Free Access
Issue
Math. Model. Nat. Phenom.
Volume 4, Number 2, 2009
Delay equations in biology
Page(s) 140 - 188
DOI https://doi.org/10.1051/mmnp/20094207
Published online 26 March 2009
  1. J. F. Andrews. A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates. Biotechnol. Bioeng., 10 (1968), 707-723. [CrossRef] [Google Scholar]
  2. R. Arditi, J.-M. Abillon, J. V. Da Silva. The effect of a time-delay in a predator-prey model. Math. Biosci., 33 (1977), 107-120. [CrossRef] [Google Scholar]
  3. M. Baptistini, P. Táboas. On the stability of some exponential polynomials. Math. Anal. Appl., 205 (1997), 259-272. [CrossRef] [MathSciNet] [Google Scholar]
  4. M. S. Bartlett. On theoretical models for competitive and predatory biological systems. Biometrika, 44 (1957), 27-42. [MathSciNet] [Google Scholar]
  5. J. R. Beddington, J. G. Cooke. Harvesting from a prey-predator complex. Ecol. Modelling, 14 (1982), 155-177. [CrossRef] [Google Scholar]
  6. R. Bellman, K. L. Cooke. Differential-difference equations. Academic Press, New York, 1963. [Google Scholar]
  7. E. Beretta, Y. Kuang. Convergence results in a well-known delayed predator-prey system. J. Math. Anal. Appl., 204 (1996), 840-853. [CrossRef] [MathSciNet] [Google Scholar]
  8. E. Beretta, Y. Kuang. Global analysis in some delayed ratio-dependent predator-prey systems. Nonlinear Anal., 32 (1998), 381-408. [CrossRef] [MathSciNet] [Google Scholar]
  9. E. Beretta, Y. Kuang. Geometric stability switch crteria in delay differential equations with delay dependent parameters. SIAM J. Math. Anal., 33(2002), 1144-1165. [Google Scholar]
  10. F. G. Boes. Stability criteria for second-order dynamical systems involving several time delays. SIAM J. Math. Anal., 26 (1995), 1306-1330. [CrossRef] [MathSciNet] [Google Scholar]
  11. F. Brauer. Stability of some population models with delay. Math. Biosci., 33 (1977), 345-358. [CrossRef] [MathSciNet] [Google Scholar]
  12. F. Brauer. Characteristic return times for harvested population models with time lag. Math. Biosci., 45 (1979), 295-311. [CrossRef] [MathSciNet] [Google Scholar]
  13. F. Brauer. Absolute stability in delay equations. J. Differential Equations, 69 (1987), 185-191. [CrossRef] [MathSciNet] [Google Scholar]
  14. F. Brauer, A. C. Soudack. Stability regions and transition phenomena for harvested predator-prey systems. J. Math. Biol., 7 (1979), 319-337. [CrossRef] [MathSciNet] [Google Scholar]
  15. F. Brauer, A. C. Soudack. Stability regions in predator-prey systems with constant-rate prey harvesting. J. Math. Biol., 8 (1979), 55-71. [CrossRef] [MathSciNet] [Google Scholar]
  16. F. Brauer, A. C. Soudack. Coexistence properties of some predator-prey systems under constant rate harvesting and stocking. J. Math. Biol., 12 (1981), 101-114. [CrossRef] [MathSciNet] [Google Scholar]
  17. M. Brelot. Sur le problème biologique héréditaiare de deux especès dévorante et dévorée. Ann. Mat. Pura Appl., 9 (1931), 58-74. [Google Scholar]
  18. A. W. Bush, A. E. Cook. The effect of time delay and growth rate inhibition in the bacterial treatment of wastewater. J. Theoret. Biol., 63 (1976), 385-395. [CrossRef] [Google Scholar]
  19. Y. Cao, H. I. Freedman. Global attractivity in time-delayed predator-prey systems. J. Austral. Math. Soc. Ser. B, 38 (1996), 149-162. [CrossRef] [MathSciNet] [Google Scholar]
  20. J. Caperon. Time lag in population growth response of isochrysis galbana to a variable nitrate environment. Ecology, 50 (1969), 188-192. [CrossRef] [Google Scholar]
  21. Y.-S. Chin. Unconditional stability of systems with time-lags. Acta Math. Sinica, 1 (1960), 125-142. [Google Scholar]
  22. K. L. Cooke, Z. Grossman. Discrete delay, distributed delay and stability switches. J. Math. Anal. Appl., 86 (1982), 592-627. [CrossRef] [MathSciNet] [Google Scholar]
  23. K. L. Cooke, P. van den Driessche. On zeros of some transcendental equations. Funkcialaj Ekvacioj, 29 (1986), 77-90. [MathSciNet] [Google Scholar]
  24. J. M. Cushing. Integrodifferential Equations and Delay Models in Population Dynamics. Springer-Verlag, Heidelberg, 1977. [Google Scholar]
