A diffusive predator-prey model in heterogeneous environment

Journal of Differential Equations

Volumn 203, Issue 2, 2004, Pages 331-364

  Du, Yihong ^a   Hsu, Sze Bi ^b  

^a UNIVERSITY OF NEW ENGLAND   (Australia)

^b NATIONAL TSING HUA UNIVERSITY   (Taiwan)

Author keywords

A priori estimate; Coexistence; Pattern; Predator prey

Indexed keywords

EID: 3343010401     PISSN: 00220396     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.jde.2004.05.010     Document Type: Article

References (27)

1. Ali S.W. Cosner C. Models for the effects of individual size and spatial scale on competition between species in heterogeneous environments Math. Biosci. 127 1995 45-76
2. Brown P.N. Decay to uniform states in ecological interactions SIAM J. Appl. Math. 38 1980 22-37
3. Cantrell R.S. Cosner C. Hutson V. Ecological models, permanence and spatial heterogeneity Rocky Mountain J. Math. 26 1996 1-35
4. Dancer E.N. Du Y. Effects of certain degeneracies in the predator-prey model SIAM J. Math. Anal. 34 2002 292-314
5. De Mottoni P. Rothe F. Convergence to homogeneous equilibrium state for generalized Volterra-Lotka systems with diffusion SIAM J. Appl. Math. 37 1979 648-663
6. Du Y. Effects of a degeneracy in the competition model, Part I. Classical and generalized steady-state solutions J. Differential Equations 181 2002 92-132
7. Du Y. Effects of a degeneracy in the competition model, Part II. Perturbation and dynamical behaviour J. Differential Equations 181 2002 133-164
8. Du Y. Realization of prescribed patterns in the competition model J. Differential Equations 193 2003 147-179
9. Du Y. Spatial patterns for population models in a heterogeneous environment Taiwanese J. Math. 8 2004 155-182
10. Du Y. Huang Q. Blow-up solutions for a class of semilinear elliptic and parabolic equations SIAM J. Math. Anal. 31 1999 1-18
11. Du Y. Li S. Positive solutions with prescribed patterns in some simple semilinear equations Differential Integral Equations 15 2002 805-822
12. N by a squeezing method involving boundary blow-up solutions J. London Math. Soc. 64 2001 107-124
13. Fraile J.M. Koch Medina P. López-Gómez J. Merino S. Elliptic eigenvalue problems and unbounded continua of positive solutions of a semilinear elliptic problem J. Differential Equations 127 1996 295-319
14. Gilbarg D. Trudinger N.S. Elliptic Partial Differential Equation of Second Order 1998 Edition 2001 Springer Berlin
15. Hainzl J. Multiparameter bifurcation of a predator-prey system SIAM J. Math. Anal. 23 1992 150-180
16. Hsu S.B. Steady states of a system of partial differential equations modeling microbial ecology SIAM J. Math. Anal. 14 1983 1130-1138
17. Hsu S.B. Hwang T.W. Global stability for a class of predator-prey systems SIAM J. Appl. Math. 55 1995 763-783
18. Hsu S.B. Hwang T.W. Uniqueness of limit cycles for a predator-prey system of Holling and Leslie type Canad. Appl. Math. Quart. 6 2 1998 91-117
19. Hutson V. Lou Y. Mischaikow K. Spatial heterogeneity of resources versus Lotka-Volterra dynamics J. Differential Equations 185 2002 97-136
20. Hutson V. Mischaikow K. Polacik P. The evolution of dispersal rates in a heterogeneous time-periodic environment J. Math. Biol. 43 2001 501-533
21. Leung A. Limiting behavior for a prey-predator model with diffusion and crowding effects J. Math. Biol. 6 1978 87-93
22. Lin C.S. Ni W.M. Takagi I. Large amplitude stationary solutions to a chemotaxis system J. Differential Equations 72 1988 1-27
23. Lopez-Gomez J. Pardo R.M. Invertibility of linear noncooperative elliptic systems Nonlinear Anal. 31 1998 687-699
24. Lou Y. Ni W.M. Diffusion vs cross-diffusion: an elliptic approach J. Differential Equations 154 1999 157-190
25. Murry J.D. Mathematical Biology 1993 Springer Berlin
26. p=0 on the compact manifolds Trans. Amer. Math. Soc. 331 1992 503-527
27. Renshaw E. Modelling Biological Populations in Space and Time 1991 Cambridge University Press Cambridge