Qualitative behaviour of a ratio-dependent predator-prey system

Nonlinear Analysis: Real World Applications

Volumn 10, Issue 3, 2009, Pages 1627-1642

  Kovács, Sándor ^a   Kiss, Krisztina ^b   Farkas, Miklós ^b  

^a EÖTVÖS LORÁND UNIVERSITY   (Hungary)

^b BUDAPEST UNIVERSITY OF TECHNOLOGY AND ECONOMICS   (Hungary)

Author keywords

Characteristic equation; Global asymptotic stability; Hopf bifurcation; Ratio dependence; Steady states; Time delay

Indexed keywords

ASYMPTOTIC ANALYSIS; ASYMPTOTIC STABILITY; SYSTEM STABILITY; TIME DELAY;

AUTONOMOUS SYSTEMS; CHARACTERISTIC EQUATION; CRITICAL VALUES; DEATH RATES; GLOBAL ASYMPTOTIC STABILITY; PERIODIC SOLUTIONS; PREDATOR - PREY SYSTEMS; PREDATOR-PREY INTERACTIONS; QUALITATIVE PROPERTIES; RATIO DEPENDENTS; RATIO-DEPENDENCE; SMALL AMPLITUDES; STEADY STATES;

HOPF BIFURCATION;

EID: 60549114318     PISSN: 14681218     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.nonrwa.2008.02.009     Document Type: Article

References (41)