  25. J. M. Cushing. Stability and maturation periods in age structured populations. In “Differential Equations and Applications in Ecology, Epidemics, and Population Problems”, S. Busenberg and K. L. Cooke (Eds.), Academic Press, New York, 1981, pp. 163-182. [Google Scholar]
  26. J. M. Cushing, M. Saleem. A predator prey model with age structure. J. Math. Biol., 14 (1982), 231-250. Erratum: 16 (1983), 305. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  27. G. Dai, M. Tang. Coexistence region and global dynamics of a harvested predator-prey system. SIAM J. Appl. Math., 58 (1998), 193-210. [CrossRef] [MathSciNet] [Google Scholar]
  28. L. S. Dai. Nonconstant periodic solutions in predator-prey systems with continuous time delay. Math. Biosci., 53 (1981), 149-157. [CrossRef] [MathSciNet] [Google Scholar]
  29. R. Datko. A procedure for determination of the exponential stability of certain differential difference equations. Quart. Appl. Math., 36 (1978), 279-292. [MathSciNet] [Google Scholar]
  30. J. Dieudonné. Foundations of modern analysis. Academic Press, New York, 1960. [Google Scholar]
  31. B. Ermentrout. Simulating, analyzing, and animating dynamical systems: a guide to XPPAUT for researchers and students. SIAM, Philadelphia, 2002. [Google Scholar]
  32. T. Faria. Stability and bifurcation for a delayed predator-prey model and the effect of diffusion. J. Math. Anal. Appl., 254 (2001), 433-463. [CrossRef] [MathSciNet] [Google Scholar]
  33. T. Faria, L. T. MagalhFormula es. Normal forms for retarded functional differential equations and applications to Bogdanov-Takens singularity. J. Differential Equations, 122 (1995), 201-224. [CrossRef] [MathSciNet] [Google Scholar]
  34. T. Faria, L. T. MagalhFormula es. Normal forms for retarded functional differential equations with parameters and applications to Hopf bifurcations. J. Differential Equations, 122 (1995), 181-200. [CrossRef] [MathSciNet] [Google Scholar]
  35. A. Farkas, M. Farkas, G. Szabó. Multiparameter bifurcation diagrams in predator-prey models with time lag. J. Math. Biol., 26 (1988), 93-103. [MathSciNet] [Google Scholar]
  36. H. I. Freedman. Deterministic Mathematical Models in Population Ecology. HIFR Consulting Ltd., Edmonton, 1987. [Google Scholar]
  37. H. I. Freedman, K. Gopalsamy. Nonoccurence of stability switching in systems with discrete delays. Canad. Math. Bull., 31 (1988), 52-58. [CrossRef] [MathSciNet] [Google Scholar]
  38. H. I. Freedman, V. S. H. Rao. The tradeoff between mutual interference and time lags in predator-prey systems. Bull. Math. Biol., 45 (1983), 991-1004. [MathSciNet] [Google Scholar]
  39. H. I. Freedman, V. S. H. Rao. Stability criteria for a system involving two time delays. SIAM J. Appl. Anal., 46 (1986), 552-560. [CrossRef] [Google Scholar]
  40. H. I. Freedman, G. S. K. Wolkowicz. Predator-prey systems with group defence: The paradox of enrichment revisited. Bull. Math. Biol., 48 (1986), 493-508. [MathSciNet] [PubMed] [Google Scholar]
  41. N. S. Goel, S. C. Maitra, E. W. Montroll. On the Volterra and other nonlinear models of interacting populations. Rev. Modern Phys., 43 (1971), 231-276. [CrossRef] [MathSciNet] [Google Scholar]
  42. K. Gopalsamy. Harmless delay in model systems. Bull. Math. Biol., 45 (1983), 295-309. [MathSciNet] [Google Scholar]
  43. K. Gopalsamy. Delayed responses and stability in two-species systems. J. Austral. Math. Soc. Ser. B, 25 (1984), 473-500. [CrossRef] [MathSciNet] [Google Scholar]
  44. K. Gopalsamy. Stability and Oscillations in Delay Differential Equations of Population Dynamics. Kluwer Academic, Dordrecht, 1992. [Google Scholar]
  45. S. Gourley, Y. Kuang. A stage structured predator-prey model and its dependence on maturation delay and death rate. J. Math. Biol., 49 (2004), 188-200. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  46. J. K. Hale, E. F. Infante, F.-S. P. Tsen. Stability in linear delay equations. J. Math. Anal. Appl., 105 (1985), 533-555. [CrossRef] [MathSciNet] [Google Scholar]
