Russian version English version
Volume 14   Issue 1   Year 2019
Oksana L. Revutskaya, Matvey P. Kulakov, Efim Ya. Frisman

Bistability and Bifurcations in Modified Nicholson-Bailey Model with Age-Structure for Prey

Mathematical Biology & Bioinformatics. 2019;14(1):257-278.

doi: 10.17537/2019.14.257.

References

 

  1. Volterra V. Matematicheskaia teoriia bor'by za sushchestvovanie (Mathematical theory of the struggle for existence). Moscow: Nauka; 2004. 288 p. (in Russ.).
  2. Nicholson A.J. Supplement: the Balance of Animal Populations. Journal of Animal Ecology. 1933;2(1):131–178. doi: 10.2307/954
  3. Nicholson A.J., Bailey V.A. The Balance of Animal Populations. Proceedings of the Zoological Society of London. 1935;105(3):551–598. doi: 10.1111/j.1096-3642.1935.tb01680.x
  4. Nicholson A.J. An outline of the dynamics of animal populations. Australian Journal of Zoology. 1954;2:9–65. doi: 10.1071/ZO9540009
  5. Rosenzweig A., MacArthur R.H. Graphical representation and stability conditions of predator-prey interaction. Amer, Natur. 1963;97:209–223. doi: 10.1086/282272
  6. Kolmogorov A.N. Problemy kibernetiki (Problems of Cybernetics). 1972;5:100–106 (in Russ.).
  7. Hassell M.P., May R.M. Stability in insect host-parasite models. Journal of Animal Ecology. 1973;42:693–726. doi: 10.2307/3133
  8. Bazykin A.D. Matematicheskaia biofizika vzaimodeistvuiushchikh populiatsii (Mathematical biophysics of interacting populations). Moscow: Nauka; 1985. 181 p. (in Russ.).
  9. Mills N.J., Getz W.M. Modelling the biological control of insect pests: a review of host-parasitoid models. Ecological Modelling. 1996;92:121–143. doi: 10.1016/0304-3800(95)00177-8
  10. Hassell M P. Host-parasitoid population dynamics. Journal of Animal Ecology. 2000;69:543–566. doi: 10.1046/j.1365-2656.2000.00445.x
  11. Svirezhev Iu.M., Logofet D.O. Ustoichivost' biologicheskikh soobshchestv (Resilience of biological communities). Moscow: Nauka; 1978. 352 p. (in Russ.).
  12. Hastings A. Age-Dependent Predation Is Not a Simple Process. II. Wolves, Ungulates, and a Discrete Time Model for Predation on Juveniles with a Stabilizing Tail. Theoretical Population Biology. 1984;26:271–282. doi: 10.1016/0040-5809(84)90033-9
  13. Fryxell J.M., Mosser A., Sinclair A.R.E., Packer C. Group formation stabilizes predator–prey dynamics. Nature. 2007;449:1041–1043. doi: 10.1038/nature06177
  14. Abbott K.C., Dwyer G. Food limitation and insect outbreaks: complex dynamics in plant–herbivore models. Journal of Animal Ecology. 2007;76:1004–1014. doi: 10.1111/j.1365-2656.2007.01263.x
  15. Odum E. Ekologiia (Fundamentals of Ecology). Vol. 2. Moscow; 1986. 376 p. (in Russ.).
  16. Nedorezov L.V., Utiupin Iu.V. Ekologiia. Seriia analiticheskikh obzorov mirovoi literatury (Ecology. A series of analytical reviews of world literature). 2011;95:1–234 (in Russ.).
  17. Nikolsky G.V. Ekologiia ryb (The ecology of fishes). Moscow; 1974. 357 p. (in Russ.).
  18. Pimlott D. Wolf predation and ungulate populations. Am. Zool. 1967;7:267–278.
  19. Tanner J. The stability and intrinsic growth rates of prey and predator populations. Ecology. 1975;56:855–867. doi: 10.2307/1936296
  20. Messier F. Ungulate population models with predation: a case study with the North American moose. Ecology. 1994;75(2):478–488. doi: 10.2307/1939551
