The relationships between equilibria and positive solutions of certain nonlinear elliptic systems

Journal of Mathematical Analysis and Applications

Volumn 209, Issue 1, 1997, Pages 154-179

  Li, Lige ^a   Cao, Guoxiang ^a,c   Liu, Yaping ^b  

^a KANSAS STATE UNIVERSITY   (United States)

^b PITTSBURG STATE UNIVERSITY   (United States)

^c US Sprint   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

BOUNDARY CONDITIONS; FUNCTIONS; NONLINEAR EQUATIONS; PARTIAL DIFFERENTIAL EQUATIONS; PROBLEM SOLVING;

NONLINEAR ELLIPTIC SYSTEMS; PHASE CURVES; POSITIVE EQUILIBRIA;

MATHEMATICAL TECHNIQUES;

EID: 0031128539     PISSN: 0022247X     EISSN: None     Source Type: Journal    
DOI: 10.1006/jmaa.1997.5354     Document Type: Article

References (34)

1. Amann H. Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Rev. 18:1976;620-709.
2. Berestycki H., Lions P. L. Some applications of the method of super and subsolutions. Lecture Notes in Math. 782:1980;. p. 16-41.
3. Blat J., Brown K. J. Bifurcation of steady-state solutions in predator-prey and competition systems. Proc. Roy. Soc. Edinburgh Sect. A. 97:1984;21-34.
4. Brown P. N. Decay to uniform states in ecological interactions. SIAM J. Appl. Math. 38:1980;22-37.
5. A. Cañada, J. L. Gámez, Prey-predator systems without restrictions on the sign of the birthrate of the predator species, Proceedings of the Third International Conference on Math Population Dynamics.
6. Cantrell R. S., Cosner C. On the uniqueness and stability of positive solutions in the Lotka-Volterra competition model with diffusion. Houston J. Math. 15:1989;341-361.
7. Cantrell R. S., Cosner C., Hutson V. Permanence in ecological systems with spatial heterogeneity. Proc. Roy. Soc. Edinburgh Sect. A. 123:1993;533-559.
8. Conway E. D., Gardner R., Smoller J. Stability and bifurcation of steady state solutions for predator-prey equations. Adv. in Appl. Math. 3:1982;288-334.
9. Conway E. D., Smoller J. Global analysis of a class of predator-prey equations. SIAM J. Appl. Math. 46:1986;630-642.
10. Dancer E. N. On positive solutions of some pairs of differential equations. Trans. Amer. Math. Soc. 284:1984;729-743.
11. Dancer E. N. On the existence and uniqueness of positive solutions for cometing species models with diffusion. Trans. Amer. Math. Soc. 326:1991;829-859.
12. DeMottoni P., Rothe F. Convergence to homogeneous equilibrium state for generalized Volterra-Lotka systems with diffusion. SIAM J. Appl. Math. 37:1979;648-663.
13. C. Eilbeck, J. E. Furter, J. López-Gómez, Coexistence in the competition model with diffusion, J. Differential Equations.
14. Ghoreishi A., Logan R. Positive solutions of a class of biological models in a heterogeneous environment. Bull. Austral. Math. Soc. 44:1991;79-94.
15. Gidas B., Ni W. M., Nirenberg L. Symmetry and related properties via the maximum principle. Comm. Math. Phys. 68:1979;209-243.
16. Gui C., Lou Y. Uniqueness and nonuniqueness of coexistence states in the Lotka-Volterra competition model. Comm. Pure Appl. Math. 47:1994;1571-1594.
17. Keller C., Lui R. Existence of steady-state solutions to predator-prey equations in a heterogeneous environment. J. Math. Anal. Appl. 123:1987;306-323.
18. 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.
19. Kufner A. Weighted Sobolev Spaces. 1986;Wiley, New York.
20. Leung A. W. Systems of Nonlinear Partial Differential Equations: Applications to Biology and Engineering. 1989;Kluwer Academic, Norwell.
21. Li L. Coexistence theorems of steady states for predator-prey interacting systems. Trans. Amer. Math. Soc. 305:1988;143-166.
22. Li L. Global positive coexistence of a nonlinear elliptic biological interacting model. Math. Biosci. 97:1989;1-15.
23. Li L., Ghoreishi A. On positive solutions of general nonlinear elliptic symbiotic interacting systems. Appl. Anal. 40:1991;281-295.
24. Li L., Liu Y. Spectral and nonlinear effects in certain elliptic systems of three variables. SIAM J. Math. Anal. 24:1993;480-498.
25. Lions J. L., Magenes E. Non-homogeneous Boundary Value Problems and Applications. 1972;Springer-Verlag, New York.
26. López-Gómez J., Pardo R. Existence and uniqueness for some competition models with diffusions. C. R. Acad. Sci. Paris Sér. I Math. 313:1991;933-938.
27. Lui R. Positive solutions of an elliptic system arising from a model in evolutionary ecology. J. Math. Biol. 29:1991;239-250.
28. McKenna P. J., Walter W. On the Dirichlet problem for elliptic systems. Appl. Anal. 21:1986;207-224.
29. Mimura M., Tabata M., Hosono Y. Multiple solutions of two point boundary value problems of Neumann type with small parameter. SIAM J. Math. Anal. 11:1980;613-631.
30. Pao C. V. Nonlinear Parabolic and Elliptic Equations. 1992;Plenum, New York.
31. W. Ruan, W. Feng, 1995, On the fixed point index and multiple steady-state solutions of reaction-diffusion system, Differential Integral Equations, 2, 371, 391.
32. Ruan W., Pao C. V. Asymptotic behavior and positive solutions of a chemical reaction diffusion system. J. Math. Anal. Appl. 169:1992;157-178.
33. Smoller J. Shock Waves and Reaction-Diffusion Equations. 1994;Springer-Verlag.
34. Zhou L., Pao C. V. Asymptotic behavior of a competition-diffusion system in population dynamics. Nonlinear Anal. 6:1982;1163-1184.