Russian version English version
Volume 15   Issue 1   Year 2020
Neverova G.P.1, Zhdanova O.L.1, Frisman E.Ya.2

Dynamics of Predator-Prey Community with Age Structures and Its Changing Due To Harvesting

Mathematical Biology & Bioinformatics. 2020;15(1):73-92.

doi: 10.17537/2020.15.73.

References

  1. Lotka A.J. Analytical theory of biological populations. Springer Science & Business Media; 1998. 220 p. doi: 10.1007/978-1-4757-9176-1
  2. Volterra V. Leçons sur la théorie mathématique de la lutte pour la vie (Lessons on the mathematical theory of the struggle for life). Paris: Gauthier-Villars; 1931 (in French).
  3. Tyutyunov Yu.V., Titova L.I. From Lotka–Volterra to Arditi–Ginzburg: 90 years of evolving trophic functions. Zhurnal Obshchei Biologii (Journal of General Biology). 2018;79(6):428–448 (in Russ.).
  4. Holling C.S. The functional response of predators to prey density and its role in mimicry and population regulation. Mem. Ent. Soc. Can. 1965;45:1–60. doi: 10.4039/entm9745fv
  5. Caswell H. Matrix Population Models: Construction, Analysis, and Interpretation. Sunderland, MA: Sinauer Associates. Inc.; 2001.
  6. Bazykin A.D. Matematicheskaia biofizika vzaimodeistvuiushchikh populiatsii (Mathematical biophysics of interacting populations). Moscow; 1985. 181 p. (in Russ.).
  7. Sabo J.L. Stochasticity, predator–prey dynamics, and trigger harvest of nonnative predators. Ecology. 2005;86(9):2329–2343. doi: 10.1890/04-1152
  8. Pacht E.V., Abakumov A.I. Uncertainty at modelling of a lake’s ecosystem. Mathematical Biology and Bioinformatics. 2011;6(1):102–114. doi: 10.17537/2011.6.102
  9. Bashkirtseva I.A., Boyarshinova P.V., Ryazanova T.V., Ryashko L.B. Analysis of noise-induced destruction of coexistence regimes in "prey-predator" population model. Computer Research and Modeling. 2016;8(4):647–660 (in Russ.). doi: 10.20537/2076-7633-2016-8-4-647-660
  10. Abramova E.P., Ryazanova T.V. Dynamic regimes of the stochastic “prey – predatory” model with competition and saturation. Computer Research and Modeling. 2019;11(3):515–531. doi: 10.20537/2076-7633-2019-11-3-515-531
  11. Aponin Yu.M., Aponina E.A. Mathematical model of predator – prey system with lower critical prey density. Computer Research and Modeling. 2009;1(1):51–56. doi: 10.20537/2076-7633-2009-1-1-51-56
  12. Xu C., Wu Y., Lu L. Permanence and global attractivity in a discrete Lotka-Volterra predator-prey model with delays. Advances in Difference Equations. 2014;1:1–5.
  13. Frisman Y.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. doi: 10.20537/2076-7633-2019-11-1-119-151
  14. Saito Y., Takeuchi Y. A time-delay model for prey-predator growth with stage structure. Canadian Applied Mathematics Quarterly. 2003;11(3):293–302.
  15. 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.
  16. Sun X.K., Huo H.F., Xiang H. Bifurcation and stability analysis in predator–prey model with a stage-structure for predator. Nonlinear Dynamics. 2009;58(3):497–513. doi: 10.1007/s11071-009-9495-y
  17. Xu R. Global dynamics of a predator–prey model with time delay and stage structure for the prey. Nonlinear Analysis: Real World Applications. 2011;12(4):2151–2162. doi: 10.1016/j.nonrwa.2010.12.029
  18. 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
  19. Kundu S., Maitra S. Dynamics of a delayed predator-prey system with stage structure and cooperation for preys. Chaos, Solitons & Fractals. 2018;114:453–460. doi: 10.1016/j.chaos.2018.07.013
  20. 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
