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ISSN 2542-1905 (Online)

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Perevaryukha A. Y. Modeling of adaptive counteraction of the induced biotic environment during the invasive process. Izvestiya VUZ. Applied Nonlinear Dynamics, 2022, vol. 30, iss. 4, pp. 436-455. DOI: 10.18500/0869-6632-2022-30-4-436-455, EDN: CRMSGN

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530.182, 004.942, 303.732.4, 577.35

Modeling of adaptive counteraction of the induced biotic environment during the invasive process

Perevaryukha A. Yu., St. Petersburg Institute for Informatics and Automation of RAS

Purpose is to develop a mathematical model for the analysis of a variant in the development of a population process with a non-trivially regulated confrontation between an invading species and a biotic environment. Relevance. The situation we are studying arises in invasive processes, but is a previously unexplored special variant of their development. The task of modeling is to describe the transition to a deep ν-shaped crisis after intensive growth. The model is based on examples of the adaptive dynamics of a bacterial colony and the suppression of mollusk populations, carriers of dangerous parasitic diseases, after targeted anti-epidemic introduction of their antagonists. Methods. In our work equations with a retarded argument in the range of parameter values that have a biological interpretation were studied. The model uses a logarithmic form of species regulation, taking into account the theoretically permissible capacity of the medium. In the equation we included the function of external influence with flexible threshold regulation relative to the current and previous population size. Results. It is shown that the proposed form of impact regulation leads to the formation of a stable adapted population after the crisis, which does not have a destructive impact on the habitat. With an increase in the reproductive potential of an invasive species, a deep crisis becomes critically dangerous. The form of the crisis passage depends on the reproductive potential, on the size of the initial group of individuals, and also on the time of activation of the adaptive counteraction from the environment. It is established that at a sufficient level of resistance, a non destructive equilibrium is established. Conclusion. The actual scenario of sudden depression of an actively spreading population with a large reproductive r-parameter, which is caused by the delayed activity of its natural antagonists, has been studied. The threshold form of biotic regulation is characteristic of insects, the abundance of which is regulated by competing species of parasitic hymenoptera. The variant of rapid phase change considered by us in the model is relevant as a description of one of the forms of developing the body’s immune response to the development of an acute infection with a significant delay. If the immune response is prematurely inhibited by the body itself, then the chronic focus of the disease persists. Examples of the dynamics of two real biological processes in experiments with biological suppression methods are given, which correspond to the invasion scenario obtained in the new model.

