Saddle-point behavior for monotone semiflows and reaction-diffusion models

Journal of Differential Equations

Volumn 203, Issue 2, 2004, Pages 313-330

  Jiang, Jifa ^a,c   Liang, Xiang ^a   Zhao, Xiao Qiang ^b  

^a UNIVERSITY OF SCIENCE AND TECHNOLOGY OF CHINA   (China)

^b MEMORIAL UNIVERSITY OF NEWFOUNDLAND   (Canada)

^c TONGJI UNIVERSITY   (China)

Author keywords

Competitive systems; Monotone semiflows; Reaction diffusion models; Saddle point behavior

Indexed keywords

EID: 3342927709     PISSN: 00220396     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.jde.2004.05.002     Document Type: Article

References (42)

1. Các N.P. Gatica J.A. Fixed point theorems for mappings in ordered Banach spaces J. Math. Anal. Appl. 71 1979 547-557
2. Capasso V. Mathematical Structures of Epidemic Systems Lecture Notes in Biomathematics Vol. 97 1993 Springer Heidelberg
3. Capasso V. Kunish K. A reaction-diffusion system arising in modelling man-environment diseases Quart. Appl. Math. 46 1988 431-450
4. Capasso V. Maddalena L. Convergence to equilibrium states for a reaction-diffusion system modelling the spatial spread of a class of bacterial and viral diseases J. Math. Biol. 13 1981 173-184
5. Capasso V. Maddalena L. Saddle point behavior for a reaction-diffusion system: application to a class of epidemic models Math. Comput. Simulation 24 1982 540-547
6. Capasso V. Wilson R.E. Analysis of a reaction-diffusion system modelling man-environment-man epidemics SIAM J. Appl. Math. 57 1997 327-346
7. Dancer E.N. Guo Z.M. Uniqueness and stability for solutions of competing species equations with large interactions Comm. Appl. Nonlinear Anal. 1 1994 19-45
8. Dancer E.N. Hess P. Stability of fixed points for order-preserving discrete-time dynamical systems J. Reine Angew. Math. 419 1991 125-139
9. Dancer E.N. Zhang Z. Dynamics of Lotka-Volterra competition systems with large interaction J. Differential Equations 182 2002 470-489
10. Deimling K. Nonlinear Functional Analysis 1988 Springer Berlin
11. Diekmann O. Heesterbeek J.A.R. Mathematical Epidemiology of Infectious Diseases 2000 Wiley Chichester, New York
12. Hale J.K. Asymptotic Behavior of Dissipative Systems Mathematical Surveys and Monographs Vol. 25 1988 American Mathematical Society Providence, RI
13. Hess P. Periodic-parabolic Boundary Value Problems and Positivity, Pitman Research Notes in Mathematics, Series 247, Longman Scientific and Technical, New York, 1991.
14. Hirsch M. Stability and convergence in strongly monotone dynamical systems J. Reine Angew. Math. 383 1988 1-53
15. Hirsch M. Systems of differential equations which are competitive or cooperative: III Competing species, Nonlinearity 1 1988 51-71
16. Ida M. Muramatsu T. Ninomiya H. Diffusion-induced extinction of a superior species in a competition system Japan J. Indust. Appl. Math. 15 1998 233-252
17. Kishimoto K. Weinberger H. The spatial homogeneity of stable equilibria of some reaction-diffusion systems on convex domains J. Differential Equations 58 1985 15-21
18. Lajmanovich A. Yorke J.A. A deterministic model for gonorrhea in a nonhomogeneous population Math. Biosci. 28 1976 221-236
19. Liang X. Jiang J. On the finite-dimensional dynamical systems with limited competition Trans. Amer. Math. Soc. 354 2002 3535-3554
20. Liang X. Jiang J. The dynamical behavior of type-K competitive Kolmogorov systems and its application to 3-dimensional type-K competitive Lotka-Volterra systems Nonlinearity 16 2003 785-801
21. Martin R.H. Smith H.L. Abstract functional-differential equations and reaction-diffusion systems Trans. Amer. Math. Soc. 321 1990 1-44
22. Mirenghi E. Equilibrium solutions of a semilinear reaction-diffusion system Nonlinear Anal. 11 1987 393-405
23. Nussbaum R.D. Eigenvectors of nonlinear positive operator and the linear Krein-Rutman theorem, in: E. Fadell, G. Fournier (Eds.), Fixed Point Theory, Lecture Notes in Mathematics, Vol. 886, Springer, New York/Berlin, 1981, pp. 309-331
24. Selgrade J.F. Asymptotic behavior of solutions to single loop positive feedback systems J. Differential Equations 38 1980 80-103
25. Shub M. Global Stability of Dynamical Systems 1987 Springer New-York/Berlin
26. Smith H.L. Cooperative systems of differential equations with concave nonlinearities Nonlinear Anal. 10 1986 1037-1052
27. Smith H.L. Competing subcommunities of mutualists and a generalized Kamke theorem SIAM J. Appl. Math. 46 1986 856-874
28. Smith H.L. Periodic solutions of periodic competitive and cooperative systems SIAM J. Math. Anal. 17 1986 1289-1318
29. Smith H.L. Periodic competitive differential equations and the discrete dynamics of competitive maps J. Differential Equations 64 1986 165-194
30. Smith H.L. Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems Mathematical Surveys and Monographs Vol. 41 1995 AMS Providence, RI
31. Smith H.L. Planar competitive and cooperative difference equations J. Differential Equations Appl. 3 1998 335-357
32. Smith H.L. Thieme H.R. Convergence for strongly order-preserving semiflows SIAM J. Math. Anal. 22 1991 1081-1101
33. Smith H.L. Thieme H.R. Stable coexistence and bi-stability for competitive systems on ordered Banach spaces J. Differential Equations 176 2001 195-222
34. Smoller. A. Wasserman J. Global bifurcation of steady state solutions J. Differential Equations 39 1981 269-290
35. Takáč P. Asymptotic behavior of discrete-time semigroups of sublinear, strongly increasing mappings with applications in biology Nonlinear Anal. 14 1990 35-42
36. Takáč P. Domains of attraction of generic ω-limit sets for strongly monotone semiflows Z. Anal. Anwendungen 10 1991 275-317
37. Takáč P. Domains of attraction of generic ω-limit sets for strongly monotone discrete-time semigroups J. Reine Angew. Math. 423 1992 101-173
38. Takáč P. Discrete monotone dynamics and time-periodic competition between two species Differential Integral Equations 10 1997 547-576
39. Terščák I. Dynamics of C1smooth strongly monotone discrete-time dynamical system, preprint.
40. Wang Y. Jiang J. Uniqueness and attractivity of the carrying simplex for discrete-time competitive dynamical systems J. Differential Equations 186 2002 611-632
41. Zhao X.-Q. Global attractivity and stability in some monotone discrete dynamical systems Bull. Austral. Math. Soc. 53 1996 305-324
42. Zhao X.-Q. Dynamical Systems in Population Biology 2003 Springer New York