  21. Khajanchi S., Banerjee S. Role of constant prey refuge on stage structure predator-prey model with ratio dependent functional response. Applied Mathematics and Computation. 2017;314:193–198. doi: 10.1016/j.amc.2017.07.017
  22. 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
  23. Abakumov A.I., Il’in O.I., Ivanko N.S. Game problems of harvesting in a biological community. Mathematical Game Theory and Applications. 2016;77:697–707.
  24. Abakumov A.I., Izrailsky Yu.G. The Harvesting Effect on a Fish Population. Mathematical Biology and Bioinformatics. 2016;11(2):191–204. doi: 10.17537/2016.11.191
  25. Walters P., Christensen V., Fulton B., Smith A.D., Hilborn R. Predictions from simple predator-prey theory about impacts of harvesting forage fishes. Ecological modelling. 2016;337:272–280. doi: 10.1016/j.ecolmodel.2016.07.014
  26. Abakumov A.I., Il’in O.I., Ivanko N.S. Game problems of harvesting in a biological community. Automation and Remote Control. 2016;77(4):697–707. doi: 10.1134/S0005117916040135
  27. Liu C., Zhang Q., Duan X. Dynamical behavior in a harvested differential-algebraic prey–predator model with discrete time delay and stage structure. Journal of the Franklin Institute. 2009;346(10):1038–1059. doi: 10.1016/j.jfranklin.2009.06.004
  28. Liu C., Zhang Q., Zhang X., Duan X. Dynamical behavior in a stage-structured differential-algebraic prey–predator model with discrete time delay and harvesting. Journal of Computational and Applied Mathematics. 2009;231(2):612–625. doi: 10.1016/j.cam.2009.04.011
  29. Caughley G. Analiz populiatsii pozvonochnykh. Moscow; 1979. 362 p. (Translation of: Caughley G. Analysis of Vertebrate Populations. John Wiley and Sons; 1977).
  30. Neverova G.P., Abakumov A.I., Frisman E.Ya. Dynamic Modes of Limited Structured Population under Age Specific Harvest. Mathematical Biology and Bioinformatics. 2017;12(2):327–342. doi: 10.17537/2017.12.327
  31. Neverova G.P., Abakumov A.I, Yarovenko I.P., Frisman E.Ya. Mode change in the dynamics of exploited limited population with age structure. Nonlinear Dynamics. 2018;94:827–844. doi: 10.1007/s11071-018-4396-6
  32. Revutskaya O.L., Neverova G.P., Frisman E.Ya. Influence of Harvest on the Dynamics of Populations with Age and Sex Structures. Mathematical Biology and Bioinformatics. 2018;13(1):270–289. doi: 10.17537/2018.13.270
  33. Agiza H.N., Elabbasy E.M., El-Metwally H., Elsadany A.A. Chaotic dynamics of a discrete prey–predator model with Holling type II. Nonlinear Analysis: Real World Applications. 2009;10(1):116–129. doi: 10.1016/j.nonrwa.2007.08.029
  34. Hu Z., Teng Z., Zhang L. Stability and bifurcation analysis of a discrete predator–prey model with nonmonotonic functional response. Nonlinear Analysis: Real World Applications. 2011;12(4):2356–2377. doi: 10.1016/j.nonrwa.2011.02.009
  35. Zhao J., Yan Y. Stability and bifurcation analysis of a discrete predator–prey system with modified Holling–Tanner functional response. Advances in Difference Equations. 2018. Article No. 402. doi: 10.1186/s13662-018-1819-0
  36. Mistro D.P., Rodrigues L.A.D., Petrovskii S. Spatiotemporal complexity of biological invasion in a space- and time-discrete predator–prey system with the strong Allee effect. Ecological Complexity. 2012;9:16–32. doi: 10.1016/j.ecocom.2011.11.004
  37. Huang T., Zhang H. Bifurcation, chaos and pattern formation in a space-and time-discrete predator–prey system. Chaos, Solitons & Fractals. 2016;91:92–107. doi: 10.1016/j.chaos.2016.05.009
