Rich dynamics of a food chain model with Hassell-Varley type functional responses

Applied Mathematics and Computation

Volumn 208, Issue 2, 2009, Pages 303-317

  Pathak, Sweta ^a   Maiti, Alakes ^b   Samanta, G P ^c  

^a Belur Girls' High School (HS)   (India)

^b PRESIDENCY UNIVERSITY   (India)

^c INDIAN INSTITUTE OF ENGINEERING SCIENCE AND TECHNOLOGY   (India)

Author keywords

Food chain model; Functional response; Stability; Time delay

Indexed keywords

DYNAMICS;

BOUNDEDNESS; COMPLEX DYNAMICS; DISCRETE TIME DELAYS; DYNAMICAL BEHAVIOURS; FOOD CHAIN MODEL; FUNCTIONAL RESPONSE; RICH DYNAMICS; STABILITY; STABILITY AND BIFURCATIONS; TRITROPHIC FOOD CHAINS;

TIME DELAY;

EID: 58849154183     PISSN: 00963003     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.amc.2008.12.015     Document Type: Review

References (51)

1. Aiello W.G., and Freedman H.I. A time-delay model of single-species growth with stage structure. Math. Biosci. 101 (1990) 139-153
2. 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) 855-869
3. Arditi R., and Ginzburg L.R. Coupling in predator-prey dynamics: ratio-dependence. J. Theor. Biol. 139 (1989) 311-326
4. Arditi R., and Saiah H. Empirical evidence of the role of heterogeneity in ratio-dependent consumption. Ecology 73 (1992) 1544-1551
5. Bence J.R., and Nisbet R.M. Space-limited recruitment in open systems: the importance of time delays. Ecology 70 (1989) 1434-1441
6. Berezovskaya F., Karev G., and Arditi R. Parametric model of the ratio-dependent predator-prey model. J. Math. Biol. 43 (2001) 221-246
7. Berreta E., and Kuang Y. Convergence results in a well-known delayed predator-prey system. J. Math. Anal. Appl. 204 (1996) 840-853
8. Berreta E., and Kuang Y. Global analysis in some delayed ratio-dependent predator-prey systems. Nonlinear Anal. TMA 32 (1998) 381-408
9. Birkhoff G., and Rota G.C. Ordinary Differential Equations (1982), Ginn, Boston
10. Boer M., Kooi B., and Kooijman S. Homoclinic and heteroclinic orbits to a cycle in a tri-trophic food chain. J. Math. Biol. (1999) 19-38
11. Chi C.W., Hsu S.B., and Wu L.L. On asymmetric May-Leonard model of three competing species. SIAM J. Appl. Math. 58 (1998) 211-226
12. Cooke K.L., Van Den Driessche P., and Zou X. Interaction of maturation delay and nonlinear birth in population and epidemic models. J. Math. Biol. 39 (1999) 332-352
13. Cushing J.M. Integrodifferential Equation and Delay Models in Population Dynamics (1977), Springer-Verlag, Heidelberg
14. S.C. Das, K.C. Barua, Scope of bio-control of pests and diseases in tea plantations, in: Proceedings of the International Conference on Tea Research: Global Perspective, Calcutta, January 11-12, 1990, pp. 119-125.
15. S.C. Das, M.C. Borthakur, B. Gope, Need for non-conventional approach in tea pest management, in: Proceedings of the 30th Tocklai Conference, 1988, pp. 80-89.
16. Freedman H.I. Deterministic Mathematical Models in Population Ecology (1980), Marcel Dekker, New Work
17. Freedman H.I., and Ruan S. Hopf bifurcation in the three-species food chain models with group defence. Math. Biosci. 111 (1992) 73-87
18. Freedman H.I., and Waltman P. Mathematical analysis of some three-species food-chain models. Math. Biosci. 33 (1977) 257-276
19. Fritzsche-Hoballah M., and Turlings T. Experimental evidence that plants under caterpillar attack may benefit from attracting parasitoids. Evol. Ecol. Res. 3 (2001) 1-13 (Das et al. (1988))
20. Gomez J., and Zamora R. Top-down effects in a tritrophic system: parasitoids enhance plant fitness. Ecology 75 (1994) 1023-1030
21. Gopalsamy K. Stability and Oscillations in Delay Differential Equations of Population Dynamics (1992), Kluwer Academic Publisher, Boston
22. Hairston N.G., Smith F.E., and Slobodkin L.B. Community structure, population control and competition. Am. Nat. 94 (1960) 421-425
