zbMATH — the first resource for mathematics

Random switching between vector fields having a common zero. (English) Zbl 1437.60044
Authors’ abstract: Let \(E\) be a finite set, \({\{F_i\}_{i\in E}}\) a family of vector fields on \({\mathbb{R}^d}\) leaving positively invariant a compact set \(M\) and having a common zero \({p\in M}\). We consider a piecewise deterministic Markov process \({(X_t,I_t)}\) on \({M\times E}\) defined by \({\dot{X}_t = F^{I_t}(X_t)}\) where \(I\) is a jump process controlled by \(X:\)
\({P(I_{t+s}=j | (X_u, I_u)_{u\leq t})=a_{ij}(X_t )s + o(s)}\) for \({i\neq j}\) on \({\{I_t=i\}}\). We show that the behaviour of \({(X,I)}\) is mainly determined by the behaviour of the linearized process \({(Y,J)}\) where \({\dot{Y}_t=A^{J_t} Y_t},\) \(A^i\) is the Jacobian matrix of \({F_i}\) at \(p\) and \(J\) is the jump process with rates \({(a_{ij} (p))}\). We introduce two quantities \(\Lambda^{-}\) and \(\Lambda^{+}\), respectively, defined as the minimal (resp., maximal) growth rate of \({\|Y_t\|}\), where the minimum (resp., maximum) is taken over all the ergodic measures of the angular process \({(\Theta, J)}\) with \({\Theta_t=Y_t/\|Y_t\|}.\) It is shown that \(\Lambda^{+}\) coincides with the top Lyapunov exponent (in the sense of ergodic theory) of \({(Y, J)}\) and that under general assumptions \({\Lambda^{-}=\Lambda^{+}}.\) We then prove that, under certain irreducibility conditions, \({X_t\to p}\) exponentially fast when \({\Lambda^{+}<0}\) and \((X,I)\) converges in distribution at an exponential rate toward a (unique) invariant measure supported by \({(M\setminus\{p\})\times E}\) when \({\Lambda^{-}>0}.\)
Some applications to certain epidemic models in a fluctuating environment (based on the differential equation due to A. Lajmanovich and J. A. Yorke [Math. Biosci. 28, 221–236 (1976; Zbl 0344.92016)]) are discussed and illustrate our results.

60J25 Continuous-time Markov processes on general state spaces
34A37 Ordinary differential equations with impulses
37H15 Random dynamical systems aspects of multiplicative ergodic theory, Lyapunov exponents
37A50 Dynamical systems and their relations with probability theory and stochastic processes
92D30 Epidemiology
Full Text: DOI Euclid arXiv
[1] Arnold, L. (1998). Random Dynamical Systems. Springer, Berlin. · Zbl 0906.34001
[2] Arnold, L., Gundlach, V. M. and Demetrius, L. (1994). Evolutionary formalism for products of positive random matrices. Ann. Appl. Probab.4 859–901. · Zbl 0818.15015
[3] Aubin, J.-P. and Cellina, A. (1984). Differential Inclusions: Set-Valued Maps and Viability Theory. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 264. Springer, Berlin. · Zbl 0538.34007
[4] Bakhtin, Y. and Hurth, T. (2012). Invariant densities for dynamical systems with random switching. Nonlinearity25 2937–2952. · Zbl 1251.93132
[5] Bakhtin, Y., Hurth, T. and Mattingly, J. C. (2015). Regularity of invariant densities for 1D systems with random switching. Nonlinearity28 3755–3787. · Zbl 1327.93396
[6] Baxendale, P. H. (1991). Invariant measures for nonlinear stochastic differential equations. In Lyapunov Exponents (Oberwolfach, 1990). Lecture Notes in Math.1486 123–140. Springer, Berlin. · Zbl 0741.60047
[7] Benaïm, M. (1998). Recursive algorithms, urn processes and chaining number of chain recurrent sets. Ergodic Theory Dynam. Systems18 53–87. · Zbl 0921.60061
[8] Benaïm, M., Colonius, F. and Lettau, R. (2017). Supports of invariant measures for piecewise deterministic Markov processes. Nonlinearity30 3400–3418. · Zbl 1384.93025
[9] Benaïm, M. and Hirsch, M. W. (1999). Differential and stochastic epidemic models. In Differential Equations with Applications to Biology (Halifax, NS, 1997). Fields Inst. Commun.21 31–44. Amer. Math. Soc., Providence, RI. · Zbl 0916.58040
[10] Benaïm, M., Hofbauer, J. and Sandholm, W. H. (2008). Robust permanence and impermanence for stochastic replicator dynamics. J. Biol. Dyn.2 180–195. · Zbl 1140.92025
[11] Benaïm, M., Le Borgne, S., Malrieu, F. and Zitt, P.-A. (2012). Quantitative ergodicity for some switched dynamical systems. Electron. Commun. Probab.17 no. 56, 14. · Zbl 1347.60118
[12] Benaïm, M., Le Borgne, S., Malrieu, F. and Zitt, P.-A. (2014). On the stability of planar randomly switched systems. Ann. Appl. Probab.24 292–311. · Zbl 1288.93090
[13] Benaïm, M., Le Borgne, S., Malrieu, F. and Zitt, P.-A. (2015). Qualitative properties of certain piecewise deterministic Markov processes. Ann. Inst. Henri Poincaré Probab. Stat.51 1040–1075. · Zbl 1325.60123
[14] Benaïm, M. and Lobry, C. (2016). Lotka–Volterra with randomly fluctuating environments or “How switching between beneficial environments can make survival harder”. Ann. Appl. Probab.26 3754–3785. · Zbl 1358.92075
[15] Benaïm, M. (2018). Stochastic persistence. Available at: https://arxiv.org/abs/1806.08450.
