A nonautonomous predator-prey system with stage structure and double time delays

Journal of Computational and Applied Mathematics

Volumn 230, Issue 1, 2009, Pages 283-299

  Wang, Jiaoyan ^a   Lu, Qishao ^a   Feng, Zhaosheng ^b  

^a BEIHANG UNIVERSITY   (China)

^b UNIVERSITY OF TEXAS PAN AMERICAN   (United States)

Author keywords

Continuation theorem; Global stability; Periodic solution; Permanence; Stage structure; Time delay

Indexed keywords

CONTINUATION THEOREM; GLOBAL STABILITY; PERIODIC SOLUTION; PERMANENCE; STAGE STRUCTURE;

SYSTEM STABILITY; TIME DELAY;

PERIODIC STRUCTURES;

EID: 67349106413     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cam.2008.11.014     Document Type: Article

References (52)

1. Hassell M.P. The Dynamics of Arthropod Predator-Prey Systems (1978), Princeton University Press, NJ
2. Kerr S.R., and Dickie L.M. The Biomass Spectrum: A Predator-Prey Theory of Aquatic Production (2001), Columbia University Press, New York
3. May R.M., Conway G.R., Hassella M.P., and Southwood T.R.E. Time delays, density dependence, and single-species oscillations. J. Anim. Ecol. 43 (1974) 747-770
4. Gurney W.S.C., Blythe S.P., and Nisbet R.M. Nicholson's glowflies revisited. Nature 287 (1980) 17-21
5. Aiellot W.G. The existence of nonoscillatory solutions to a generalized, nonautonomous, delay logistic equation. J. Math. Anal. Appl. 149 (1990) 114-123
6. Fishera M.E., and Goh B.S. Stability results for delayed-recruitment models in population dynamics. J. Math. Biol. 19 (1984) 147-156
7. Freedman H.I., and Gopalsamyg K. Global stability in time-delayed single species dynamics. Bull. Math. Biol. 48 (1986) 485-492
8. Ma S.Q., Feng Z., and Lu Q.S. A two-parameter geometrical criteria for delay differential equations. Discrete Contin. Dyn. Syst. Ser. B 5 (2008) 397-413
9. Kuang Y. Delay Differential Equations with Applications in Population Dynamics (1993), Academic Press, Boston
10. Fowler A.C. Mathematical Models in the Applied Sciences. Cambridge Texts in Applied Mathematics (1997), Cambridge University Press, Cambridge
11. Landahl H.D., and Hanson B.D. A three stage population model with cannibalism. Bull. Math. Biol. 37 (1975) 11-17
12. Barclay H.J., and Van Driessche P. A model for a single species with two life history stages and added mortality. Ecol. Model. 11 (1980) 157-166
13. Wood S.N., Blythe S.P., Gurney W.S.C., and Nisbeti R.M. Instability in mortality estimation schemes related to stage structured population data. Math. Med. Biol. 6 (1989) 47-68
14. Brauer F., and Ma Z.E. Stability of stage-structured population models. J. Math. Anal. Appl. 126 (1987) 301-315
15. Freedman H.I., and Wu J. Persistence and global asymptotic stability of single species dispersal models with stage structure. Quart. Appl. Math. 49 (1991) 351-371
16. Cai L.M., and Song X.Y. Permanence and stability of a predator-prey system with stage structure for predator. J. Comput. Appl. Math. 201 (2007) 356-366
17. Zhang X.A., Chen L.S., and Neumann A.U. The stage-structured predator-prey model and optimal harvesting policy. Math. Biosci. 168 (2000) 201-210
18. Xu R., and Wang Z.Q. Periodic solutions of a nonautonomous predator-prey system with stage structure and time delays. J. Comput. Appl. Math. 196 (2006) 70-86
19. Song X.Y., and Chen L.S. Optimal harvesting and stability for a two-species competitive system with stage structure. Math. Biosci. 170 (2001) 173-186
20. Zhang Z.Q. Periodic solutions of a predator-prey system with stage-structures for predator and prey. J. Math. Anal. Appl. 302 (2005) 291-305
21. Omari J., and Gourley S.A. Dynamics of a stage-structured population model incorporating a state-dependent maturation delay. Nonlinear. Anal. 6 (2005) 13-33
22. Gourley S.A., and Kuang Y. A stage structured predator-prey model and its dependence on maturation delay and death rate. J. Math. Biol. 49 (2004) 188-200
23. Omari J., and Gourley S.A. Stability and traveling fronts in Lotka-Volterra competition models with stage structure. SIAM J. Appl. Math. 63 (2003) 2063-2086
24. Ou L.M., Luo G.L., Jiang Y.L., and Li Y.P. The asymptotic behaviors of a stage-structured autonomous predator-prey system with time delay. J. Math. Anal. Appl. 283 (2003) 534-548