23. Hastings A. Delays in recruitment at different trophic levels: effects on stability. J. Math. Biol. 21 (1984) 35-44
24. Hastings A., and Powell T. Chaos in a three-species food chain. Ecology 72 (1991) 896-903
25. Hassell M.P., and Varley G.C. New inductive population model and its bearing on biological control. Nature 223 (1969) 1133-1137
26. Hsu S.B., Hwang T.W., and Kuang Y. Rich dynamics of a ratio-dependent one prey two predator model. J. Math. Biol. 43 (2001) 377-396
27. Hsu S.B., Hwang T.W., and Kuang Y. A ratio-dependent food cain model and its application to biological control. Math. Biosci. 181 (2003) 55-83
28. Hsu S.B., Hwang T.W., and Kuang Y. Global dynamics of a predator-prey model with Hassell-Varley type functional response. J. Math. Biol. (2007)
29. Jost C., Arino O., and Arditi R. About deterministic extinction in ratio-dependent predator-prey models. Bull. Math. Biol. 61 (1999) 19-32
30. Kabir S.E. Non-conventional methods of pest control in tea (Camellia sinensis L.). In: Ghoshal S.K., and Ray D. (Eds). Recent Advances in Animal Science Research vol. 1 (2001), Orion Press International, West Bengal 163-167
31. Kuang Y. Delay Differential Equations with Applications in Population Dynamics (1993), Academic Press, New York
32. Kuznetsov Y., and Rinaldi S. Remarks on food chain dynamics. Math. Biosci. 134 (1996) 1-33
33. Kuznetsov Y., De Feo O., Rinaldi S., and Belyakov. Homoclinic bifurcation in a tritrophic food chain model. SIAM J. Appl. Math. 62 (2001) 462-487
34. Li B., and Kuang Y. Simple food chain in a chemostat with distinct removal rates. J. Math. Anal. Appl. 242 (2000) 75-92
35. Liu W. Criterion of Hopf bifurcations without using eigenvalues. J. Math. Anal. Appl. 182 (1994) 250-256
36. Lotka A. Elements of Physical Biology (1925), Williams and Wilkins, Baltimore
37. Macdonald N. Biological Delay Systems: Linear Stability Theory (1989), Cambridge University Press, Cambridge
38. Maiti A., Jana M.M., and Samanta G.P. Deterministic and stochastic analysis of a ratio-dependent predator-prey system with delay. Nonlinear Anal.: Modell. Contr. 12 (2007) 383-398
39. Maiti A., and Samanta G.P. Rich dynamics of a time-delayed food chain model. J. Biol. Syst. 14 (2006) 387-412
40. May R.M. Stability and Complexity in Model Ecosystems (1974), Princeton University Press, Princeton
41. McCann K., and Yodzis P. Bifurcation structure of a three-species food chain model. Theor. Popul. Biol. 48 (1995) 93-125
42. Mochizuki M. Effectiveness and pesticide susceptibility of the pyrethroid-resistant predatory mite Amblyseius womersleyi in the integrated pest management of tea pests. BioControl 48 (2003) 207-221
43. Murray J.D. Mathematical Biology (1993), Springer-Verlag, New York
44. Reeve J.D. Predation and bark beetle dynamics. Oecologia 112 (1997) 48-54
45. Rosenzweig M.L. Paradox of enrichment: destabilization of exploitation ecosystems in ecological time. Science 171 (1969) 385-387
46. Ruan S. A three-trophic-level model of plankton dynamics with nutrient recycling. Can. Appl. Math. Quart. 1 (1993) 529-553
47. Takeuchi Y., Oshime Y., and Matsuda H. Persistence and periodic orbits of a three-competitor model with refuges. Math. Biosci. 108 (1992) 105-125
48. Van Loon J., Boer G.De., and Dicke M. Parasitoid-plant mutualism: parasitoid attack of herbivore increases plant reproduction. Entomol. Exp. Appl. 97 (2000) 219-227
49. Volterra V. Variazioni e fluttuazioni del numero di individui in specie animali conviventi. Mem. Accd. Lincei. 2 (1926) 31-113
50. Wangersky P.J., and Cunnigham W.J. Time lag in prey-predator population models. Ecology 38 (1957) 136-139
51. Xiao D., and Ruan S. Global dynamics of a ratio-dependent predator-prey system. J. Math. Biol. 43 (2001) 268-290