  1. Kowarik I. Time lags in biological invasions with regard to the success and failure of alien species. In: Pysek P, Prach K, Rej’anek M, Wade M, editors. Plant Invasions - General Aspects and Special ˇ Problems. Amsterdam: SPB Academic Publishing; 1995. P. 15–38.
  2. Arim M, Abades RS, Neill PE, Marquet PA. Spread dynamics of invasive species. Proceedings of the National Academy of Sciences. 2006;103(2):374–378. DOI: 10.1073/pnas.0504272102.
  3. Sakai AK, Allendorf FW, Holt JS, Lodge DM, Molofsky J, With KA, Baughman S, Cabin RJ, Cohen JE, Ellstrand NC, McCauley DE, O’Neil P, Parker IM, Thompson JN, Weller SG. The population biology of invasive species. Annu. Rev. Ecol. Syst. 2001;32:305–332. DOI: 10.1146/annurev.ecolsys.32.081501.114037.
  4. Bonser SP. High reproductive efficiency as an adaptive strategy in competitive environments. Functional Ecology. 2013;27(4):876–885. DOI: 10.1111/1365-2435.12064.
  5. Gushing JM. Volterra integrodifferential equations in population dynamics. In: Iannelli M, editor. Mathematics of Biology. Vol. 80 of C.I.M.E. Summer Schools. Berlin, Heidelberg: Springer; 2010. P. 81–148. DOI: 10.1007/978-3-642-11069-6_2.
  6. Hutchinson GE. Circular causal systems in ecology. Ann. N. Y. Acad. Sci. 1948;50(4):221–246. DOI: 10.1111/j.1749-6632.1948.tb39854.x.
  7. Utida S. Population fluctuation, an experimental and theoretical approach. Cold Spring Harbor Symposia on Quantitative Biology. 1957;22:139–151. DOI: 10.1101/SQB.1957.022.01.016.
  8. Wright EM. A non-linear difference-differential equation. Journal fur die reine und angewandte Mathematik. 1955;1955(194):66–87. DOI: 10.1515/crll.1955.194.66.
  9. May RM, Conway GR, Hassell MP, Southwood TRE. Time delays, density-dependence and single-species oscillations. J. Anim. Ecol. 1974;43(3):747–770. DOI: 10.2307/3535.
  10. Verhulst PF. Deuxieme memoire sur la loi d’accroissement de la population. Memoires del’Academie Royale des Sciences, des Lettres et des Beaux-Arts de Belgique. 1847;20:1–32 (in French).
  11. Peleg M, Corradini MG, Normand MD. The logistic (Verhulst) model for sigmoid microbial growth curves revisited. Food Research International. 2007;40(7):808–818. DOI: 10.1016/j.foodres.2007.01.012.
  12. Sales LP, Hayward MW, Loyola R. What do you mean by “niche”? Modern ecological theories are not coherent on rhetoric about the niche concept. Acta Oecologica. 2021;110:103701. DOI: 10.1016/j.actao.2020.103701.
  13. Severtsov AS. Relationship between fundamental and realized ecological niches. Biology Bulletin Reviews. 2013;3(3):187–195. DOI: 10.1134/S2079086413030080.
  14. Glyzin DS, Kaschenko SA, Polstyanov AS. Spatially inhomogeneous periodic solutions in the Hutchinson equation with distributed saturation. Modeling and Analysis of Information Systems. 2011;18(1):37–45 (in Russian).
  15. Yumagulov MG, Yakshibaeva DA. Operator method for studying small self-oscillations in systems with aftereffect. Bulletin of Samara State University. Natural Science Series. 2013;(9/2(110)):37–42 (in Russian).
  16. Kolesov AY, Kolesov YS. Relaxation oscillations in mathematical models of ecology. Proc. Steklov Inst. Math. 1995;199:1–126.
  17. Pertsev NV, Loginov KK, Topchii VA. Analysis of an epidemic mathematical model based on delay differential equations. Journal of Applied and Industrial Mathematics. 2020;14(2):396–406. DOI: 10.1134/S1990478920020167.
  18. Daneev AV, Lakeev AV, Rusanov VA, Plesnev PA. On differential-non-autonomous representation integrative activity of neuropopulation bilinear second-order model with a delay. Izvestia of Samara Scientific Center of the Russian Academy of Sciences. 2021;23(2):115–126 (in Russian). DOI: 10.37313/1990-5378-2021-23-2-115-126.
  19. Smith JM. Mathematical Ideas in Biology. Cambridge: Cambridge University Press; 1968. 168 p. DOI: 10.1017/CBO9780511565144.
  20. Finley C, Oreskes N. Maximum sustained yield: a policy disguised as science. ICES Journal of Marine Science. 2013;70(2):245–250. DOI: 10.1093/icesjms/fss192.
  21. Kashchenko IS, Kashchenko SA. Dynamics of equation with two delays modelling the number of population. Izvestiya VUZ. Applied Nonlinear Dynamics. 2019;27(2):21–38 (in Russian). DOI: 10.18500/0869-6632-2019-27-2-21-38.
  22. Gopalsamy K, Liu P. Persistence and global stability in a population model. Journal of Mathematical Analysis and Applications. 1998;224(1):59–80. DOI: 10.1006/jmaa.1998.5984.
  23. Liu Y, Wei J. Bifurcation analysis in delayed Nicholson blowflies equation with delayed harvest. Nonlinear Dynamics. 2021;105(2):1805–1819. DOI: 10.1007/s11071-021-06651-5.
  24. Hale JK, Waltman P. Persistence in infinite-dimensional systems. SIAM Journal on Mathematical Analysis. 1989;20(2):388–395. DOI: 10.1137/0520025.
  25. Kolesov AY, Mishchenko EF, Rozov NK. A modification of Hutchinson’s equation. Computational Mathematics and Mathematical Physics. 2010;50(12):1990–2002. DOI: 10.1134/S0965542510120031.
  26. Erdakov LN, Savichev VV, Chernyshova ON. Quantitative assessment of population cyclicity in animals. Biology Bulletin Reviews. 1990;51(5):661–668 (in Russian).