  21. Kucherenko S.P. Zveri u sebia doma (Beasts at home). Khabarovsk: Publishing House. 1979. 432 p. (in Russ.).
  22. Kang Y., Armbruster D. Noise and seasonal effects on the dynamics of plant-herbivore models with monotonic plant growth functions. International Journal of Biomathematics. 2011;4(3):255–274. doi: 10.1142/S1793524511001234
  23. Kang Y., Armbruster D., Kuang Y. Dynamics of a plant-herbivore model. Journal of Biological Dynamics. 2008;2. Issue 2:89–101. doi: 10.1080/17513750801956313
  24. Weide V., Varriale M.C., Hilker F.M. Hydra effect and paradox of enrichment in discrete-time predator-prey models. Mathematical Biosciences. 2018. doi: 10.1016/j.mbs.2018.12.010
  25. Gourley S.A., Kuang Y. A stage structured predator-prey model and its dependence on maturation delay and death rate. Journal of Mathematical Biology. 2004;49(2):188–200.
  26. Abrams P.A., Quince C. The impact of mortality on predator population size and stability in systems with stage-structured prey. Theoretical Population Biology. 2005;68(4):253–266. doi: 10.1016/j.tpb.2005.05.004
  27. Chakraborty K., Jana S., Kar T.K. Global dynamics and bifurcation in a stage structured prey-predator fishery model with harvesting. Applied Mathematics and Computation. 2012;218(18):9271–9290. doi: 10.1016/j.amc.2012.03.005
  28. Bhattacharyya J., Pal S. Stage-Structured Cannibalism in a Ratio-Dependent System with Constant Prey Refuge and Harvesting of Matured Predator. Differential Equations and Dynamical Systems. 2016;24(3):345–366. doi: 10.1007/s12591-016-0299-5
  29. Shapiro A.P., Luppov S.P. Recurrent equations in the theories of population biology. Moscow: Nauka; 1983. 132 p. (in Russ.).
  30. Wikan A. From chaos to chaos. An analysis of a discrete age-structured prey-predator model. Journal of Mathematical Biology. 2001;43(6):471–500. doi: 10.1007/s002850100101
  31. Wikan A. An Analysis of Discrete Stage-Structured Prey and Prey-Predator Population Models. Discrete Dynamics in Nature and Society. 2017;2017. Article ID 9475854. doi: 10.1155/2017/9475854
  32. Tang S., Chen L. A discrete predator-prey system with age-structure for predator and natural barriers for prey. Mathematical Modelling and Numerical Analysis. 2001;35(4):675–690. doi: 10.1051/m2an:2001102
  33. Neverova G.P., Zhdanova O.L., Frisman E.Ya. Modeling the dynamics of predator-prey community with age structures. Mathematical Biology and Bioinformatics. 2019;14(1):77–93 (in Russ.). doi: 10.17537/2019.14.77
  34. Xiao Y., Cheng D., Tang S. Dynamic complexities in predator–prey ecosystem models with age-structure for predator. Chaos, Solitons and Fractals. 2002;14:1403–1411. doi: 10.1016/S0960-0779(02)00061-9
  35. Agarwal M., Devi S. Persistence in a ratio-dependent predator-prey-resource model with stage structure for prey. International Journal of Biomathematics. 2010;3(3):313–336. doi: 10.1142/S179352451000101X
  36. Frisman E.Y., Kulakov M.P., Revutskaya O.L., Zhdanova O.L., Neverova G.P. The key approaches and review of current researches on dynamics of structured and interacting populations. Computer Research and Modeling. 2019;11(1):119–151 (in Russ.). doi: 10.20537/2076-7633-2019-11-1-119-151
  37. Frisman E.Y., Skaletskaia E.I. Strange attractors in elementary models for dynamics of the quantity of biological populations. Review of Applied and Industrial Mathematics. 1994;1(6):988.(in Russ.).
