Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

Absolute stability, conditional stability and bifurcation in Kolmogorov-type predator-prey systems with discrete delays


Author: Shigui Ruan
Journal: Quart. Appl. Math. 59 (2001), 159-173
MSC: Primary 34K20; Secondary 34K18, 92D25
DOI: https://doi.org/10.1090/qam/1811101
MathSciNet review: MR1811101
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Abstract | References | Similar Articles | Additional Information

Abstract: The dynamics of delayed systems depend not only on the parameters describing the models but also on the time delays from the feedback. A delay system is absolutely stable if it is asymptotically stable for all values of the delays and conditionally stable if it is asymptotically stable for the delays in some intervals. In the latter case, the system could become unstable when the delays take some critical values and bifurcations may occur. We consider three classes of Kolmogorov-type predator-prey systems with discrete delays and study absolute stability, conditional stability and bifurcation of these systems from a global point of view on both the parameters and delays.


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  • [1] M. Baptistini and P. Táboas, On the stability of some exponential polynomials, J. Math. Anal. Appl. 205, 259-272 (1997) MR 1426993
  • [2] M. S. Bartlett, On theoretical models for competitive and predatory biological systems, Biometrika 44, 27-42 (1957) MR 0086727
  • [3] R. Bellman and K. L. Cooke, Differential-Difference Equations, Academic Press, New York, 1963 MR 0147745
  • [4] E. Beretta and Y. Kuang, Convergence results in a well-known delayed predator-prey system, J. Math. Anal. Appl. 204, 840-853 (1996) MR 1422776
  • [5] F. G. Boese, Stability criteria for second-order dynamical systems involving several time delays, SIAM J. Math. Anal. 26, 1306-1330 (1995) MR 1347422
  • [6] F. Brauer, Characteristic return times for harvested population models with time lag, Math. Biosci. 45, 295-311 (1979) MR 538430
  • [7] F. Brauer, Absolute stability in delay equations, J. Differential Equations 69, 185-191 (1987) MR 899158
  • [8] M. Brelot, Sur le problème biologique héréditaire de deux espéces dévorante et dévoré, Ann. Mat. Pura Appl. 9, 58-74 (1931)
  • [9] Y. Cao and H. I. Freedman, Global attractivity in time-delayed predator-prey systems, J. Austral. Math. Soc. Ser. B 38, 149-162 (1996) MR 1414356
  • [10] Y.-S. Chin, Unconditional stability of systems with time-lags, Acta Math. Sinica 1, 125-142 (1960) MR 0114031
  • [11] K. L. Cooke and Z. Grossman, Discrete delay, distributed delay and stability switches, J. Math. Anal. Appl. 86, 592-627 (1982) MR 652197
  • [12] K. L. Cooke and P. van den Driessche, On zeros of some transcendental equations, Funkcialaj Ekvacioj 29, 77-90 (1986) MR 865215
  • [13] J. M. Cushing, Integrodifferential Equations and Delay Models in Population Dynamics, Springer-Verlag, Heidelberg, 1977 MR 0496838
  • [14] L. S. Dai, Nonconstant periodic solutions in predator-prey systems with continuous time delay, Math. Biosci. 53, 149-157 (1981) MR 613620
  • [15] R. Datko, A procedure for determination of the exponential stability of certain differential difference equations, Quart. Appl. Math. 36, 279-292 (1978) MR 508772
  • [16] J. Dieudonné, Foundations of Modern Analysis, Academic Press, New York, 1960 MR 0120319
  • [17] A. Farkas, M. Farkas, and G. Szabó, Multiparameter bifurcation diagrams in predator-prey models with time lag, J. Math. Biol. 26, 93-103 (1988) MR 929973
  • [18] H. I. Freedman and K. Gopalsamy, Nonoccurrence of stability switching in systems with discrete delays, Canad. Math. Bull. 31, 52-58 (1988) MR 932613