  27. Erdakov LN, Moroldoev IV. Variability of long-term cyclicity in the population dynamics of the northern red-backed vole (Myodes Rutilus (Pallas, 1779). Principles of the Ecology. 2017;(4): 26–36 (in Russian). DOI: 10.15393/
  28. Whitfield J. Why cycling lemmings crash. Nature. 2000. DOI: 10.1038/news000601-10.
  29. Perevaryukha AY. Scenarios of the passage of the «population bottleneck» by an invasive species in the new model of population dynamics. Izvestiya VUZ. Applied Nonlinear Dynamics. 2018;26(5):63–80 (in Russian). DOI: 10.18500/0869-6632-2018-26-5-63-80.
  30. Perevaryukha AY. Transition from relaxation oscillations to pseudoperiodic trajectory in the new model of population dynamics. Izvestiya VUZ. Applied Nonlinear Dynamics. 2017;25(2):51–62 (in Russian). DOI: 10.18500/0869-6632-2017-25-2-51-62. 
  31. Bazykin AD, Aponina EA. Model of an ecosystem of three trophic levels taking into account the existence of a lower critical density of the producer population. Problems of Ecological Monitoring and Modeling of Ecosystems. 1981;4:186–203 (in Russian).
  32. Rozenberg GS. Warder Clyde Allee and the principle of special aggregation. Samarskaya Luka: Problems of Regional and Global Ecology. 2020;29(3):77–88 (in Russian). DOI: 10.24411/2073- 1035-2020-10335.
  33. Gause GF. The Struggle for Existence. Baltimore: Williams and Wilkins; 1934. 163 p.
  34. Owren B, Zennaro M. Order barriers for continuous explicit Runge-Kutta methods. Mathematics of Computation. 1991;56(194):645–661. DOI: 10.2307/2008399.
  35. Rosenberg GS. On the history of the model of logistic growth. Bulletin Samarskaya Luka. 2006;(18):188–193 (in Russian).
  36. Buck JC, Hechinger RF, Wood AC, Stewart TE, Kuris AM, Lafferty KD. Host density increases parasite recruitment but decreases host risk in a snail–trematode system. Ecology. 2017;98(8): 2029–2038. DOI: 10.1002/ecy.1905.
  37. Colledge S, Conolly J, Crema E, Shennan S. Neolithic population crash in northwest Europe associated with agricultural crisis. Quaternary Research. 2019;92(3):686–707. DOI: 10.1017/qua.2019.42.
  38. Maina GM, Kinuthia JM, Mutuku MW, Mwangi IN, Agola EL, Kutima HL, Mkoji GM. Regulatory influence of Procambarusclarkii, Girad (Decapoda: Cambaridae) on schistosome-transmitting snails in lotic habitats within the River Athi Basin, Kenya. International Journal of Marine Biology and Research. 2017;2(1):1–7. DOI: 10.15226/24754706/2/1/00113.
  39. Lenski RE. Dynamics of interactions between bacteria and virulent bacteriophage. In: Marshall KC, editor. Advances in Microbial Ecology. Vol. 10 of Advances in Microbial Ecology. Boston, MA: Springer; 1988. P. 1–44. DOI: 10.1007/978-1-4684-5409-3_1.
  40. Deer Habitat Carrying Capacity [Internet]. Forest and Wildlife Research Center Report. Mississippi State, MS: Mississippi State University; 2013. Available from: deer-habitat-carrying-capacity.php.
  41. Dubrovskaya VA, Perevaryukha AY, Trofimova IV. Model of dynamics of structured subpopulations of sturgeon fish in the Caspian Sea takes into account deviations in the rate of development of immature fish. Journal of the Belarusian State University. Biology. 2017;(3):76–86 (in Russian).
  42. Kuznetsov VA, Knott GD. Modeling tumor regrowth and immunotherapy. Mathematical and Computer Modelling. 2001;33(12–13):1275–1287. DOI: 10.1016/S0895-7177(00)00314-9.
  43. Mikhailov VV, Perevaryukha AY, Reshetnikov YS. Model of fish population dynamics with calculation of individual growth rate and hydrological situation scenarios. Information and Control Systems. 2018;(4):31–38. DOI: 10.31799/1684-8853-2018-4-31-38.
  44. Graham AL, Tate AT. Host Defense: Are we immune by chance? eLife. 2017;6:e32783. DOI: 10.7554/eLife.32783.
  45. Perevaryukha AY. A continuous model of three scenarios of the infection process with delayed immune response factors. Biophysics. 2021;66(2):327–348. DOI: 10.1134/S0006350921020160.
  46. Naikhin AN, Losev IV. The impact of conservative and hypervariable immunodominant epitopes in internal proteins of the influenza a virus on cytotoxic T-cell immune responses. Problems of Virology. 2015;60(1):11–16 (in Russian).
  47. Perevaryukha AY. Delay in the regulation of population dynamics — cellular automaton model. Dynamical Systems. 2017;7(2):157–165 (in Russian).
  48. Nikitina AV, Leontyev AL. Hydrophysical modeling of the caspian sea based on the model of variable density. Herald Of Computer And Information Technologies. 2018;(6(168)):12–19 (in Russian). DOI: 10.14489/vkit.2018.06.pp.012-019.
  49. Shabunin AV. SIRS-model with dynamic regulation of the population: Probabilistic cellular automata approach. Izvestiya VUZ. Applied Nonlinear Dynamics. 2019;27(2):5–20 (in Russian). DOI: 10.18500/0869-6632-2019-27-2-5-20.
  50. Perevaryukha AY. Modeling of oscillating population dynamics of aquatic organisms in the «resource–consumer» system using cellular automata. Izvestiya VUZ. Applied Nonlinear Dynamics. 2020;28(1):62–76 (in Russian). DOI: 10.18500/0869-6632-2020-28-1-62-76.