Geodesic
The stochastic sensitivity function method in analysis of the piecewise-smooth model of population dynamics
Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, Tome 53 (2019), pp. 36-47.

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This work is devoted to the application of the stochastic sensitivity function method to attractors of a piecewise-smooth one-dimensional map describing the dynamics of the population size. The first stage of the study is a parametric analysis of possible modes of the deterministic model: the definition of zones of existence of stable equilibria and chaotic attractors. The theory of critical points is used to determine the parametric boundaries of a chaotic attractor. In the case where the system is influenced by a random effect, based on the technique of the stochastic sensitivity function, a description of the spread of random states around the equilibrium and chaotic attractor is carried out. A comparative analysis of the influence of parametric and additive noise on the attractors of the system is conducted. Using the technique of confidence intervals, probabilistic mechanisms of extinction of a population under the influence of random disturbances are studied. Changes in the parametric boundaries of the existence of a population under the impact of a random perturbation are analyzed.
Keywords: piecewise-smooth map, population dynamics, stochastic sensitivity. 
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     title = {The stochastic sensitivity function method in analysis of the piecewise-smooth model of population dynamics},
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A. V. Belyaev; T. V. Ryazanova. The stochastic sensitivity function method in analysis of the piecewise-smooth model of population dynamics. Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, Tome 53 (2019), pp. 36-47. http://geodesic.mathdoc.fr/item/IIMI_2019_53_a3/

[1] Turchin P., Complex population dynamics: a theoretical/empirical synthesis, Princeton University Press, 2003 | DOI | MR | Zbl

[2] Lande R., Engen S., Saether B.-E., Stochastic population dynamics in ecology and conservation, Oxford University Press, 2003 | DOI

[3] Chesson P., “Stochastic population models”, Ecological Heterogeneity, Ecological Studies (Analysis and Synthesis), 86, Springer, New York, 1991, 123–143 | DOI

[4] Gadrich T., Katriel G., “A mechanistic stochastic Ricker model: analytical and numerical investigations”, International Journal of Bifurcation and Chaos, 26:4 (2016), 1650067 | DOI | MR | Zbl

[5] May R. M., “Simple mathematical models with very complicated dynamics”, Nature, 261:5560 (1976), 459–467 | DOI | Zbl

[6] Murray J. D., “Discrete population models for a single species”, Interdisciplinary Applied Mathematics, Springer, New York, 1993, 44–78 | DOI

[7] Higgins R. J., Kent C. M., Kocic V. L., Kostrov Y., “Dynamics of a nonlinear discrete population model with jumps”, Applicable Analysis and Discrete Mathematics, 9:2 (2015), 245–270 | DOI | MR | Zbl

[8] Avrutin V., Sushko I., “A gallery of bifurcation scenarios in piecewise smooth 1d map”, Global Analysis of Dynamic Models in Economics and Finance, Springer, Berlin-Heidelberg, 2012, 369–395 | DOI | MR

[9] Sushko I., Gardini L., Avrutin V., “Nonsmooth one-dimensional maps: some basic concepts and definitions”, Journal of Difference Equations and Applications, 22:12 (2016), 1816–1870 | DOI | MR | Zbl

[10] Avrutin V., Gardini L., Schanz M., Sushko I., “Bifurcations of chaotic attractors in one-dimensional piecewise smooth maps”, International Journal of Bifurcation and Chaos, 24:8 (2014), 1440012 | DOI | MR | Zbl

[11] Laurea M. B., Champneys A. R., Budd C. J., Kowalczyk P., Piecewise-smooth dynamical systems, Springer, London, 2008 | DOI | MR

[12] Bashkirtseva I., “Crises, noise, and tipping in the Hassell population model”, Chaos: An Interdisciplinary Journal of Nonlinear Science, 28:3 (2018), 033603 | DOI | MR | Zbl

[13] Bashkirtseva I., “Preventing noise-induced extinction in discrete population models”, Discrete Dynamics in Nature and Society, 2017 (2017), 1–10 | DOI | MR

[14] Bashkirtseva I., Ryashko L., “Stochastic sensitivity analysis of noise-induced order-chaos transitions in discrete-time systems with tangent and crisis bifurcations”, Physica A: Statistical Mechanics and its Applications, 467 (2017), 573–584 | DOI | MR | Zbl

[15] Bashkirtseva I., Nasyrova V., Ryashko L., “Analysis of noise effects in a map-based neuron model with Canard-type quasiperiodic oscillations”, Communications in Nonlinear Science and Numerical Simulation, 63 (2018), 261–270 | DOI | MR

[16] Bashkirtseva I., Nasyrova V., Ryashko L., “Noise-induced bursting and chaos in the two-dimensional Rulkov model”, Chaos, Solitons and Fractals, 110 (2018), 76–81 | DOI | MR | Zbl

[17] Bashkirtseva I. A., Ryashko L. B., “Stochastic sensitivity of regular and multi-band chaotic attractors in discrete systems with parametric noise”, Physics Letters A, 381:37 (2017), 3203–3210 | DOI | MR | Zbl

[18] Bashkirtseva I., Ryashko L., “Approximating chaotic attractors by period-three cycles in discrete stochastic systems”, International Journal of Bifurcation and Chaos, 25:10 (2015), 1550138 | DOI | MR | Zbl