Some uniqueness and exact multiplicity results for a predator-prey model

Transactions of the American Mathematical Society

Volumn 349, Issue 6, 1997, Pages 2443-2475

  Yihong, D U ^a,c   Lou, Yuan ^b,c  

^a Statistics And Computing Science ^*  (Australia)

^b UNIVERSITY OF MINNESOTA   (United States)

^c UNIVERSITY OF CHICAGO   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 21744459836     PISSN: 00029947     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (29)

1. K.J. Brown, Nontrivial solutions of predator-prey systems with small diffusion, Nonlinear Anal., 11 (1987), 685-689. MR 88135073
2. J. Blat and K.J. Brown, Global bifurcation of positive solutions in some systems of elliptic equations, SIAM J. Math. Anal. 17 (1986), 1339-1353. MR 87k35077
3. A. Casai, J.C. Eilbeck and J. Lopez-Gomez, Existence and uniqueness of coexistence states for a predator-prey model with diffusion, Differential Integral Equations 7 (1994), 411-439. MR 95f92009
4. S.N. Chow and J.K. Hale, Methods of Bifurcation Theory, Springer-Verlag, Berlin, 1982. MR 84e58019
5. E.D. Conway, R. Gardner and J. Smoller, Stability and bifurcation of steady-state solutions for predator-prey equations, Adv. Appl. Math. 3 (1982), 288-334. MR 83m35023
6. D.S. Cohen and T.W. Laetsch, Nonlinear boundary value problems suggested by chemical reactor theory, J. Differential Equations 7 (1970), 217-226. MR 413994
7. M.G. Crandall and P.H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis 8 (1971), 321-340. MR 445836
8. M.G. Crandall and P.H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal. 52 (1973), 161-180. MR 495962
9. E.N. Dancer, On the indices of fixed points of mappings in cones and applications, J. Math. Anal. Appl. 91 (1983), 131-151. MR 84d58020
10. E.N. Dancer, On positive solutions of some pairs of differential equations, Trans. Amer. Math. Soc. 284 (1984), 729-743. MR 85i35056
11. E.N. Dancer, On positive solutions of some pairs of differential equations, II, J. Differential Equations 60 (1985), 236-258. MR 87i35070
12. E.N. Dancer, Counterexamples to some conjectures on the number of solutions of nonlinear equations, Math. Ann. 272 (1985), 421-440. MR 86j35056
13. E.N. Dancer, On the number of positive solutions of weakly nonlinear elliptic equations when a parameter is large, Proc. London Math. Soc. 53 (1986), 429-452. MR 88c35061
14. E.N. Dancer, On uniqueness and stability for solutions of singularly perturbated predator-prey type equations with diffusion, J. Differential Equations 102 (1993), 1-32. MR 94e35075
15. E.N. Dancer, Multiple fixed points of positive mappings, J. Reine .ngew. Math. 371 (1986), 46-66. MR 88b58020
16. E.N. Dancer and Y. Du, Competing species equations with diffusion, large interactions and jumping nonlinearities, J. Differential Equations 114 (1994), 434-475. MR 95i35296
17. Y. Du, Uniqueness, multiplicity and stability for positive solutions of a pair of reaction- diffusion equations, Proc. Roy. Soc. Edinburgh 126A (1996), 777-809.
18. D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics 840, Springer-Verlag, Berlin, York, 1981. MR 83j35084
19. T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin, 1966. MR 343324
20. P. Korman and A.W. Leung, A general monotone scheme for elliptic systems with applications to ecological models, Proc. Roy. Soc. Edinburgh 102A (1986), 315-325. MR 87m35008
21. N. Lakos, Existence of steady-state solutions for a one-predator-two-prey system, SIAM J. Math. Anal. 21 (1990), 647-659. MR 91f35114
22. L. Li, Coexistence theorems of steady-states for predator-prey interacting systems, Trans. Amer. Math. Soc. 305 (1988), 143-166. MR 89b35042
23. L. Li, On the uniqueness and ordering of steady- states of predator-prey systems, Proc. Roy. Soc. Edinburgh. 110A (1989), 295-303. MR 89j92066
24. J. Lopez-Gomez and R. Pardo, Existence and uniqueness of coexistence states for the predatorprey model with diffusion the scalar case, Differential Integral Equations 6 (1993), 1025-1031. MR 94i34027
25. C.V. Pao, On nonlinear reaction-diffusion systems, J. Math. Anal. Appl. 87 (1982), 165-198. MR 83135094
26. C.V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. MR 94c35002
27. P.H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Functional Analysis 7 (1971), 487-513. MR 46745
28. Y. Yamada, Stability of steady-states for prey-predator diffusion equations with homogeneous Dirichlet conditions, SIAM J. Math. Anal. 21 (1990), 327-345. MR 91f35140
29. Q.X. Ye and Z.Y. Li, Introduction to Reaction-Diffusion Equations (in Chinese), Science Press, Beijing, 1990.