
8. B. Dubey, B. Das, J. Hassain, A predator-prey interaction model with self and cross-diffusion, Ecol. Modelling 141 (2002) 67-76.
9. C. Gui, Y. Lou, Uniqueness and nonuniqueness of coexistence states in the Lotka-Volterra competition model, Comm. Pure. Appl. Math. 47 (1994) 1571-1594.
10. L. Li, Coexistence theorems of steady states for predator prey interacting system, Trans. Amer. Math. Soc. 305 (1988) 143-166.
11. L. Li, On positive solutions of a nonlinear equilibrium boundary value problem, J. Math. Anal. Appl. 138 (1989) 537-549.
12. J. López-Gómez, Nonlinear eigenvalues and global bifurcation to the search of positive solutions for general Lotka-Volterra reaction diffusion systems with two species, Differential Integral Equations 7 (1994) 1427-1452.
13. J. López-Gómez, R. Pardo, Coexistence regions in Lotka-Volterra models with diffusion, Nonlinear Anal. TMA 19 (1992) 11-28.
14. J. López-Gómez, R. Pardo, Existence and uniqueness of coexistence states for the predator-prey model with diffusion, Differential Integral Equations 6 (1993) 1025-1031.
15. K. Nakashima, Y. Yamada, Positive steady states for prey-predator models with cross-diffusion, Adv. Differential Equations 6 (1996) 1099-1122.
16. A. Okubo, L.A. Levin, Diffusion and ecological problems: modern perspective, Interdisciplinary Applied Mathematics, 2nd Edition, Vol. 14, Springer, New York, 2001.
17. C.V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.
18. R.H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal. 7 (1971) 487-513.
19. K. Ryu, I. Ahn, Positive steady-states for two interacting species models with linear self-cross diffusions, Discrete Contin. Dynam. Systems 9 (2003) 1049-1061.
20. N. Shigesada, K. Kawasaki, E. Teramoto, Spatial segregation of interacting species, J. Theoret. Biol. 79 (1979) 83-99.
21. Y. Yamada, Stability of steady states for prey-predator diffusion equations with homogeneous Dirichlet conditions, SIAM J. Math. Anal. 21 (1990) 327-345.