  38. Huang T., Zhang H., Yang H., Wang N., Zhang F. Complex patterns in a space-and time-discrete predator-prey model with Beddington-DeAngelis functional response. Communications in Nonlinear Science and Numerical Simulation. 2017;43:182–199. doi: 10.1016/j.cnsns.2016.07.004
  39. Zhong J., Yu Z. Qualitative properties and bifurcations of Mistro–Rodrigues–Petrovskii model. Nonlinear Dynamics. 2018;91(4):2063–2075. doi: 10.1007/s11071-017-3932-0
  40. Reimer J.R., Brown H., Beltaos-Kerr E., de Vries G. Evidence of intraspecific prey switching: stage-structured predation of polar bears on ringed seals. Oecologia. 2018;189(1):133–148. doi: 10.1007/s00442-018-4297-x
  41. Wikan A., Kristensen Ø. Prey-Predator Interactions in Two and Three Species Population Models. Discrete Dynamics in Nature and Society. 2019;2019:1–14. doi: 10.1155/2019/9543139
  42. Wilmers C.C., Post E., Hastings A. The anatomy of predator-prey dynamics in a changing climate. Journal of Animal Ecology. 2007;76(6):1037–1044. doi: 10.1111/j.1365-2656.2007.01289.x
  43. 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
  44. Revutskaya O.L., Kulakov M.P., Frisman E.Ya. Bistability and Bifurcations in Modified Nicholson-Bailey Model with Age-Structure for Prey. Mathematical Biology and Bioinformatics. 2019;14(1):257–278. doi: 10.17537/2019.14.257
  45. Kang Y., Armbruster D., Kuang Y. Dynamics of a plant-herbivore model. Journal of Biological Dynamics. 2008;2(2):89–101. doi: 10.1080/17513750801956313
  46. 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
  47. Wikan A. An analysis of discrete stage-structured prey and prey-predator population models. Discrete Dynamics in Nature and Society. 2017;2017. doi: 10.1155/2017/9475854
  48. Basson M., Fogarty M.J. Harvesting in discrete-time predator-prey systems. Mathematical biosciences. 1997;141(1):41–74. doi: 10.1016/S0025-5564(96)00173-3
  49. Chen B., Chen J. Complex dynamic behaviors of a discrete predator–prey model with stage structure and harvesting. International Journal of Biomathematics. 2017;10(1):1750013. doi: 10.1142/S1793524517500139
  50. Zhdanova O.L., Neverova G.P., Frisman E.Ya. Modeling the dynamics of the predator-prey community based on the age structure of the interacting species. Information Science and Control Systems. 2018;4(58):34–45 (in Russ.). doi: 10.17537/icmbb18.56
  51. 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. doi: 10.17537/2019.14.77
  52. Neverova G.P., Zhdanova O.L., Bapan Ghosh, Frisman E.Ya. Dynamics of a discrete-time stage-structured predator–prey system with Holling type II response function. Nonlinear dynamics. 2019;98(1):427–446. doi: 10.1007/s11071-019-05202-3
  53. Angerbjorn A., Tannerfeldt M., Erlinge S. Predator–prey relationships: arctic foxes and lemmings. Journal of Animal Ecology. 1999;68(1):34–49. doi: 10.1046/j.1365-2656.1999.00258.x
  54. Kuznetsov A.P., Savin A.V., Sedova Y.V., Tyuryukina L.V. Bifurcation of Images. Saratov: Press Center Ltd “Nauka”, 2012. 196 p. (in Russ.).
  55. Hersteinsson P., Macdonald D.W. Diet of Arctic foxes (Alopex lagopus) in Iceland. J. Zool. 1996;240:457–474. doi: 10.1111/j.1469-7998.1996.tb05298.x
Table of Contents Original Article
Math. Biol. Bioinf.
2020;15(1):73-92
doi: 10.17537/2020.15.73
published in Russian

Abstract (rus.)
Abstract (eng.)
Full text (rus., pdf)
References Translation into English
Math. Biol. Bioinf.
2020, 15(Suppl):t35-t51
doi: 10.17537/2020.15.t35

Full text (eng., pdf)

 

  Copyright IMPB RAS © 2005-2024