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All issues Volume 15 (2020) Math. Model. Nat. Phenom., 15 (2020) 60 References
Ecology and evolution
Open Access
Issue
Math. Model. Nat. Phenom.
Volume 15, 2020
Ecology and evolution
Article Number 60
Number of page(s) 27
DOI https://doi.org/10.1051/mmnp/2020042
Published online 03 December 2020
  1. E.H. Abed, H.O. Wang and R.C. Chen, Stabilization of period–doubling bifurcation and implications for control of chaos. Physica D 70 (1994) 154–164. [CrossRef] [Google Scholar]
  2. F. Barraquand, S. Louca, K.C. Abbott, et al., Moving forward in circles: challenges and opportunities in modelling population cycles. Ecol. Lett. 20 (2017) 1074–1092. [CrossRef] [Google Scholar]
  3. V.A. Bailey and A.J. Nicholson, The balance of animal populations. Proc. Zool. Soc. Lond. 3 (1935) 551–598. [Google Scholar]
  4. E. Bešo, S. Kalabušić, N. Mujić and E. Pilav, Neimark–Sacker bifurcation and stability of a certain class of a host–parasitoid models with a host refuge effect. Int. J. Bifurcat. Chaos 29 (2019) 195169. [Google Scholar]
  5. E. Bešo, N. Mujić, S. Kalabušić, E. Pilav, Stability of a certain class of a host–parasitoid models with a spatial refuge effect. J. Biol. Dyn. 14 (2020) 1–31. [CrossRef] [PubMed] [Google Scholar]
  6. J.R. Beddington, C.A. Free and J.H. Lawton, Dynamic complexity in predator–prey models framed in difference equations. Nature 225 (1975) 58–60. [CrossRef] [Google Scholar]
  7. M. Begon, C. Townsend, L. John, R. Colin and L.H. John, Ecology: From Individuals to Ecosystems. Oxford, Blackwell Publishing Ltd (2006). [Google Scholar]
  8. F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology. New York, Springer (2012). [CrossRef] [Google Scholar]
  9. F. Brauer, Z. Feng and C. Castillo-Chavez, Discrete epidemic models. Math. Biosci. Eng. 7 (2010) 1–15. [CrossRef] [PubMed] [Google Scholar]
  10. J. Carr, Application of Center Manifold Theory. Springer–Verlag, New York (1981). [CrossRef] [Google Scholar]
  11. G. Chen, J. Fang, Y. Hong and H. Qin, Controlling Hopf bifurcations: discrete–time systems. Discrete Dyn. Nat. Soc. 5 (2000) 29–33. [CrossRef] [Google Scholar]
  12. G. Chen and X. Yu, On time–delayed feedback control of chaotic systems. IEEE Trans. Circ. Sys. 46 (1999) 767–772. [CrossRef] [Google Scholar]
  13. U. Daugaard, O.L. Petchey and F. Pennekamp, Warming can destabilise predator—prey interactions by shifting the functional response from Type III to Type II. J. Anim. Ecol. 88 (2019) 1575–1586. [CrossRef] [PubMed] [Google Scholar]
  14. J.P. DeLong, T.C. Hanley and D.A. Vasseur, Predator—prey dynamics and the plasticity of predator body size. Funct. Ecol. 28 (2014) 487–493. [CrossRef] [Google Scholar]
  15. Q. Din, Complexity and chaos control in a discrete-time prey—predator model. Commun. Nonlinear Sci. Numer. Simul. 49 (2017) 113–134. [CrossRef] [Google Scholar]
  16. Q. Din, Stability, Bifurcation analysis and chaos control for a predator–prey system. J. Vib. Control 25 (2019) 612–626. [CrossRef] [Google Scholar]
  17. Q. Din, A novel chaos control strategy for discrete-time Brusselator models. J. Math. Chem. 56 (2018) 3045–3075. [CrossRef] [Google Scholar]
  18. M. Fan and K. Wang, Periodic solutions of a discrete time nonautonomous ratio–dependent predator—prey system. Math. Comput. Model. 35 (2002) 951–961. [CrossRef] [Google Scholar]
  19. J. Guckenheimer and P. Holmes, Nonlinear Oscillations Dynamical Systems, and Bifurcations of Vector Fields. Springer–Verlag, New York (1983). [CrossRef] [Google Scholar]
  20. K.P. Hadeler and I. Gerstmann, The discrete Rosenzweig model. Math. Biosci. 98 (1990) 49–72. [CrossRef] [Google Scholar]