  47. J. K. Hale, S. M. Verduyn Lunel. Introduction to functional differential equations. Springer-Verlag, New York, 1993. [Google Scholar]
  48. B. D. Hassard, N. D. Kazarinoff, Y.-H. Wan. Theory and applications of Hopf bifurcation. Cambridge University Press, London, 1981. [Google Scholar]
  49. A. Hastings. Age-dependent predation is not a simple process: I. continuous time models. Theoret. Pop. Biol., 23 (1983), 347-362. [Google Scholar]
  50. A. Hastings. Delays in recruitment at different trophic levels: effects on stability. J. Math. Biol., 21 (1984), 35-44. [MathSciNet] [PubMed] [Google Scholar]
  51. X.-Z. He. Stability and delays in a predator-prey system. J. Math. Anal. Appl., 198 (1996), 355-370. [CrossRef] [MathSciNet] [Google Scholar]
  52. X.-Z. He. The Lyapunov functionals for delay Lotka-Volterra-type models. SIAM J. Appl. Math., 58 (1998), 1222-1236. [CrossRef] [MathSciNet] [Google Scholar]
  53. W. L. Hogarth, J. Norbury, I. Cunning, K. Sommers. Stability of a predator-prey model with harvesting. Ecol. Modelling, 62 (1992), 83-106. [CrossRef] [Google Scholar]
  54. W. Huang. Algebraic criteria on the stability of the zero solutions of the second order delay differential equations. J. Anhui University, (1985), 1–7. [Google Scholar]
  55. J. A. Hutchings, R. A. Myers. What can be learned from the collapse of a renewable resource? Atlantic code, Gadus morhua, of Newfoundland and Labrador. Can. J. Fish. Aquat. Sci., 51 (1994), 2126-2146. [CrossRef] [Google Scholar]
  56. Y. Kuang. Delay differential equations with applications in population dynamics. Academic Press, New York, 1993. [Google Scholar]
  57. Y. A. Kuznetsov. Elements of applied bifurcation theory. Applied Mathematical Sciences 112, Springer-Verlag, New York, 1995. [Google Scholar]
  58. S. Liu, L. Chen, R. Agarwal. Recent progress on stage-structured population dynamics. Math. Computer Model.,36 (2002), 1319-1360. [Google Scholar]
  59. Z. Liu, R. Yuan. Stability and bifurcation in a delayed predator-prey system with Beddinton-DeAngelis functional response. J. Math. Anal. Appl., 296 (2004), 521-537. [CrossRef] [MathSciNet] [Google Scholar]
  60. Z. Lu, W. Wang. Global stability for two-species Lotka-Volterra systems with delay. J. Math. Anal. Appl., 208 (1997), 277-280. [CrossRef] [MathSciNet] [Google Scholar]
  61. Z. Ma. Stability of predation models with time delay. Applicable Anal., 22 (1986), 169-192. [CrossRef] [Google Scholar]
  62. J. M. Mahaffy. A test for stability of linear differential delay equations. Quart. Appl. Math., 40 (1982), 193-202. [MathSciNet] [Google Scholar]
  63. A. Martin, S. Ruan. Predator-prey models with delay and prey harvesting. J. Mathematical Biology, 43 (2001), 247-267. [CrossRef] [Google Scholar]
  64. R. M. May. Time delay versus stability in population models with two and three trophic levels. Ecology, 4 (1973), 315-325. [CrossRef] [Google Scholar]
  65. N. MacDonald. Time lags in biological models. Springer-Verlag, Heidelberg, 1978. [Google Scholar]
  66. R. A. Myers, J. A. Hutchings, N. J. Barrowman. Why do fish stocks collapse? The example of cod in Atlantic Canada. Ecol. Appl., 7 (1997), 91-106. [CrossRef] [Google Scholar]
  67. R. A. Myers, B. Worm. Rapid worldwide depletion of large predatory fish communities. Nature, 423 (2003), 280-283. [CrossRef] [PubMed] [Google Scholar]
  68. M. R. Myerscough, B. F. Gray, W. L. Hogarth, J. Norbury. An analysis of an ordinary differential equation model for a two-species predator-prey system with harvesting and stocking. J. Math. Biol., 30 (1992), 389-411. [MathSciNet] [Google Scholar]
  69. S. Nakaoka, Y. Saito, Y. Takeuchi. Stability, delay, and chaotic behavior in a Lotka-Volterra predator-prey system. Math. Biosci. Engineer., 3 (2006), 173-187. [Google Scholar]
  70. L. Nunney. The effect of long time delays in predator-prey systems. Theoret. Pop. Biol., 27 (1985), 202-221. [CrossRef] [PubMed] [Google Scholar]
  71. L. Nunney. Absolute stability in predator-prey models. Theoret. Pop. Biol., 28 (1985), 209-232. [CrossRef] [Google Scholar]