  • [19] H. I. Freedman and V. S. H. Rao, The tradeoff between mutual interference and time lags in predator-prey systems, Bull. Math. Biol. 45, 991-1004 (1983) MR 727356
  • [20] H. I. Freedman and V. S. H. Rao, Stability criteria for a system involving two time delays, SIAM J. Appl. Anal. 46, 552-560 (1986) MR 849081
  • [21] N. S. Goel, S. C. Maitra, and E. W. Montroll, On the Volterra and other nonlinear models of interacting populations, Rev. Modern Phys. 43, 231-276 (1971) MR 0484546
  • [22] K. Gopalsamy, Harmless delay in model systems, Bull. Math. Biol. 45, 295-309 (1983) MR 708998
  • [23] K. Gopalsamy, Delayed responses and stability in two-species systems, J. Austral. Math. Soc. Ser. B 25, 473-500 (1984) MR 734969
  • [24] K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer Academic, Dordrecht, 1992 MR 1163190
  • [25] J. K. Hale, E. F. Infante, and F.-S. P. Tsen, Stability in linear delay equations, J. Math. Anal. Appl. 105, 533-555 (1985) MR 778486
  • [26] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993 MR 1243878
  • [27] B. D. Hassard, N. D. Kazarinoff, and Y.-H. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, London, 1981 MR 603442
  • [28] A. Hastings, Age-dependent predation is not a simple process: I. Continuous time models, Theoret. Pop. Biol. 23, 347-362 (1983) MR 711912
  • [29] A. Hastings, Delays in recruitment at different trophic levels: Effects on stability, J. Math. Biol. 21, 35-44 (1984) MR 770711
  • [30] X.-Z. He, Stability and delays in a predator-prey system, J. Math. Anal. Appl. 198, 355-370 (1996) MR 1376269
  • [31] W. Huang, Algebraic criteria on the stability of the zero solutions of the second order delay differential equations, J. Anhui University, 1-7 (1985)
  • [32] G. E. Hutchinson, Circular cause systems in ecology, Ann. New York Acad. Sci. 50, 221-246 (1948)
  • [33] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, 1993 MR 1218880
  • [34] Z. Lu and W. Wang, Global stability for two-species Lotka-Volterra systems with delay, J. Math. Anal. Appl. 208, 277-280 (1997) MR 1440357
  • [35] Z. Ma, Stability of predation models with time delay, Applicable Analysis 22, 169-192 (1986) MR 860988
  • [36] J. M. Mahaffy, A test for stability of linear differential delay equations, Quart. Appl. Math. 40, 193-202 (1982) MR 666674
  • [37] R. M. May, Time delay versus stability in population models with two and three trophic levels, Ecology 4, 315-325 (1973)
  • [38] N. MacDonald, Time Lags in Biological Models, Springer-Verlag, Heidelberg, 1978 MR 521439
  • [39] L. Nunney, The effect of long time delays in predator-prey systems, Theoret. Pop. Biol. 27, 202-221 (1985) MR 797394
  • [40] L. Nunney, Absolute stability in predator-prey models, Theoret. Pop. Biol. 28, 209-232 (1985) MR 809778
  • [41] S. Ruan and J. Wei, On the zeros of transcendental functions with applications to stability of delay differential equations, preprint.
  • [42] G. Stépán, Great delay in a predator-prey model, Nonlinear Analysis 10, 913-929 (1986) MR 856874
  • [43] V. Volterra, Leçons sur la théorie mathématique de la lutte pour la vie, Gauthier-Villars, Paris, 1931
  • [44] P. J. Wangersky and W. J. Cunningham, Time lag in prey-predator population models, Ecology 38, 136-139 (1957)
  • [45] T. Zhao, Y. Kuang, and H. L. Smith, Global existence of periodic solutions in a class of delayed Gause-type predator-prey systems, Nonlinear Analysis 28, 1373-1394 (1997) MR 1428657

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DOI: https://doi.org/10.1090/qam/1811101
Article copyright: © Copyright 2001 American Mathematical Society

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