  21. A. Hastings, Population Biology: Concepts and Models. Springer, New York (1997). [CrossRef] [Google Scholar]
  22. M.P. Hassell, The dynamics of arthropod predator—prey systems. Princeton University Press, Princeton (1978). [Google Scholar]
  23. M.P. Hassell and G.C. Varley, New inductive population model for insect parasites and its bearing on biological control. Nature 223 (1969) 1133–1137. [CrossRef] [PubMed] [Google Scholar]
  24. J. Hofbauer, V. Hutson and W. Jansen, Coexistence for systems governed by difference equations of Lotka–Volterra type. J. Math. Biol. 25 (1987) 553–570. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  25. W.T. Jamieson, On the global behaviour of May’s host–parasitoid model. J. Differ. Equ. Appl. 25 (2019) 583–596. [CrossRef] [Google Scholar]
  26. V. Kaitala, J. Ylikarjula, M. Heino, Chaos in functional response host-parasitoid ecosystem models. Chaos Soliton Fract. 197 (1999) 331–341. [Google Scholar]
  27. B.E. Kendall, C.J. Briggs, W.W. Murdoch et al., Why do populations cycle? A synthesis of statistical and mechanistic modeling approaches. Ecology 80 (1999) 1789–1805. [CrossRef] [Google Scholar]
  28. Y.A. Kuznetsov, Elements of Applied Bifurcation Theory. Springer-Verlag, New York (1997). [Google Scholar]
  29. V. Krivan and A. Priyadarshi, L–shaped prey isocline in the Gause predator—prey experiments with a prey refuge. J. Theor. Biol. 370 (2015) 21–26. [CrossRef] [PubMed] [Google Scholar]
  30. H. Liu, K. Zhang, Y. Ye, Y. Wei and M. Ma, Dynamic complexity and bifurcation analysis of a host–parasitoid model with Allee effect and Holling type III functional response. Adv. Differ. Equ. 2019 (2019) 507. [CrossRef] [Google Scholar]
  31. X. Liu, Y. Chu and Y. Liu, Bifurcation and chaos in a host–parasitoid model with a lower bound for the host. Adv. Differ. Equ. 2018 (2018) 31. [CrossRef] [Google Scholar]
  32. P. Liu and S.N. Elaydi, Discrete competitive and cooperative models of Lotka–Volterra type. J. Comput. Anal. Appl. 3 (2001) 53–73. [Google Scholar]
  33. X. Liu and D. Xiao, Complex dynamic behaviors of a discrete–time predator—prey system. Chaos Soliton Fract. 32 (2007) 80–94. [CrossRef] [Google Scholar]
  34. S.J. Lv and M. Zhao, The dynamic complexity of a host–parasitoid model with a lower bound for the host. Chaos Soliton Fract. 36 (2008) 911–919. [CrossRef] [Google Scholar]
  35. X.S. Luo, G.R. Chen, B.H. Wang, and J.Q. Fang, Hybrid control of period–doubling bifurcation and chaos in discrete nonlinear dynamical systems. Chaos Soliton Fract. 18 (2003) 775–783. [CrossRef] [Google Scholar]
  36. R.M. May, Host–parasitoid systems in patchy environments: a phenomenological model. J. Animal Ecol. 47 (1978) 833–844. [CrossRef] [Google Scholar]
  37. R.M. May, M.P. Hassell, R.M. Anderson and D.W. Tonkyn, Density dependence in host-parasitoid models. J. Animal Ecol. 50 (1981) 855–865. [CrossRef] [Google Scholar]
  38. W.W. Murdoch, C.J. Briggs and R.M. Nisbet, Consumer–Resource Dynamics. Princeton and Oxford, Princeton University Press (2003). [Google Scholar]
  39. M. Neubert and M. Kot, The subcritical collapse of predator populations in discrete time predator—prey models. Math. Biosci. 110 (1992) 45–66. [CrossRef] [Google Scholar]
  40. K. Ogata, Modern Control Engineering, 2nd edition. Prentice-Hall, Englewood, New Jersey (1997). [Google Scholar]
  41. E. Ott, C. Grebogi and J.A. Yorke, Controlling chaos. Phys. Rev. Lett. 64 (1990) 1196–1199. [NASA ADS] [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  42. J.L. Ren and L.P. Yu, Codimension–Two Bifurcation Chaos and control in a discrete–time information diffusion model. J. Nonlinear Sci. 26 (2016) 1895–1931. [CrossRef] [Google Scholar]