25. Wang W., Mulone G., Salemi F., and Salone V. Permanence and stability of a stage-structured predator-prey model. J. Math. Anal. Appl. 262 (2001) 499-528
26. Liu S.Q., Kouche M., and Tatar N. Permanence extinction and global asymptotic stability in a stage structured system with distributed delays. J. Math. Anal. Appl. 301 (2005) 187-207
27. Magnusson K.G. Destabilizing effect of cannibalism on a structured predator-prey system. Math. Biosci. 155 (1999) 61-75
28. Aiello W.G., Freedman H.I., and Wu J. Analysis of a model representing stage-structured population growth with state-dependent time delay. SIAM J. Appl. Math. 52 (1992) 885-869
29. Aiello W.G., and Freedman H.I. A time-delay model of single-species growth with stage structure. Math. Biosci. 101 (1990) 139-153
30. Beretta E., and Kuang Y. Global analyses in some delayed ratio-dependent predator-prey systems. Nonlinear. Anal. 32 (1998) 381-408
31. Berezovskaya F., Karev G., and Arditi R. Parametric analysis of the ratio-dependent predator-prey model. J. Math. Biol. 43 (2001) 221-246
32. Xiao D.M., and Zhang Z. On the uniqueness and nonexistence of limit cycles for predator-prey systems. Nonlinearity 16 (2003) 1185-1201
33. Arino O., abdllaoui A.E., Mikram J., and Chattopadhyay J. Infection in prey population may act as a biological control in ratio-dependent predator-prey models. Nonlinearity 17 (2004) 1101-1116
34. Hsu S.B., Hwang T.W., and Kuang Y. Global analysis of the MichaelisMenten ratio-dependent predator-prey system. J. Math. Biol. 42 (2001) 489-506
35. Hsu S.B., Hwang T.W., and Kuang Y. A ratio-dependent food chain model and its applications to biological control. Math. Biosci. 181 (2003) 55-83
36. Jost C., Arino O., and Arditi R. About deterministic extinction in ratio-dependent predator-prey models. Bull. Math. Biol. 61 (1999) 19-32
37. Kuang Y., and Beretta E. Global qualitative analysis of a ratio-dependent predator-prey system. J. Math. Biol. 36 (1998) 389-406
38. Xu R., and Chen L. Persistence and global stability for n-species ratio-dependent predator-prey system with time delays. J. Math. Anal. Appl. 275 (2002) 27-43
39. Xu R., Chaplain M.A.J., and Davidson F.A. Permanence and periodicity of a delayed ratio-dependent predator-prey model with stage structure. J. Math. Anal. Appl. 303 (2005) 602-621
40. Kats S.C., Kats L.B., and Anzalone C.B. Aggression by non-native crayfish deters breeding in California newts. Conserv. Biol. 11 (1997) 793-796
41. Ma Z.E. Stability of predation models with time delays. Appl. Anal. 22 (1986) 169-192
42. Ma S.Q., Lu Q.S., and Feng Z. Double Hopf bifurcation for van der Pol-Duffing oscillator with parametric delay feedback control. J. Math. Anal. Appl. 338 (2008) 993-1007
43. Wang W.D., and Ma Z.E. Harmless delays for uniform persistence. J. Math. Anal. Appl. 158 (1991) 256-268
44. Huo H.F. Permanence and global attractivity of delay diffusive prey-predator systems with the Michaelis-Menten functional response. Comput. Math. Appl. 49 (2005) 407-416
45. Zhang J.R., and Chen L.S. Permanence and global stability for two-species cooperative system with delays in two-patch environment. Math. Comput. Modeling. 32 (1996) 17-27
46. Gains R.E., and Mawhin J.L. Coincidence Degree and Nonlinear Differential Equations (1977), Spring, Berlin
47. Castro A., Chhetri M., and Shivaji R. Stability analysis of positive solutions to classes of reaction-diffusion systems. Differential Integral Equations 17 (2004) 391-406
48. May M.M. Stability and Complexity in Model Ecosystems (Princeton Landmarks in Biology) (2001), Princeton University Press, NJ
49. Murdoch W.W., Briggs C.J., and Nisbet R.M. Consumer-Resource Dynamics (2003), Princeton University Press, NJ
50. Arditi R., and Saiah H. Empirical evidence of the role of heterogeneity in ratio-dependent consumption. Ecology 73 (1992) 1544-1551
51. Gutierrez A.P. The physiological basis of ratio-dependent predator-prey theory: A metabolic pool model of Nicholsons blowflies as an example. Ecology 73 (1992) 1552-1563
52. Mills M.G.L., and Shenkt T.M. predator-prey relationships: The impact of lion predation on wildebeest and zebra populations. J. Anim. Ecol. 61 (1992) 693-702