  72. Y. Qu, J. Wei. Bifurcation analysis in a time-delay model for prey-predator growth with stage-structure. Nonlinear Dynamics, 49 (2007), 285-294. [CrossRef] [MathSciNet] [Google Scholar]
  73. G. G. Ross. A difference-differential model in population dynamics. J. Theoret. Biol., 37 (1972), 477-492. [CrossRef] [PubMed] [Google Scholar]
  74. S. Ruan. Absolute stability, conditional stability and bifurcation in Kolmogorov-type predator-prey systems with discrete delays. Quart. Appl. Math., 59 (2001), 159-173. [MathSciNet] [Google Scholar]
  75. S. Ruan. Delay differential equations in single species dynamics. In “Delay Differential Equations with Applications,” O. Arino, M. Hbid and E. Ait Dads (Eds.), NATO Science Series II: Mathematics, Physics and Chemistry, Vol. 205, Springer, Berlin, 2006, pp. 477-517. [Google Scholar]
  76. S. Ruan, J. Wei. On the zeros of transcendental functions with applications to stability of delay differential equations. Dynam. Contin. Discr. Impuls. Syst., 10 (2003), 863-874. [Google Scholar]
  77. S. Ruan, D. Xiao. Global analysis in a predator-prey system with nonmonotonic functional response. SIAM J, Appl. Math., 61 (2001), 1445-1472. [Google Scholar]
  78. W. Sokol, J. A. Howell. Kinetics of phenol oxidation by washed cells. Biotechnol. Bioeng., 23 (1980), 2039-2049. [CrossRef] [Google Scholar]
  79. Y. Song, Y. Peng, J. Wei. Bifurcations for a predator-prey system with two delays. J. Math. Anal. Appl., 337 (2008), 466-479. [CrossRef] [MathSciNet] [Google Scholar]
  80. Y. Song, J. Wei. Local Hopf bifurcation and global periodic solutions in a delayed predator-prey system. J. Math. Anal. Appl., 301 (2005), 1-21. [CrossRef] [MathSciNet] [Google Scholar]
  81. G. Stépán. Great delay in a predator-prey model. Nonlinear Anal., 10 (1986), 913-929. [CrossRef] [MathSciNet] [Google Scholar]
  82. P. Táboas. Periodic solutions of a planar delay equation. Proc. Roy. Soc. Edinburgh, 116A (1990), 85-101. [Google Scholar]
  83. V. Volterra. Variazionie fluttuazioni del numbero d'individui in specie animali conviventi. Mem. Acad. Lincei., 2 (1926), 31-113. [Google Scholar]
  84. V. Volterra. Lecons sur la théorie mathematique de la lutte pour la vie. Gauthier-Villars, Paris, 1931. [Google Scholar]
  85. W. Wang, L. Chen. A predator-prey system with stage-structure for predators. Computers Math. Appl., 33 (1997), No. 8, 83-91. [NASA ADS] [CrossRef] [EDP Sciences] [MathSciNet] [PubMed] [Google Scholar]
  86. P. J. Wangersky, W. J. Cunningham. Time lag in prey-predator population models. Ecology, 38 (1957), 136-139. [CrossRef] [Google Scholar]
  87. G. S. K. Wolkowicz. Bifurcation analysis of a predator-prey system involving group defence. SIAM J. Appl. Math., 48 (1988), 592-606. [CrossRef] [MathSciNet] [Google Scholar]
  88. J. Wu. Symmetric functional differential equations and neural networks with memory. Trans. Amer. Math. Soc., 350 (1998), 4799-4838. [CrossRef] [MathSciNet] [Google Scholar]
  89. J. Xia, Z. Liu, R. Yuan, S. Ruan. The effects of harvesting and time delay on predator-prey systems with Holling type II functional response. SIAM J. Appl. Math. (revised). [Google Scholar]
  90. D. Xiao, W. Li. Stability and bifurcation in a delayed ratio-dependent predator-prey system. Proc. Edinburgh Math. Soc., 46A (2003), 205-220. [Google Scholar]
  91. D. Xiao, S. Ruan. Bogdanov-Takens bifurcations in predator-prey systems with constant rate harvesting. Fields Institute Communications, 21 (1999), 493-506. [Google Scholar]
  92. D. Xiao, S. Ruan. Multiple bifurcations in a delayed predator-prey system with nonmonotonic functional response. J. Differential Equations, 176 (2001), 494-510. [CrossRef] [MathSciNet] [Google Scholar]
  93. X.-P. Yan, W.-T. Li. Hopf bifurcation and global periodic solutions in a delayed predator-prey system. Appl. Math. Computat., 177 (2006), 427-445. [CrossRef] [Google Scholar]
  94. T. Zhao, Y. Kuang, H. L. Smith. Global existence of periodic solutions in a class of delayed Gause-type predator-prey systems. Nonlinear Anal., 28 (1997), 1373-1394. [CrossRef] [Google Scholar]

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