Multiple periodic solutions of a delayed stage-structured predator-prey model with non-monotone functional responses

Applied Mathematical Modelling

Volumn 31, Issue 9, 2007, Pages 1947-1959

  Xia, Yonghui ^a   Cao, Jinde ^b   Cheng, Sui Sun ^c  

^a FUZHOU UNIVERSITY   (China)

^b SOUTHEAST UNIVERSITY   (China)

^c NATIONAL TSING HUA UNIVERSITY   (Taiwan)

Author keywords

Non autonomous predator prey model; Non monotone functional responses; Periodic solutions; Stage structure

Indexed keywords

ANIMALS; DATA STRUCTURES; ECOLOGY; ECOSYSTEMS; ESTIMATION; MATHEMATICAL MODELS;

MAWHIN'S COINCIDENCE DEGREE; NON-MONOTONE FUNCTIONAL RESPONSES; PERIODIC SOLUTIONS; PREDATOR-PREY MODEL;

TIME VARYING SYSTEMS;

ANIMALS; DATA STRUCTURES; ECOLOGY; ECOSYSTEMS; ESTIMATION; MATHEMATICAL MODELS; TIME VARYING SYSTEMS;

EID: 34047093805     PISSN: 0307904X     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.apm.2006.08.012     Document Type: Article

References (39)

1. Chen L. Mathematical Models and Methods in Ecology (1998), Science Press, Beijing (in Chinese)
2. Fan M., Wang Q., and Zou X.F. Dynamics of a nonautonomous ratio-dependent predator-prey system. Proc. Roy. Soc. Edin. A 133 (2003) 97-118
3. Fan M., and Wang K. Positive periodic solutions of a periodic integro-differential competition system with infinite delays. Z. Angew. Math. Mech. 81 3 (2001) 197-203
4. Fan M., and Wang K. Global existence of positive periodic solutions of periodic predator-prey system with infinite delay. J. Math. Anal. Appl. 262 (2001) 1-11
5. Ma Z., and Wang W. Asymptotic behavior of predator-prey system with time dependent coefficients. Appl. Anal. 34 (1989) 79-90
6. Gopalsamy K. Stability and Oscillations in Delay Differential Equations of Population Dynamics (1992), Kluwer Academic, Dordrecht
7. He X.Z. Stability and delays in a predator-prey system. J. Math. Anal. Appl. 198 (1996) 355-370
8. Goh B.S. Global stability in two species systems. J. Math. Biol. 3 (1976) 313-318
9. Hastings A. Global stability in two species interactions. J. Math. Biol. 5 (1978) 399-403
10. Kuang Y. Delay Differential Equations with Application in Population Dynamics (1993), Academic Press, New York
11. Beretta E., and Kuang Y. Convergence results in a well-known delayed predator-prey system. J. Math. Anal. Appl. 204 (1996) 840-853
12. Fan M., Wong P.J.Y., and Agarwal R.P. Periodicity and stability in periodic n-species Lotka-Volterra competition system with feedback controls and deviating arguments. Acta Math. Sin. 19 (2003) 801-822
13. Fan M., Wong P.J.Y., and Agarwal R.P. Periodicity in a class of nonautonomous scalar equations with deviating arguments and applications to population models. Dyn. Syst. 19 (2004) 279-301
14. Lan K.Q. Multiple positive solutions of conjugate boundary value problems with singularities. Appl. Math. Comput. 147 (2004) 461-474
15. Jiang D. Multiple positive solutions for boundary value problems of second-order delay differential equations. Appl. Math. Lett. 15 (2002) 575-583
16. Lu S., and Ge W. Existence of positive periodic solutions for neutral population model with multiple delays. Appl. Math. Comput. 153 (2004) 885-902
17. Liu Y. Multiple positive solutions of nonlinear singular boundary value problem for fourth-order equations. Appl. Math. Lett. 17 (2004) 747-757
18. Xia Y.H., Chen F., Chen A., and Cao J. Existence and global attractivity of an almost periodic ecological model. Appl. Math. Comput. 157 2 (2004) 449-475
19. Y.H. Xia, J. Cao, M. Lin, Discrete-time analogues of predator-prey models with monotonic or nonmonotonic functional responses, Nonlinear Anal., in press, doi:10.1016/j.nonrwa.2006.06.007.
20. Xia Y.H., and Cao J. Almost periodicity in an ecological model with M-predators and N-preys by "pure-delay type" system. Nonlinear Dyn. 39 3 (2005) 275-304
21. Xia Y.H., and Cao J. Global attractivity of a periodic ecological model with m-predators and n-preys by "pure-delay type" system. Comput. Appl. Math. 49 (2006) 1-21
22. Xia Y.H., Lin M., and Cao J. The existence of almost periodic solutions of certain perturbation systems. J. Math. Anal. Appl. 310 1 (2005) 81-96
23. Xia Y.H., and Cao J. Almost periodic solutions for an ecological model with infinite delays. Proceedings of Edinburgh Math. Soc. A 49 (2006) 1-21
24. Bazykin A.D. Nonlinear Dynamics of Interacting Populations (1998), World Scientific, Singapore
25. Berryman A.A. The origins and evolution of predator-prey theory. Ecology 75 (1992) 1530-1535
26. Freedman H.I. Deterministic Mathematical Models in Population Ecology (1980), Marcel Dekker, New York
27. Kot M. Elements of Mathematical Ecology (2001), Cambridge University Press, New York
28. Holling C.S. The functional response of predator to prey density and its role in mimicry and population regulation. Men. Ent. Sec. Can. 45 (1965) 1-60
29. Bush A.W., and Cook A.E. The effect of time delay and growth rate inhibition in the bacterial treatment of wastewater. J. Theor. Biol. 63 (1976) 385-395
30. Chen Y. Multiple periodic solution of delayed predator-prey systems with type IV functional responses. Nonlinear Anal. 5 (2004) 45-53
31. Aiello W.G., and Freedman H.I. A time delay model of single-species growth with stage structure. Math. Biosci. 101 (1990) 139-153
32. 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
33. 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) 885-889
34. Brauer F., and Ma Z. Stability of stage-structured population models. J. Math. Anal. Appl. 126 (1987) 301-315
35. Wang W., and Chen L. A predator-prey system with stage structure for predator. Comput. Math. Appl. 33 (1997) 83-91
36. Magnusson K.G. Destabilizing effect of cannibalism on a structured predator-prey system. Math. Biosci. 155 (1999) 61-75
37. 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
38. Liu S., Chen L., and Liu Z. Extinction and permanence in nonautonomous competitive system with stage structure. J. Math. Anal. Appl. 274 (2002) 667-684
39. Gains R.E., and Mawhin J.L. Coincidence Degree and Nonlinear Differential Equations (1977), Spring, Berlin