## The Annals of Applied Probability

### Quasi-stationary distributions for randomly perturbed dynamical systems

#### Abstract

We analyze quasi-stationary distributions $\{\mu^{\varepsilon}\}_{\varepsilon>0}$ of a family of Markov chains $\{X^{\varepsilon}\}_{\varepsilon>0}$ that are random perturbations of a bounded, continuous map $F:M\to M$, where $M$ is a closed subset of $\mathbb{R}^{k}$. Consistent with many models in biology, these Markov chains have a closed absorbing set $M_{0}\subset M$ such that $F(M_{0})=M_{0}$ and $F(M\setminus M_{0})=M\setminus M_{0}$. Under some large deviations assumptions on the random perturbations, we show that, if there exists a positive attractor for $F$ (i.e., an attractor for $F$ in $M\setminus M_{0}$), then the weak* limit points of $\mu_{\varepsilon}$ are supported by the positive attractors of $F$. To illustrate the broad applicability of these results, we apply them to nonlinear branching process models of metapopulations, competing species, host-parasitoid interactions and evolutionary games.

#### Article information

Source
Ann. Appl. Probab., Volume 24, Number 2 (2014), 553-598.

Dates
First available in Project Euclid: 10 March 2014

https://projecteuclid.org/euclid.aoap/1394465365

Digital Object Identifier
doi:10.1214/13-AAP923

Mathematical Reviews number (MathSciNet)
MR3178491

Zentralblatt MATH identifier
1334.60137

#### Citation

Faure, Mathieu; Schreiber, Sebastian J. Quasi-stationary distributions for randomly perturbed dynamical systems. Ann. Appl. Probab. 24 (2014), no. 2, 553--598. doi:10.1214/13-AAP923. https://projecteuclid.org/euclid.aoap/1394465365