[16] Benaïm, M., Hurth, T. and Strickler, E. (2018). A user-friendly condition for exponential ergodicity in randomly switched environments. Electron. Commun. Probab.23 44. · Zbl 1397.60106
[17] Benaïm, M. and Schreiber, S. (2009). Persistence of structured populations in random environments. Theor. Popul. Biol.76 19–34. · Zbl 1213.92057
[18] Chesson, P. L. (1982). The stabilizing effect of a random environment. J. Math. Biol.15 1–36. · Zbl 0505.92021
[19] Chueshov, I. (2002). Monotone Random Systems Theory and Applications. Lecture Notes in Math.1779. Springer, Berlin. · Zbl 1023.37030
[20] Cloez, B. and Hairer, M. (2015). Exponential ergodicity for Markov processes with random switching. Bernoulli21 505–536. · Zbl 1330.60094
[21] Crauel, H. (1984). Lyapunov numbers of Markov solutions of linear stochastic systems. Stochastics14 11–28. · Zbl 0564.60058
[22] Davis, M. H. A. (1984). Piecewise-deterministic Markov processes: A general class of nondiffusion stochastic models. J. Roy. Statist. Soc. Ser. B46 353–388. With discussion. · Zbl 0565.60070
[23] Fainshil, L., Margaliot, M. and Chigansky, P. (2009). On the stability of positive linear switched systems under arbitrary switching laws. IEEE Trans. Automat. Control54 897–899. · Zbl 1367.93431
[24] Freidlin, M. I. and Wentzell, A. D. (2012). Random Perturbations of Dynamical Systems, 3rd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 260. Springer, Heidelberg. Translated from the 1979 Russian original by Joseph Szücs. · Zbl 1267.60004
[25] Garay, B. M. and Hofbauer, J. (2003). Robust permanence for ecological differential equations, minimax, and discretizations. SIAM J. Math. Anal.34 1007–1039. · Zbl 1026.37006
[26] Gurvits, L., Shorten, R. and Mason, O. (2007). On the stability of switched positive linear systems. IEEE Trans. Automat. Control52 1099–1103. · Zbl 1366.93436
[27] Hairer, M., Mattingly, J. C. and Scheutzow, M. (2011). Asymptotic coupling and a general form of Harris’ theorem with applications to stochastic delay equations. Probab. Theory Related Fields149 223–259. · Zbl 1238.60082
[28] Has’minskiĭ, R. Z. (1960). Ergodic properties of recurrent diffusion processes and stabilization of the solution of the Cauchy problem for parabolic equations. Teor. Veroyatn. Primen.5 196–214.
[29] Hening, A., Nguyen, D. H. and Yin, G. (2018). Stochastic population growth in spatially heterogeneous environments: The density-dependent case. J. Math. Biol.76 697–754. · Zbl 1392.92076
[30] Hirsch, M. W. (1994). Positive equilibria and convergence in subhomogeneous monotone dynamics. In Comparison Methods and Stability Theory (Waterloo, ON, 1993). Lecture Notes in Pure and Applied Mathematics162 169–188. Dekker, New York. · Zbl 0822.47051
[31] Hofbauer, J. and Schreiber, S. J. (2004). To persist or not to persist? Nonlinearity17 1393–1406. · Zbl 1055.37071
[32] Lagasquie, G. (2016). A note on simple randomly switched linear systems. arXiv preprint, arXiv:1612.01861.
[33] Lajmanovich, A. and Yorke, J. A. (1976). A deterministic model for gonorrhea in a nonhomogeneous population. Math. Biosci.28 221–236. · Zbl 0344.92016
[34] Lawley, S. D., Mattingly, J. C. and Reed, M. C. (2014). Sensitivity to switching rates in stochastically switched ODEs. Commun. Math. Sci.12 1343–1352. · Zbl 1329.60295
[35] Malrieu, F. (2015). Some simple but challenging Markov processes. Ann. Fac. Sci. Toulouse Math. (6) 24 857–883. · Zbl 1333.60185
[36] Meyn, S. and Tweedie, R. L. (2009). Markov Chains and Stochastic Stability, 2nd ed. Cambridge Univ. Press, Cambridge. · Zbl 1165.60001
[37] Mierczyński, J. (2015). Lower estimates of top Lyapunov exponent for cooperative random systems of linear ODEs. Proc. Amer. Math. Soc.143 1127–1135. · Zbl 1311.34123
[38] Rami, M. A., Bokharaie, V. S., Mason, O. and Wirth, F. R. (2014). Stability criteria for SIS epidemiological models under switching policies. Discrete Contin. Dyn. Syst. Ser. B19 2865–2887. · Zbl 1327.92054
[39] Roth, G. and Schreiber, S. J. (2014). Persistence in fluctuating environments for interacting structured populations. J. Math. Biol.69 1267–1317. · Zbl 1351.37216
[40] Schreiber, S. J. (2000). Criteria for \(C^{r}\) robust permanence. J. Differential Equations162 400–426. · Zbl 0956.34038
[41] Schreiber, S. J. (2012). Persistence for stochastic difference equations: A mini-review. J. Difference Equ. Appl.18 1381–1403. · Zbl 1258.39010
[42] Schreiber, S. J., Benaïm, M. and Atchadé, K. A. S. (2011). Persistence in fluctuating environments. J. Math. Biol.62 655–683. · Zbl 1232.92075
[43] Seneta, E. · Zbl 0471.60001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.