On the symbiotic Lotka-Volterra model with diffusion and transport effects

Journal of Differential Equations

Volumn 160, Issue 1, 2000, Pages 175-262

  Delgado, M ^a   Lopez Gomez J ^b   Suarez A ^a  

^a UNIVERSITY OF SEVILLE   (Spain)

^b UNIVERSIDAD COMPLUTENSE DE MADRID   (Spain)

Author keywords

Blowing up for a priori bounds in systems; Fixed point index in cones; Local and global bifurcation theory; Singular perturbations; Singularity theory

Indexed keywords

EID: 0033634578     PISSN: 00220396     EISSN: None     Source Type: Journal    
DOI: 10.1006/jdeq.1999.3655     Document Type: Article

References (35)

1. Ali S. W., Cosner C. On the uniqueness of the positive steady state for Lotka-Volterra models with diffusion. J. Math. Anal. Appl. 168:1992;329-341.
2. Amann H. Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Rev. 18:1976;620-709.
3. Amann H., López-Gómez J. A priori bounds and multiple solutions for superlinear indefinite elliptic problems. J. Differential Equations. 146:1998;336-374.
4. Brezis H., Nirenberg L. Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Comm. Pure Appl. Math. 36:1983;437-477.
5. Cañada A., Gámez J. L. Positive solutions of nonlinear elliptic systems. Math. Models Methods Appl. Sci. 3:1993;823-837.
6. Cantrell R. S., Cosner C. On the steady-state problem for the Lotka-Volterra competition model with diffusion. Houston J. Math. 13:1987;337-352.
7. Crandall M. G., Rabinowitz P. H. Bifurcation from simple eigenvalues. J. Funct. Anal. 8:1971;321-340.
8. Dancer E. N. The effect of domain shape on the number of positive solutions of certain nonlinear equations. J. Differential Equations. 74:1988;120-156.
9. Dancer E. N., Hess P. Stability of fixed points for order preserving discrete time dynamical systems. J. Reine Angew. Math. 419:1991;125-139.
10. Eilbeck J. C., Furter J. E., López-Gómez J. Coexistence in the competition model with diffusion. J. Differential Equations. 107:1994;96-139.
11. Furter J. E., López-Gómez J. On the existence and uniqueness of coexistence states for the Lotka-Volterra competition model with diffusion and spatially dependent coefficients. Nonlinear Anal. 25:1995;363-398.
12. Furter J. E., López-Gómez J. Diffusion-mediated permanence problem for an heterogeneus Lotka-Volterra competition model. Proc. Roy. Soc. Edinburgh Sect. A. 127:1997;281-336.
13. Gidas B., Sprück J. A priori bounds for positive solutions of nonlinear elliptic equations. Comm. Partial Differential Equations. 6:1981;883-901.
14. Hess P. Periodic-Parabolic Boundary Value Problems and Positivity. Pitman Res. Notes Math. Ser. 1991;Longman, Harlow.
15. Hirsch M. Stability and convergence in strongly monotone dynamical systems. J. Reine Angew. Math. 383:1988;1-58.
16. Korman P. Dynamics of the Lotka-Volterra systems with diffusion. Appl. Anal. 44:1992;191-207.
17. Korman P., Leung A. On the existence and uniqueness of positive steady-states in the Volterra-Lotka ecological models with diffusion. Appl. Anal. 26:1987;145-160.
18. Kato T. Perturbation Theory for Linear Operators. 1975;Springer-Verlag, Berlin.
19. Leung A. A study of three species prey-predator reaction-diffusions by monotone schemes. J. Math. Anal. Appl. 100:1984;583-604.
20. Li L., Ghoreishi A. On positive solutions of general nonlinear elliptic symbiotic interacting systems. Appl. Anal. 40:1991;281-295.
21. López-Gómez J. Nonlinear eigenvalues and global bifurcation: Application to the search of positive solutions for general Lotka-Volterra reaction-diffusion systems with two species. Differential Integral Equations. 7:1994;1427-1452.
22. López-Gómez J. The maximum principle and the existence of principal eigenvalues for some linear weighted boundary value problems. J. Differential Equations. 127:1996;263-294.
23. López-Gómez J. Positive periodic solutions of Lotka-Volterra reaction-diffusion systems. Differential Integral Equations. 5:1992;55-72.
24. López-Gómez J., Molina-Meyer M. The maximum principle for cooperative weakly coupled elliptic systems and some applications. Differential Integral Equations. 7:1994;383-398.
25. López-Gómez J., Pardo R. Coexistence regions in Lotka-Volterra models with diffusion. Nonlinear Anal. 19:1992;11-28.
26. López-Gómez J., Sabina J. C. Coexistence sates and global attactivity for some convective diffusive competing species models. Trans. Amer. Math. Soc. 347:1995;3797-3833.
27. Lou Y. Necessary and sufficient condition for the existence of positive solutions of certain cooperative system. Nonlinear Anal. 26:1996;1079-1095.
28. McKenna P. J., Walter W. On the Dirichlet problem for elliptic systems. Appl. Anal. 21:1986;207-224.
29. De Pagter B. Irreducible compact operators. Math. Z. 192:1986;149-153.
30. Pao C. V. Nonlinear Parabolic and Elliptic Equations. 1992;University Press, New York.
31. Pinsky R. G. Positive Harmonic Functions and Diffusion. Cambridge Studies in Advanced Mathematics. 45:1995;Cambridge University Press, Cambridge.
32. Schaefer H. H. Banach Lattices and Positive Operators. 1974;Springer-Verlag, Berlin.
33. Stein E. M. Singular Integrals and Differentiability Properties of Functions. 1970;Princeton Univ. Press, Princeton.
34. Walter W. The minimum principle for elliptic systems. Appl. Anal. 47:1992;1-6.
35. Zhengyuan L., de Mottoni P. Bifurcation for some systems of cooperative and predator-prey type. J. Partial Differential Equations. 5:1992;25-36.