  38. Dajo R. Osnovy ekologii (Basics of Ecology). Moscow: Progress; 1975. 416 p.(in Russ.).
  39. Inchausti P., Ginzburg L.R. Small mammals cycles in northern Europe: patterns and evidence for the maternal effect hypothesis. Journal of Animal Ecology. 1998;67:180–194. doi: 10.1046/j.1365-2656.1998.00189.x
  40. Ferriere R., Gatto M. Chaotic population dynamics can result from natural selection. Proceedings: Biological Sciences. 1993;251(1330):33–38.
  41. Frisman E.Ya., Neverova G.P., Revutskaya O.L., Kulakov M.P. Dynamic modes of two-age population model. Izvestiya VUZ. 2010;18(2):113–130 (in Russ.). doi: 10.18500/0869-6632-2010-18-2-113-130
  42. Frisman E.Ya., Neverova G.P., Revutskaya O.L. Complex Dynamics of the Population with a Simple Age Structure. Ecological Modelling. 2011;222:1943–1950. doi: 10.1016/j.ecolmodel.2016.09.005
  43. Neverova G. P., Frisman E. Ya. Comparative analysis of different types of density regulation of birth rate affecting population changes of structured populations. Information Science and Control Systems. 2015;1(43):41–53.(in Russ.).
  44. Revutskaya O.L., Neverova G.P., Kulakov M.P., Frisman E.Ya. Model of age-structured population dynamics: stability, multistability, and chaos. Nelin. Dinam. 2016;12(4):591–603 (in Russ.). doi: 10.20537/nd1604004
  45. Kon R. Multiple attractors in host-parasitoid interactions: Coexistence and extinction. Mathematical Biosciences. 2006;201(1–2):172–183. doi: 10.1016/j.mbs.2005.12.010
  46. Xiao Y., Tang S. The effect of initial density and parasitoid intergenerational survival rate on classical biological control. Chaos, Solitons and Fractals. 2008;37:1048–1058. doi: 10.1016/j.chaos.2006.10.002
  47. Huang J., Liu S., Ruan S., Xiao D. Bifurcations in a discrete predator–prey model with nonmonotonic functional response. J. Math. Anal. Appl. 2018;464:201–230. doi: 10.1016/j.jmaa.2018.03.074
  48. Pridnya M.V. State of the populations of European and American chestnut related to cryphonecrosis and ways of their sanitation. Electronic Journal “Issledovano v Rossii” “Researched in Russia”. 2003(32):330–339 (in Russ.).
  49. Popov A.P., Tsvetkov I.L., Belov A.A., Konichev A.S., Ivanushkina N.E., Kochkina G.A., Ozerskaya S.M. Molecular genetic identification of the phytopathogenic fungus Cryphonectria parasitica. Microbiology (Mikrobiologiya). 2010;79(2):223-228. doi: 10.1134/S0026261710020141
  50. Riznichenko G.Yu. Lektsii po matematicheskim modeliam v biologii (Lectures on mathematical models in biology). Moscow-Izhevsk: SRC "Regular and chaotic dynamics"; 2011. 560 p. (in Russ.).
  51. Beglyarov G.A., Smirnova A.A., Batalova T.S., Markelov G.A., Petrova T.M. Khimicheskaia i biologicheskaia zashchita rastenii (Chemical and biological plant protection). Ed. G.A. Beglyarov. Moscow: Kolos; 1983. 351 p. (in Russ.).
  52. Wang B., Ferro D.N., Hosmer D.W. Effectiveness of Trichogramma ostriniae and T. nubilale for controlling the European corn borer, Ostrinia nubilalis in sweet corn. Entomología Experimentalis et Applicata. 1999;91:297–303. doi: 10.1046/j.1570-7458.1999.00496.x
Table of Contents Original Article
Math. Biol. Bioinf.
2019;14(1):257-278
doi: 10.17537/2019.14.257
published in Russian

Abstract (rus.)
Abstract (eng.)
Full text (rus., pdf)
References

 

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