  43. C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics and Chaos. Boca Raton, New York (1999). [Google Scholar]
  44. F.J. Romeiras, C. Grebogi, E. Ott and W.P. Dayawansa, Controlling chaotic dynamical systems. Physica D 58 (1992) 165–192. [CrossRef] [Google Scholar]
  45. B. Rosenbaum and B.C. Rall, Fitting functional responses: direct parameter estimation by simulating differential equations. Methods Ecol. Evol. 9 (2018) 2076–2090. [CrossRef] [Google Scholar]
  46. M.L. Rosenzweig and R.H. MacArthur, Graphical representation and stability conditions of predator—prey interactions. Am. Nat. 97 (1963) 209–223. [CrossRef] [Google Scholar]
  47. P.S. Salomon and W. Stolte, Predicting the population dynamics in Amoebophrya parasitoids and their dinoflagellate hosts using a mathematical model. Mar. Ecol. Prog. Ser. 419 (2010) 1–10. [CrossRef] [Google Scholar]
  48. S.Y. Tang and L.S. Chen, Chaos in functional response host-parasitoid ecosystem models. Chaos Soliton Fract. 13 (2002) 875–884. [CrossRef] [Google Scholar]
  49. J.T. Tanner, The stability and the intrinsic growth rates of prey and predator populations. Ecology 56 (1975) 855–867. [Google Scholar]
  50. A.D. Taylor, Parasitiod competition and the dynamics of host-parasitiod models. Am. Nat. 132 (1988) 417–436. [CrossRef] [Google Scholar]
  51. P. Turchin, Complex Population Dynamics. A Theoretical/Empirical Synthesis. Princeton, Princeton University Press (2003). [Google Scholar]
  52. E. van Velzen and U. Gaedke, Reversed predator—prey cycles are driven by the amplitude of prey oscillations. Ecol. Evol. 8 (2018) 6317–6329. [CrossRef] [PubMed] [Google Scholar]
  53. Y.H. Wan, Computation of the stability condition for the Hopf bifurcation of diffeomorphism on R2 . SIAM.J. Appl. Math. 34 (1978) 167–175. [CrossRef] [Google Scholar]
  54. G.L. Wen, D.L. Xu, and J.H. Xie, Controlling Hopf bifurcations of discrete–time systems in resonance. Chaos Soliton Fract. 23 (2005) 1865–1877. [CrossRef] [Google Scholar]
  55. W. Wendi and L. Zhengyi, Global stability of discrete models of Lotka–Volterra type. Nonlinear Anal. Theor. 35 (1999) 1019–1030. [CrossRef] [Google Scholar]
  56. V. Weide, M.C. Varriale and F.M. Hilker, Hydra effect and paradox of enrichment in discrete–time predator—prey models. Math. Biosci. 310 (2019) 120–127. [CrossRef] [Google Scholar]
  57. S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer–Verlag, New York (2003). [Google Scholar]
  58. D. Wu and H. Zhao, Global qualitative analysis of a discrete host–parasitoid model with refuge and strong Allee effects. Math. Method. Appl. Sci. 41 (2018) 2039–2062. [CrossRef] [Google Scholar]
  59. C.L. Xu and M.S. Boyce, Dynamic complexities in a mutual interference host-parasitoid model. Chaos Soliton Fract. 24 (2005) 175–182. [CrossRef] [Google Scholar]
  60. L.-G. Yuan and Q.-G. Yang, Bifurcation, invariant curve and hybrid control in a discrete–time predator—prey system. Appl. Math. Model. 39 (2015) 2345–2362. [CrossRef] [Google Scholar]
  61. X. Zhang, Q.L. Zhang and V. Sreeram, Bifurcation analysis and control of a discrete harvested prey–predator system with Beddington–DeAngelis functional response. J. Franklin Inst. 347 (2010) 1076–1096. [CrossRef] [Google Scholar]
  62. L. Zhu and M. Zhao, Dynamic complexity of a host–parasitoid ecological model with the Hassell growth function for the host. Chaos Soliton Fract. 39 (2009) 1259–1269. [CrossRef] [Google Scholar]

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