1. Akcakaya H.R., Arditi R., and Ginzburg L.R. Ratio-dependent prediction: an abstraction that works. Ecology 76 (1995) 995-1004
2. Arditi R., and Berryman A.A. The biological paradox. Trends Ecol. Evol. 6 (1991) 32
3. Ariditi R., and Grinzburg L.R. Coupling in predator-prey dynamics: Ratio-dependence. J. Theoret. Biology 139 (1989) 311-326
4. Bellman R., and Cooke K. Differential Difference Equations (1963), Academic Press
5. Berezovskaya F., Karev G., and Arditi R. Parametric analysis of the ratio-dependent predator-prey model. J. Math. Biol. 43 (2001) 221-246
6. Berryman A.A. The origins and evolution of predator-prey theory. Ecology 73 (1992) 1530-1535
7. Boese F.G. Stability with respect to the delay: On a paper of K. L. Cooke and P. van den Driessche. J. Math. Anal. Appl. 228 (1998) 293-321
8. Cavani M., and Farkas M. Bifurcations in a predator-prey model with memory and diffusion. I: Andronov-Hopf bifurcation. Acta Math. Hungar. 63 3 (1994) 213-229
9. Cavani M., Lizana M., and Smith H.L. Stable Periodic Orbits for a Predator-Prey Model with Delay. J. Math. Anal. Appl. 249 (2000) 324-339
10. Cooke L., and van den Driessche P. On zeroes of some transcendental equations. Funkcialaj Ekvacioj 29 (1986) 77-90
11. Cooke L., and Grossman Z. Discrete delay, distributed delay and stability switches. Funkcialaj Ekvacioj 29 (1986) 77-90
12. Cosner C., DeAngelis D.L., Ault J.S., and Olson D.B. Effects of spatial grouping on the functional response of predators. Theoret. Popul. Biol. 56 1 (1999) 65-75
13. DeAngelis D.L., Goldstein R.A., and O'Neill R.V. A model for trophic interaction. Ecology 56 (1975) 881-892
14. Fan Y.-H., and Li W.-T. Permanence in delayed ratio-dependent predator-prey models with monotonic functional responses. Nonlinear Anal. RWA 8 2 (2007) 424-434
15. Farkas M. Periodic Motions (1994), Springer Verlag, Berlin, Heidelberg, New York
16. Freedman H.I., and Sree Hari Rao V. Stability criteria for a system involving two time delays. SIAM J. Appl. Math. 46 4 (1986) 552-560
17. Freedman H.I., and Mathsen R.M. Persistence in predator-prey systems with ratio-dependent predator influence. Bull. Math. Biol. 55 (1993) 817-827
18. Hale J. Theory of Functional Differential Equations (1977), Springer Verlag, Berlin, Heidelberg, New York
19. Hassard B.D., Kazarinoff N.D., and Wan Y.-H. Theory and Applications of Hopf Bifurcation (1981), Cambridge University Press, Cambridge
20. Hale J.K., Infante E.F., and Tsen F.-S.P. Stability in linear delay equations. J. Math. Anal. Appl. 105 2 (1985) 533-555
21. Hayes N.D. Roots of the transcendental equation associated with a certain difference-differential equation. J. London Math. Soc. 25 (1950) 226-232
22. Hsu S.B., Hwang T.W., and Kuang Y. Global analysis of the Michaelis-Menten-type ratio-dependent predator-prey system. J. Math. Biol. 42 (2001) 489-506
23. Hsu S.B., Hwang T.W., and Kuang Y. Rich dynamics of a ratio-dependent one-prey two-predators model. J. Math. Biol. 43 (2001) 377-396
24. Jost C., Arino O., and Arditi R. About deterministic extinction in ratio-dependent predator-prey models. Bull. Math. Biol. 61 (1999) 19-32
25. Kolmanovskii N.N., and Nosov V.R. Stability of Functional Differential Equations (1986), Academic Press, London
26. Kolmogorov A.N. Sulla teoria di Volterra della lotta per l'esistence. G. Inst. Ital. Attuari, Mir. 7 1 (1936) 74-80
27. Kolmogorov A.N. Qualitative Analysis of Mathematical Models of Populations. Problems of Cybernetics vol. 25 (1972), Moscow, Nauka 100-106
28. Kovács S. Stability of a delayed ratio-dependent predator-prey system. Miskolc Math. Notes 4 2 (2003) 127-134
29. Krall A.M. Stability Techniques for Continuous Linear Systems (1968), Gordon and Breach, London
30. Kuang Y., and Beretta E. Global qualitative analysis of a ratio-dependent predator-prey system. J. Math. Biol. 36 (1998) 389-406
31. May R. Stabilitity and Complexity in Model Ecosystems. Monographs in Population Biology (1973), Princeton University Press, Princeton
32. MacDonald N. Time lags in biological models. Lecture Notes in Biomathematics (1978), Springer-Verlag, Berlin, Heidelberg, New York
33. Neubert M.G., Klepac P., and van den Driessche P. Stabilizing dispersal delays in predator-prey metapopulation models. Theoret. Popul. Biol. 61 3 (2002) 339-347
34. Rosenzweig M.L. Paradox of enrichment: Destabilization of exploitation ecosystems in ecilogical time. Science 171 (1969) 385-387
35. Smoller J. Shock Wawes and Reaction-Diffusion Equations (1983), Springer Verlag, Berlin, Heidelberg, New York
36. Stépán G. Retarded dynamical systems: stability and characteristic functions. Pitman Research Notes in Mathematics Series vol. 210 (1989)
37. Yu.M. Svirezhev, D.O. Logofet, Stability of Biological Communities, Mir, Moscow 1983
38. Tang Y., Beretta E., and Solimano F. Stability analysis of a Volterra predator-prey system with two delays. Canad. Appl. Math. Quart. 9 1 (2001) 75-97
39. Thingstad T.F., and Langeland I. Dynamics of chemostat culture: The effect of a delay in cell response. J. Theoret. Biol. 48 1 (1974) 149-159
40. Wu S., and Ren G. Delay-independent stability criteria for a class of retarded dynamical systems with two delays. J. Sound Vibration 270 4-5 (2004) 625-638
41. Xiao D., and Ruan S. Global dynamics of a ratio-dependent predator-prey system. J. Math. Biol. 43 (2001) 268-290