#### References

• Andronov, A., Pontryagin, L. and Witt, A. (1933). On the statistical investigation of dynamical systems. J. Exp. Theor. Phys. 3 165–180.
• Arjas, E., Nummelin, E. and Tweedie, R. L. (1980). Semi-Markov processes on a general state space: $\alpha$-theory and quasistationarity. J. Austral. Math. Soc. Ser. A 30 187–200.
• Barbour, A. D. (1976). Quasi-stationary distributions in Markov population processes. Adv. in Appl. Probab. 8 296–314.
• Buckley, F. M. and Pollett, P. K. (2010). Limit theorems for discrete-time metapopulation models. Probab. Surv. 7 53–83.
• Caswell, H. (2001). Matrix Population Models. Sinauer, Sunderland, MA.
• Chan, T. (1998). Large deviations and quasi-stationarity for density-dependent birth-death processes. J. Austral. Math. Soc. Ser. B 40 238–256.
• Conley, C. (1978). Isolated Invariant Sets and the Morse Index. CBMS Regional Conference Series in Mathematics 38. Amer. Math. Soc., Providence, RI.
• Coolen-Schrijner, P. and van Doorn, E. A. (2006). Quasi-stationary distributions for a class of discrete-time Markov chains. Methodol. Comput. Appl. Probab. 8 449–465.
• Cressman, R. (2003). Evolutionary Dynamics and Extensive Form Games. MIT Press Series on Economic Learning and Social Evolution 5. MIT Press, Cambridge, MA.
• Cressman, R., Krivan, V. and Garay, J. (2004). Ideal free distributions, evolutionary games, and population dynamics in multiple-species environments. Am. Nat. 164 473–489.
• Cushing, J. M., Levarge, S., Chitnis, N. and Henson, S. M. (2004). Some discrete competition models and the competitive exclusion principle. J. Difference Equ. Appl. 10 1139–1151.
• Darroch, J. N. and Seneta, E. (1965). On quasi-stationary distributions in absorbing discrete-time finite Markov chains. J. Appl. Probab. 2 88–100.
• Dembo, A. and Zeitouni, O. (1993). Large Deviation Techniques and Applications. Springer, New York.
• Earn, D. J., Levin, S. A. and Rohani, P. (2000). Coherence and conservation. Science 290 1360–1364.
• Earn, D. J. D. and Levin, S. A. (2006). Global asymptotic coherence in discrete dynamical systems. Proc. Natl. Acad. Sci. USA 103 3968–3971.
• Ferrari, P. A., Kesten, H., Martinez, S. and Picco, P. (1995). Existence of quasi-stationary distributions. A renewal dynamical approach. Ann. Probab. 23 501–521.
• Freidlin, M. I. and Wentzell, A. D. (1984). Random Perturbations of Dynamical Systems. Springer, New York.
• Garay, B. M. and Hofbauer, J. (2003). Robust permanence for ecological differential equations, minimax, and discretizations. SIAM J. Math. Anal. 34 1007–1039.
• Getz, W. M. and Mills, N. J. (1996). Host-parasitoid coexistence and egg-limited encounter rates. Am. Nat. 148 333–347.
• Gidea, M., Meiss, J. D., Ugarcovici, I. and Weiss, H. (2011). Applications of KAM theory to population dynamics. J. Biol. Dyn. 5 44–63.
• Godfray, H. C. J. (1994). Parasitoids. Princeton Univ. Press, Princeton, NJ.
• Gosselin, F. (2001). Asymptotic behavior of absorbing Markov chains conditional on nonabsorption for applications in conservation biology. Ann. Appl. Probab. 11 261–284.
• Hassell, M. P. (1978). The Dynamics of Arthropod Predator–Prey Systems. Monographs in Population Biology 13. Princeton Univ. Press, Princeton, NJ.
• Hassell, M. P. (2000). The Spatial and Temporal Dynamics of Host-Parasitoid Interactions. Oxford Univ. Press, Oxford.
• Hassell, M. P., May, R. M., Pacala, S. W. and Chesson, P. L. (1991). The persistence of host-parasitoid associations in patchy environments. I. A general criterion. Am. Nat. 138 586–583.
• Hastings, A. (1993). Complex interactions between dispersal and dynamics: Lessons from coupled logistic equations. Ecology 74 1362–1372.
• Hastings, A. (1997). Population Biology: Concepts and Models. Springer, New York.
• Hastings, A. and Botsford, L. W. (2006). Persistence of spatial populations depends on returning home. Proc. Natl. Acad. Sci. USA 103 6067–6072.
• Hofbauer, J. and Sigmund, K. (1998). Evolutionary Games and Population Dynamics. Cambridge Univ. Press, Cambridge.
• Hofbauer, J. and Sigmund, K. (2003). Evolutionary game dynamics. Bull. Amer. Math. Soc. (N.S.) 40 479–519.
• Högnäs, G. (1997). On the quasi-stationary distribution of a stochastic Ricker model. Stochastic Process. Appl. 70 243–263.
• Imhof, L. A. and Nowak, M. A. (2006). Evolutionary game dynamics in a Wright–Fisher process. J. Math. Biol. 52 667–681.
• Imhof, L. A. and Nowak, M. A. (2010). Stochastic evolutionary dynamics of direct reciprocity. Proceedings of the Royal Society B: Biological Sciences 277 463–468.
• Jacobs, F. and Schreiber, S. J. (2006). Random perturbations of dynamical systems with absorbing states. SIAM J. Appl. Dyn. Syst. 5 293–312.
• Kermack, W. O. and McKendrick, A. G. (1927). A contribution to the mathematical theory of epidemics. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 115 700–721.
