Scenarios of Critical Outbreak of Invasive Species in New Modification of Gompertz EquationScenarios of Critical Outbreak of Invasive Species in New Modification of Gompertz Equation

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system’s guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	Dear authors! Submission of all materials is carried out only electronically through Online Submission System in personal account. DOI: 10.23671/VNC.2019.1.27734 Scenarios of Critical Outbreak of Invasive Species in New Modification of Gompertz EquationScenarios of Critical Outbreak of Invasive Species in New Modification of Gompertz Equation Perevarukha, A. Yu. Vladikavkaz Mathematical Journal 2019. Vol. 21. Issue 1. Abstract: The paper discusses the problem of modeling the variants of the development of situations of extreme type in the population process that can arise due to the propagation of alien species. For mathematical formalization of phenomena, equations with a delay argument are used. In the above-mentioned environmental context, it is interesting to consider not the occurrence of cycles or the properties of stable oscillation modes in the solution of such equations. We urgently need to search for specific transitional scenarios of population dynamics. A series of modifications based on the Gompertz equation is proposed successively, as proved to be more suitable for improvement than the Hutchinson or Nicholson models. In models involving the function of the resistance of the biotic environment, scenarios of the death of the population after the outbreak were obtained. An alternative variant of the numerical scenario is the formation of a stable small group with the passage of the permissible barrier number of adults in the population. The resulting computational scenarios have a practical interpretation in the analysis of the possible development of events after the introduction of dangerous new species into conservative ecosystems. Improved by the original complement and model for an explicitly critical low \(L\)-quantity, flexible correction of the properties of the population dynamics under the action of the Allee effect, than the function \(\dot N=F(N^2)\times(N(t)-L)\) from the well-known Bazykin model. The resulting model scenarios are similar for a group of invasive and dangerous infectious processes, which undermine our idea that cybernetic regulatory mechanisms take precedence over ecological species specificity of alien populations. Keywords: delayed equations, extreme states of populations, transitional regimes, cycles, modeling of~invasive processes, alien species, cybernetics of biological warfare. Language: Russian Download the full text For citation: Perevarukha, A. Yu. Scenarios of Critical Outbreak of Invasive Species in New Modification of Gompertz Equation, Vladikavkaz Math. J., 2019, vol. 21, no. 1, pp. 51-61 (in Russian). DOI 10.23671/VNC.2019.1.27734 + References 1. Perevarukha, A. Yu. Scenario of Involuntary Destruction of a Population in a Modified Hutchinson Equation, Vladikavkaz Math. J., 2017, vol. 19, no. 1, pp. 58-69 (in Russian). DOI: 10.23671/VNC.2018.4.9168. 2. Hutchinson, G. An Introduction to Population Ecology, New Haven, Yale Univ. Press, 1978, 234 p. 3. Kolesov, A. Yu., Mishchenko, E. F. and Rozov, N. Kh. A Modification of Hutchinson’s Equation, Computational Mathematics and Mathematical Physics, 2010, vol. 50, no. 12, pp. 1990-2002. DOI: 10.1134/S0965542510120031. 4. Glyzin, S. D. Mathematical Model of Nicholson’s Experiment, Modeling and Analysis of Information Systems, 2017, vol. 24, no. 3, pp. 365-386 (in Russian). DOI: 10.18255/1818-1015-2017-3-365-386. 5. Odum, H. T. Systems Ecology, N.Y., Wiley, 1983, 644 p. 6. Ruan, S. Delay Differential Equations in Single Species Dynamics, Delay Differential Equations and Appl., Berlin, Springer, 2006, pp. 477-517. DOI: 10.1007/1-4020-3647-7_11. 7. Gopalsamy, K., Kulenovic, M. and Ladas, G. Time Lags in a "Food-Limited" Population Model, Applicable Analysis, 1988, vol. 31, no. 3, pp. 225-237. DOI: 10.1080/00036818808839826. 8. Glyzin, S. D. A Registration of Age Groups for the Hutchinson's Equation, Modeling and Analysis of Information Systems, 2007, vol. 14, no. 3, pp. 29–42 (in Russian). 9. Glyzin, S. D., Kolesov, A. Yu. and Rozov, N. Kh. Extremal Dynamics of the Generalized Hutchinson Equation, Computational Mathematics and Mathematical Physics, 2009, vol. 49, no. 1, pp. 71-83. DOI: 10.1134/S0965542509010059. 10. Laird, A. K. Dynamics of Tumor Growth, British Journal of Cancer, 1964, vol. 18, no. 3, pp. 490-502. DOI: 10.1038/bjc.1964.55. 11. Piotrowska, J. and Forysz, U. The Nature of Hopf Bifurcation for the Gompertz Model with Delays, Mathematical and Computer Modelling, 2011, vol. 54, no. 9-10, pp. 2183-2198. DOI: 10.1016/j.mcm.2011.05.027. 12. Gause, G. F. The Struggle for Existence, Baltimore, Williams & Wilkins, 1934, 163 p. 13. Ludwig, D., Jones, D. and Holling, S. Qualitative Analysis of Insect Outbreak Systems: The Spruce Budworm and Forest, The Journal of Animal Ecology, 1978, vol. 47, no. 1, pp. 315-332. 14. Bazykin, A. D. Nonlinear Dynamics of Interacting Populations, London, WSP, 1998, 198 p. 15. Hutchings, J. A. Renaissance of a Caveat: Allee Effects in Marine Fish, ICES Journal of Marine Science, 2014, vol. 71, no. 8, pp. 2152-2157. DOI: 10.1093/icesjms/fst179. 16. URL: https://www.upf.edu/web/virology-unit/virus-host. ← Contents of issue


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