Coexistence states of a predator-prey system with non-monotonic functional response

Nonlinear Analysis: Real World Applications

Volumn 8, Issue 3, 2007, Pages 769-786

  Ko, Wonlyul ^a   Ryu, Kimun ^a  

^a Korea University   (South Korea)

Author keywords

Bifurcation theory; Coexistence state; Fixed point index; Group defense; Monod Haldane function; Upper lower solution

Indexed keywords

BIFURCATION (MATHEMATICS); BOUNDARY CONDITIONS; DIFFERENTIAL EQUATIONS; SYSTEM STABILITY;

COEXISTENCE STATE; FIXED POINT INDEX; GROUP DEFENSE; MONOD-HALDANE FUNCTION; PREDATOR-PREY SYSTEMS; UPPER-LOWER SOLUTION;

SYSTEMS ANALYSIS;

EID: 33846252316     PISSN: 14681218     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.nonrwa.2006.03.003     Document Type: Article

References (27)

1. Andrews J.F. A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates. Biotech. Bioeng. 10 (1968) 707-723
2. Blat J., and Brown K.J. Bifurcation of steady-state solutions in predator-prey and competition systems. Proc. R. Soc. Edinburgh Sect. A 97 (1984) 21-34
3. 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
4. Brown P.N. Decay to uniform states in ecological interactions. SIAM J. Appl. Math. 38 (1980) 22-37
5. Cano-Casanova S. Existence and structure of the set of positive solutions of a general class of sublinear elliptic non-classical mixed boundary value problems. Nonlinear Anal. Ser. A 49 3 (2002) 361-430
6. Cano-Casanova S., and López-Gómez J. Properties of the principal eigenvalues of a general class of non-classical mixed boundary value problems. J. Differential Equations 178 1 (2002) 123-211
7. Casal A., Eilbeck J.C., and López-Gómez J. Existence and uniqueness of coexistence states for a predator-prey model with diffusion. Differential Integr. Equations 7 (1994) 411-439
8. Cosner C., and Lazer A.C. Stable coexistence states in the Volterra-Lotka competition model with diffusion. SIAM J. Appl. Math. 44 (1984) 1112-1132
9. Crandall M.G., and Rabinowitz P.H. Bifurcation from simple eigenvalues. J. Funct. Anal. 8 (1971) 321-340
10. Dancer E.N. On the indices of fixed points of mappings in cones and applications. J. Math. Anal. Appl. 91 (1983) 131-151
11. Du Y., and Lou Y. Some uniqueness and exact multiplicity results for a predator-prey model. Trans. Am. Math. Soc. 349 6 (1997) 2443-2475
12. Du Y., and Lou Y. S-shaped global bifurcation curve and Hopf bifurcation of positive solutions to a predator-prey model. J. Differential Equations 144 2 (1998) 390-440
13. Du Y., and Lou Y. Qualitative behaviour of positive solutions of a predator-prey model: effects of saturation. Proc. R. Soc. Edinburgh Sect. A 131 2 (2001) 321-349
14. Freedman H.I., and Wolkowicz G.S.K. Predator-prey systems with group defence: the paradox of enrichment revisited. Bull. Math. Biol. 48 (1986) 493-508
15. Ghoreishi A., and Logan R. Positive solutions of a class of biological models in a heterogeneous environment. Bull. Aust. Math. Soc. 44 (1991) 79-94
16. Kato T. Perturbation Theory for Linear Operators (1966), Springer, New York
17. Li L. Coexistence theorems of steady states for predator-prey interacting systems. Trans. Am. Math. Soc. 305 (1998) 143-166
18. Pang P.Y.H., and Wang M. Non-constant positive steady states of a predator-prey system with non-monotonic functional response and diffusion. Proc. London Math. Soc. (3) 88 1 (2004) 135-157
19. Pao C.V. Nonlinear Parabolic and Elliptic Equations (1992), Plenum Press, New York
20. Rinaldi S., Muratori S., and Kuznetsov Y.A. Multiple attractors, catastrophes and chaos in seasonally perturbed predator-prey communities. Bull. Math. Biol. 55 (1993) 15-35
21. Ruan W., and Feng W. On the fixed point index and multiple steady-state solutions of reaction-diffusion coefficients. Differential Integr. Equations 8 (1995) 371-391
22. Ruan S., and Xiao D. Global analysis in a predator-prey system with nonmonotonic functional response. SIAM J. Appl. Math. 61 4 (2000/01) 1445-1472
23. Ryu K., and Ahn I. Coexistence theorem of steady states for nonlinear self-cross-diffusion systems with competitive dynamics. J. Math. Anal. Appl. 283 (2003) 46-65
24. Ryu K., and Ahn I. Positive solutions to ratio-dependent predator-prey interacting systems. J. Differential Equations 218 1 (2005) 117-135
25. Smoller J. Shock Waves and Reaction-Diffusion Equations (1983), Springer, New York
26. Wang M., Li Z.Y., and Ye Q.X. Existence of positive solutions for semilinear elliptic system. School on Qualitative Aspects and Applications of Nonlinear Evolution Equations, Trieste, 1990 (1991), World Scientific, River Edge, NJ 256-259
27. Zhu H., Campbell S., and Wolkowicz G. Bifurcation analysis of a predator-prey system with nonmonotonic functional response. SIAM J. Appl. Math. 63 2 (2002) 636-682