Stochastic bifurcations and tipping phenomena of insect outbreak systems driven by α-stable Lévy processes

Shenglan Yuan¹ ; Yang Li² ; Zhigang Zeng³

¹ Institut für Mathematik, Universität Augsburg, 86135 Augsburg, Germany
² School of Automation, Nanjing University of Science and Technology, Nanjing 210094, China
³ School of Artificial Intelligence and Automation, Huazhong University of Science and Technology, Wuhan 430074, China

Mathematical modelling of natural phenomena, Tome 17 (2022), article no. 34.

Résumé

In this work, we mainly characterize stochastic bifurcations and tipping phenomena of insect outbreak dynamical systems driven by α-stable Lévy processes. In one-dimensional insect outbreak model, we find the fixed points representing refuge and outbreak from the bifurcation curves, and carry out a sensitivity analysis with respect to small changes in the model parameters, the stability index and the noise intensity. We perform the numerical simulations of dynamical trajectories using Monte Carlo methods, which contribute to looking at stochastic hysteresis phenomenon. As for two-dimensional insect outbreak system, we identify the global stability properties of fixed points and express the probability density function by the stationary solution of the nonlocal Fokker-Planck equation. Through numerical modelling, the phase portrait makes it possible to detect critical transitions among the stable states. It is then worth describing stochastic bifurcation associated with tipping points induced by noise through a careful examination of the dynamical paths of the insect outbreak system with external forcing. The results give new insight into the sensitivity of ecosystems to realistic environmental changes represented by stochastic perturbations.

Détail

DOI : 10.1051/mmnp/2022037

Bibliographie

[1] L. Arnold , Random dynamical systems. Springer, Berlin (2013).

[2] P. Ashwin, S. Wieczorek, R. Vitolo, P. Cox Tipping points in open systems: Bifurcation, noise-induced and rate-independent examples in the climate system Philos. Trans. R. Soc. A 2012 1166 1184

[3] H.L. Barker, L.M. Holeski, R.L. Lindroth Genotypic variation in plant traits shapes herbivorous insect and ant communities on a foundation tree species PloS one 2018 0200954

[4] S. Boccaletti, J. Kurths, G. Osipov, D.L. Valladares, C.S. Zhou The synchronization of chaotic systems Phys. Rep 2002 1 101

[5] C. Boettiger, A. Hastings From patterns to predictions Nature 2013 157 158

[6] C. Ciemer, R. Winkelmann, J. Kurths, N. Boers Impact of an AMOC weakening on the stability of the southern Amazon rainforest Eur. Phys. J. Spec. Top 2021 1 9

[7] A. Das, G.P. Samanta A prey-Cpredator model with refuge for prey and additional food for predator in a fluctuating environment Physica A 2020 122844

[8] J. Duan , An introduction to stochastic dynamics. Cambridge University Press (2015).

[9] T. Gao, J. Duan, X. Li Fokker-Planck equations for stochastic dynamical systems with symmetric Levy motions Appl. Math. Comput 2016 1 20

[10] M. Gladwell , The tipping point: How little things can make a big difference. Little Brown (2006).

[11] D. Goulson The insect apocalypse, and why it matters Curr. Biol 2019 967 971

[12] T. Grafke, E. Vanden-Eijnden Non-equilibrium transitions in multiscale systems with a bifurcating slow manifold J. Stat. Mech 2017 093208

[13] R.C. Hilborn , Chaos and nonlinear dynamics: an introduction for scientists and engineers. Oxford University Press on Demand (2000).

[14] H. Kielhofer , Bifurcation theory: an introduction with applications to partial differential equations. Springer Science Business Media (2011).

[15] T.M. Lenton, H. Held, E. Kriegler Tipping elements in the Earth’s climate system Proc. Nati. Acad. Sci 2008 1786 1793

[16] Y. Li, J. Duan, X. Liu, Y. Zhang Most probable dynamics of stochastic dynamical systems with exponentially light jump fluctuations Chaos 2020 063142

[17] D. Ludwig, D.D. Jones, C.S. Holling Qualitative analysis of insect outbreak systems: the spruce budworm and forest J. Anim. Ecol 1978 315 332

[18] A.C.J. Luo and H. Merdan (Eds.), Mathematical Modeling and Applications in Nonlinear Dynamics. Springer International Publishing (2016).

[19] S. Lynch , Dynamical systems with applications using MATLAB. Birkhäuser, Boston (2004).

[20] A. Markos , T. Dimitris , S. Michalis and P. Charalambos , Stochastic processes and insect outbreak systems: Application to olive fruit fly. WSEAS Press (2007) 98–103.

[21] G.A. Meyer, J.A. Senulis, J.A. Reinartz Effects of temperature and availability of insect prey on bat emergence from hibernation in spring J. Mammal 2016 1623 1633

[22] C. Robinet, A. Roques Direct impacts of recent climate warming on insect populations Integr. Zool 2010 132 142

[23] G. Samorodnitsky and M.S. Taqqu , Stable non-Gaussian random processes - Stochastic models with infinite variance - Stochastic modeling. Chapman Hall, New York (1994).

[24] K.I. Sato , Levy processes and infinitely divisible distributions. Cambridge University Press (1999).

[25] M.E. Saunders Ups and downs of insect populations Nat. Ecol. Evol 2019 1616 1617

[26] M. Scheffer , Critical transitions in nature and society. Princeton University Press (2020).

[27] S.H. Strogatz , Nonlinear dynamics and chaos with applications to physics, biology, chemistry, and engineering. CRC Press (2018).

[28] J.M.T. Thompson, J. Sieber Predicting climate tipping as a noisy bifurcation: a review Int. J. Bifurcation Chaos 2011 399 423

[29] S.F. Ward, R.D. Moon, D.A. Herms, B.H. Aukema Determinants and consequences of plant-insect phenological synchrony for a non-native herbivore on a deciduous conifer: implications for invasion success Oecologia 2019 867 878

[30] S. Wiggins , Introduction to applied nonlinear dynamical systems and chaos, 2nd edn., Springer, New York (2003).

[31] F. Wong, J.J. Collins Evidence that coronavirus superspreading is fat-tailed Proc. Nati. Acad. Sci 2020 29416 29418

[32] S. Yuan, Z. Zeng, J. Duan Stochastic bifurcation for two-time-scale dynamical system with a-stable Levy noise J. Stat. Mech 2021 033204

[33] S. Yuan , Code, Github, https://github.com/ShenglanYuan/Stochastic-bifurcation-and-tipping-phenomena-of-insect-outbreak-dynamical-systems-driven-by-stable (2021).

[34] Y. Zheng, L. Serdukova, J. Duan, J. Kurths Transitions in a genetic transcriptional regulatory system under Levy motion Sci. Rep 2016 1 12