Geodesic
Scenarios of critical outbreak of invasive species in new modification of Gompertz equation
Vladikavkazskij matematičeskij žurnal, Tome 21 (2019) no. 1, pp. 51-61 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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. 
@article{VMJ_2019_21_1_a4,
     author = {A. Yu. Perevarukha},
     title = {Scenarios of critical outbreak of invasive species in new modification of {Gompertz} equation},
     journal = {Vladikavkazskij matemati\v{c}eskij \v{z}urnal},
     pages = {51--61},
     year = {2019},
     volume = {21},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMJ_2019_21_1_a4/}
}
TY  - JOUR
AU  - A. Yu. Perevarukha
TI  - Scenarios of critical outbreak of invasive species in new modification of Gompertz equation
JO  - Vladikavkazskij matematičeskij žurnal
PY  - 2019
SP  - 51
EP  - 61
VL  - 21
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/VMJ_2019_21_1_a4/
LA  - ru
ID  - VMJ_2019_21_1_a4
ER  - 
%0 Journal Article
%A A. Yu. Perevarukha
%T Scenarios of critical outbreak of invasive species in new modification of Gompertz equation
%J Vladikavkazskij matematičeskij žurnal
%D 2019
%P 51-61
%V 21
%N 1
%U http://geodesic.mathdoc.fr/item/VMJ_2019_21_1_a4/
%G ru
%F VMJ_2019_21_1_a4
A. Yu. Perevarukha. Scenarios of critical outbreak of invasive species in new modification of Gompertz equation. Vladikavkazskij matematičeskij žurnal, Tome 21 (2019) no. 1, pp. 51-61. http://geodesic.mathdoc.fr/item/VMJ_2019_21_1_a4/

[1] Perevarukha A. Yu., “Scenario of Involuntary Destruction of a Population in a Modified Hutchinson Equation”, Vladikavkaz Math. J., 19:1 (2017), 58–69 (in Russian) | DOI | MR

[2] Hutchinson G., An Introduction to Population Ecology, Yale Univ. Press, New Haven, 1978, 234 pp. | MR | Zbl

[3] Kolesov A. Yu., Mishchenko E. F., Rozov N. Kh., “A modification of Hutchinson's equation”, Comp. Math. and Math. Phys., 50:12 (2010), 1990–2002 | DOI | MR | Zbl

[4] Glyzin S. D., “Mathematical Model of Nicholson's Experiment”, Modeling and Analysis of Information Systems, 24:3 (2017), 365–386 (in Russian) | DOI | MR

[5] Odum H. T., Systems Ecology, Wiley, N. Y., 1983, 644 pp.

[6] Ruan S., “Delay differential equations in single species dynamics”, Delay Differential Equations and Appl., Springer, Berlin, 2006, 477–517 | DOI | MR

[7] Gopalsamy K., Kulenovic M., Ladas G., “Time lags in a «food-limited» population model”, Applicable Analysis, 31:3 (1988), 225–237 | DOI | MR | Zbl

[8] Glyzin S. D., “A Registration of Age Groups for the Hutchinson's Equation”, Modeling and Analysis of Information Systems, 14:3 (2007), 29–42 (in Russian) | MR

[9] Glyzin S. D., Kolesov A. Yu., Rozov N. Kh., “Extremal dynamics of the generalized Hutchinson equation”, Comp. Math. and Math. Phys., 49:1 (2009), 71–83 | DOI | MR | Zbl

[10] Laird A. K., “Dynamics of tumor growth”, British J. of Cancer., 18:3 (1964), 490–502 | DOI

[11] Piotrowska J., Forysz U., “The nature of Hopf bifurcation for the Gompertz model with delays”, Mathematical and Computer Modelling, 54:9–10 (2011), 2183–2198 | DOI | MR | Zbl

[12] Gause G. F., The Struggle for Existence, Williams Wilkins, Baltimore, 1934, 163 pp.

[13] Ludwig D., Jones D., Holling S., “Qualitative Analysis of Insect Outbreak Systems: The Spruce Budworm and Forest”, The Journal of Animal Ecology, 47:1 (1978), 315–332 | DOI

[14] Bazykin A. D., Nonlinear Dynamics of Interacting Populations, WSP, London, 1998, 198 pp. | MR

[15] Hutchings J. A., “Renaissance of a caveat: Allee effects in marine fish”, ICES J. of Marine Science, 71:8 (2014), 2152–2157 | DOI

[16] https://www.upf.edu/web/virology-unit/virus-host