Persistence and extinction in two species reaction-diffusion systems with delays

Journal of Differential Equations

Volumn 156, Issue 1, 1999, Pages 71-92

  Ruan, Shigui ^a   Zhao, Xiao Qiang ^b  

^a DALHOUSIE UNIVERSITY   (Canada)

^b INSTITUTE OF APPLIED MATHEMATICS   (China)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0033587464     PISSN: 00220396     EISSN: None     Source Type: Journal    
DOI: 10.1006/jdeq.1998.3599     Document Type: Article

References (44)

1. Britton B. F. Spatial structure and periodic travelling waves in an integro-differential reaction-diffusion population model. SIAM J. Appl. Math. 50:1990;1663-1688.
2. Butler G. J., Freedman H. I., Waltman P. Uniformly persistent systems. Proc. Amer. Math. Soc. 96:1986;425-430.
3. Butler G. J., Waltman P. Persistence in dynamical systems. J. Differential Equations. 63:1986;255-263.
4. Cantrell R. S., Cosner C. Practical persistence in ecological models via comparison methods. Proc. Royal Soc. Edinburgh A. 126:1996;247-272.
5. Cantrell R. S., Cosner C. Models for predator-prey systems at multiple scales. SIAM Rev. 38:1996;256-286.
6. Cantrell R. S., Cosner C., Hutson V. Permanence in ecological systems with spatial heterogeneity. Proc. Royal Soc. Edinburgh A. 123:1993;533-559.
7. Cosner C. Persistence (permanence), compressivity, and practical persistence in some reaction-diffusion models from ecology. Comparison Methods and Stability Theory. 1994;Dekker, New York. p. 101-115.
8. Dancer E., Hess P. Stability of fixed points for order-preserving discrete-time dynamical systems. J. Reine Angew. Math. 419:1991;125-139.
9. Feng W., Lu X. Harmless delays for permanence in a class of population models with diffusion effects. J. Math. Anal. Appl. 206:1997;547-566.
10. Freedman H. I., Ruan S. Uniform persistence in functional differential equations. J. Differential Equations. 115:1995;173-192.
11. Freedman H. I., Ruan S., Tang M. Uniform persistence and flows near a closed positively invariant set. J. Dynam. Differential Equations. 6:1994;583-600.
12. Freedman H. I., So J. W.-H. Persistence in discrete semi-dynamical systems. SIAM J. Math. Anal. 20:1989;930-932.
13. Freedman H. I., Zhao X.-Q. Global asymptotics in some quasimonotone reaction-diffusion systems with delays. J. Differential Equations. 137:1997;340-362.
14. Furter J. E., López-Gómez J. Diffusion-mediated permanence problem for a heterogeneous Lotka-Volterra competition model. Proc. Roy. Soc. Edinburgh A. 127:1997;281-336.
15. Garroni M. G., Menaldi J. L. Green's Function for Second Order Parabolic Integro-differential Problems. 1992;Longman Sci. Tech. Harlow.
16. Gourley S. A., Bartuccelli M. V. Parameter domains for instability of uniform states in systems with many delays. J. Math. Biol. 35:1997;843-867.
17. Gourley S. A., Britton N. F. A predator-prey reaction-diffusion system with nonlocal effects. J. Math. Biol. 34:1996;297-333.
18. Hale J. K. Large diffusivity and asymptotic behavior in parabolic systems. J. Math. Anal. Appl. 118:1986;455-466.
19. Hale J. K. Asymptotic Behavior of Dissipative Systems. 1988;Am. Math. Soc. Providence.
20. Hale J. K., Waltman P. Persistence in infinite dimensional systems. SIAM J. Math. Anal. 20:1989;388-395.
21. Hess P. Periodic-Parabolic Boundary Value Problems and Positivity. Pitman Research Notes in Mathethmatics Series. 247:1991;Longman Sci. Tech. Harlow.
22. Hirsch M. W. The dynamical systems approach to differential equations. Bull. Amer. Math. Soc. 11:1984;1-64.
23. Hirsch M. W. Stability and convergence in strongly monotone dynamical systems. J. Reine Angew. Math. 383:1988;1-53.
24. Hofbauer J., So J. W.-H. Uniform persistence and repellors for maps. Proc. Amer. Math. Soc. 107:1989;1137-1142.
25. Huang W. Global dynamics for a reaction-diffusion equation with time delay. J. Differential Equations. 143:1998;293-326.
26. Hutson V., Schmitt K. Permanence and the dynamics of biological systems. Math. Biosci. 111:1992;1-71.
27. Kuang Y., Smith H. L. Convergence in Lotka-Volterra type diffusive delay systems without dominating instantaneous negative feedbacks. J. Austral. Math. Soc. Ser. B. 34:1993;471-493.
28. López-Gómez J. On the structure of the permanence region for competing species models with general diffusivities and transport effects. Discrete Contin. Dynam. Systems. 2:1996;525-542.
29. Lu X. Persistence and extinction in a competition-diffusion system with time delays. Canad. Appl. Math. Quart. 2:1994;231-246.
30. Martin R. H., Smith H. L. Abstract functional differential equations and reaction-diffusion systems. Trans. Amer. Math. Soc. 321:1990;1-44.
31. Martin R. H., Smith H. L. Reaction-diffusion systems with time delays: Monotonicity, invariance, comparison and convergence. J. Reine Angew. Math. 413:1991;1-35.
32. Mischaikow K., Smith H. L., Thieme H. R. Asymptotically autonomous semiflows. Trans. Amer. Math. Soc. 347:1995;1669-1685.
33. Pao C. V. Dynamics of nonlinear parabolic systems with time delays. J. Math. Anal. Appl. 198:1996;751-779.
34. Smith H. L. Monotone Dynamical Systems. 1995;Am. Math. Soc. Providence.
35. Smith H. L., Waltman P. The theory of the Chemostat. 1994;Cambridge Univ. Press, Cambridge.
36. Thieme H. R. Convergence results and Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations. J. Math. Biol. 30:1992;755-763.
37. Thieme H. R. Persistence under relaxed point-dissipativity (with application to an endemic model). SIAM J. Math. Anal. 24:1993;407-435.
38. Travis C. C., Webb G. F. Existence and stability for partial functional differential equations. Trans. Amer. Math. Soc. 200:1974;395-418.
39. Wu J. Theory and Applications of Partial Functional Differential Equations. 1996;Springer-Verlag, New York.
40. Yamada Y. Stability of steady states for prey-predator diffusion equations with homogeneous Dirichlet conditions. SIAM J. Math. Anal. 21:1990;327-345.
41. Yang F., Ruan S. A generalization of the Butler-McGehee lemma and its applications in persistence theory. Differential Integral Equations. 9:1996;1321-1330.
42. Zhao X.-Q. Uniform persistence and periodic coexistence states in infinite-dimensional periodic semiflows with applications. Canad. Appl. Math. Quart. 3:1995;473-495.
43. Zhao X.-Q. Asymptotic behavior for asymptotically periodic semiflows with applications. Comm. Appl. Nonlinear Anal. 3:1996;43-66.
44. Zhao X.-Q., Hutson V. Permanence in Kolmogrov periodic predator-prey models with diffusion. Nonlinear Anal. 23:1994;651-668.