• Kifer, Y. (1988). Random Perturbations of Dynamical Systems. Birkhäuser, Boston, MA.
• Kifer, Y. (1989). Attractors via random perturbations. Comm. Math. Phys. 121 445–455.
• Kifer, Y. (1990). A discrete-time version of the Wentzell–Freidlin theory. Ann. Probab. 18 1676–1692.
• Kijima, M. (1992). On the existence of quasi-stationary distributions in denumerable $R$-transient Markov chains. J. Appl. Probab. 29 21–36.
• Klebaner, F. C., Lazar, J. and Zeitouni, O. (1998). On the quasi-stationary distribution for some randomly perturbed transformations of an interval. Ann. Appl. Probab. 8 300–315.
• Kon, R., Saito, Y. and Takeuchi, Y. (2004). Permanence of single-species stage-structured models. J. Math. Biol. 48 515–528.
• Kozlovski, O. S. (2003). Axiom A maps are dense in the space of unimodal maps in the $C^{k}$ topology. Ann. of Math. (2) 157 1–43.
• Lasserre, J. B. and Pearce, C. E. M. (2001). On the existence of a quasistationary measure for a Markov chain. Ann. Probab. 29 437–446.
• Leslie, P. H. and Gower, J. C. (1958). The properties of a stochastic model for two competing species. Biometrika 45 316–330.
• Lotka, A. J. (1925). Elements of Physical Biology. Williams and Witkins, Baltimore, MD.
• May, R. M. (1995). Necessity and chance: Deterministic chaos in ecology and evolution. Bull. Amer. Math. Soc. (N.S.) 32 291–308.
• Maynard Smith, J. (1974). The theory of games and the evolution of animal conflicts. J. Theoret. Biol. 47 209–221.
• Nicholson, A. J. and Bailey, V. A. (1935). The balance of animal populations. Proc. Zool. Soc. London 551–598.
• Nowak, M. A., Sasaki, A., Taylor, C. and Fudenberg, D. (2004). Emergence of cooperation and evolutionary stability in finite populations. Nature 428 646–650.
• Nummelin, E. and Arjas, E. (1976). A direct construction of the $R$-invariant measure for a Markov chain on a general state space. Ann. Probab. 4 674–679.
• Park, T. (1948). Experimental studies of interspecies competition. Ecol. Monogr. 18 265–308.
• Park, T. (1954). Experimental studies of the interspecies competition. II. Temperature, humidity and competition in two species of Tribolium. Physiol. Zool. 27 177–238.
• Przytycki, F. (1976). Anosov endomorphisms. Studia Math. 58 249–285.
• Ramanan, K. and Zeitouni, O. (1999). The quasi-stationary distribution for small random perturbations of certain one-dimensional maps. Stochastic Process. Appl. 84 25–51.
• Ricker, W. E. (1954). Stock and recruitment. J. Fish. Bd. Can. 11 559–623.
• Ruelle, D. (1981). Small random perturbations of dynamical systems and the definition of attractors. Comm. Math. Phys. 82 137–151.
• Schaefer, H. H. (1974). Banach Lattices and Positive Operators. Springer, New York.
• Schreiber, S. J. (2006a). Host-parasitoid dynamics of a generalized Thompson model. J. Math. Biol. 52 719–732.
• Schreiber, S. J. (2006b). Persistence despite perturbations for interacting populations. J. Theoret. Biol. 242 844–852.
• Schreiber, S. J. (2007). Periodicity, persistence, and collapse in host-parasitoid systems with egg limitation. J. Biol. Dyn. 1 273–288.
• Schreiber, S. J. (2010). Interactive effects of temporal correlations, spatial heterogeneity and dispersal on population persistence. Proceedings of the Royal Society B: Biological Sciences 277 1907–1914.
• Schreiber, S., Fox, L. and Getz, W. (2000). Coevolution of contrary choices in host-parasitoid systems. Am. Nat. 155 637–648.
• Seneta, E. and Vere-Jones, D. (1966). On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states. J. Appl. Probab. 3 403–434.
• Sinaĭ, J. G. (1972). Gibbs measures in ergodic theory. Russian Math. Surveys 27 21–69.
• Sinervo, B. and Lively, C. (1996). The rock-paper-scissors game and the evolution of alternative male strategies. Nature 380 240–243.
• Thompson, W. R. (1924). La théorie mathématique de l’action des parasites entomophages et le facteur du hassard. Ann. Fac. Sci. Marseille 2 69–89.
• Tweedie, R. L. (1974). Quasi-stationary distributions for Markov chains on a general state space. J. Appl. Probab. 11 726–741.
• van Strien, S. J. (1981). On the bifurcations creating horseshoes. In Dynamical Systems and Turbulence, Warwick 1980 (Coventry, 1979/1980). Lecture Notes in Math. 898 316–351. Springer, Berlin.
• Volterra, V. (1926). Fluctuations in the abundance of a species considered mathematically. Nature 118 558–560.
• Wysham, D. B. and Hastings, A. (2008). Sudden shifts in ecological systems: Intermittency and transients in the coupled Ricker population model. Bull. Math. Biol. 70 1013–1031.
• Zeeman, E. C. (1980). Population dynamics from game theory. In Global Theory of Dynamical Systems (Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1979). Lecture Notes in Math. 819 471–497. Springer, Berlin.