# Random fluctuations around a stable limit cycle in a stochastic system with parametric forcing

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## Abstract

Many real populations exhibit stochastic behaviour that appears to have some periodicity. In terms of populations, these time series can occur as limit cycles that arise through seasonal variation of parameters such as, e.g., disease transmission rate. The general mathematical context is that of a stochastic differential system with periodic parametric forcing whose solution is a stochastically perturbed limit cycle. Earlier work identified the power spectral density (PSD) features of these fluctuations by computation of the autocorrelation function of the stochastic process and its transform. Here, we present an alternative analysis which shows that the structure of the fluctuations around the limit cycle is analogous to that of fluctuations about a fixed point. Furthermore, we show that these fluctuations can be expressed, approximately, as a factorization which reveals the combined frequencies of the limit cycle and the stochastic perturbation. This result, based on a new limit theorem near a Hopf point, yields an understanding of the previously found features of the PSD. Further insights are obtained from the corresponding stochastic equations for phase and amplitude.

## Keywords

Disease recurrence Epidemiology Floquet theory Limit cycle with noise Ornstein–Uhlenbeck process Periodic forcing Seasonal forcing Stochastic fluctuations Sustained oscillations## Mathematics Subject Classification

60H10 34F05 34F15## Notes

### Acknowledgements

This research was partially funded by NSERC (Discovery Grant program, Application Number RGPIN-2016-05277)) (RT), and by the Institute for Biodiversity, Resilience, and Ecosystem Services at UBC Okanagan. Towards completion of this research, MAM was supported by the Ph.D. Fellowship Program of the University of the Philippines System.

## References

- Alonso D, McKane AJ, Pascual M (2007) Stochastic amplification in epidemics. J R Soc Interface 4(14):575–582CrossRefGoogle Scholar
- Altizer S, Dobson A, Hosseini P, Hudson P, Pascual M, Rohani P (2006) Seasonality and the dynamics of infectious diseases. Ecol Lett 9(4):467–484CrossRefGoogle Scholar
- Aron JL, Schwartz IB (1984) Seasonality and period-doubling bifurcations in an epidemic model. J Theor Biol 110(4):665–679. https://doi.org/10.1016/S0022-5193(84)80150-2
**(ISSN 0022-5193)**MathSciNetCrossRefGoogle Scholar - Bartlett MS (1957) Measles periodicity and community size. J R Stat Soc Ser A (Gen) 120(1):48–70CrossRefGoogle Scholar
- Baxendale PH, Greenwood PE (2011) Sustained oscillations for density dependent Markov processes. J Math Biol 63(3):433–457. https://doi.org/10.1007/s00285-010-0376-2
**(ISSN 0303-6812)**MathSciNetCrossRefzbMATHGoogle Scholar - Black AJ (2010) The stochastic dynamics of epidemic models. Ph.D. thesis, The University of Manchester, Manchester, UKGoogle Scholar
- Black AJ, McKane AJ (2010) Stochastic amplification in an epidemic model with seasonal forcing. J Theor Biol 267(1):85–94MathSciNetCrossRefzbMATHGoogle Scholar
- Black AJ, McKane AJ (2012) Stochastic formulation of ecological models and their applications. Trends Ecol Evol 27(6):337–345CrossRefGoogle Scholar
- Boland RP, Galla T, McKane AJ (2008) How limit cycles and quasi-cycles are related in systems with intrinsic noise. J Stat Mech Theory Exp 2008(09):P09001CrossRefGoogle Scholar
- Boland RP, Galla T, McKane AJ (2009) Limit cycles, complex Floquet multipliers, and intrinsic noise. Phys Rev E (3) 79(5):051131, 13. https://doi.org/10.1103/PhysRevE.79.051131
**(ISSN 1539-3755)**MathSciNetCrossRefGoogle Scholar - Gardiner C (1990) Handbook of stochastic methods for physics, chemistry and the natural sciences, 2nd edn. Springer, Berlin
**(ISBN 978-0387156071)**zbMATHGoogle Scholar - Grimshaw R (1991) Nonlinear ordinary differential equations, vol 2. CRC Press, Boca RatonzbMATHGoogle Scholar
- Keeling MJ, Rohani P, Grenfell BT (2001) Seasonally forced disease dynamics explored as switching between attractors. Phys D Nonlinear Phenom 148(3):317–335CrossRefzbMATHGoogle Scholar
- Klausmeier CA (2008) Floquet theory: a useful tool for understanding nonequilibrium dynamics. Theor Ecol 1(3):153–161CrossRefGoogle Scholar
- Kurtz TG (1978) Strong approximation theorems for density dependent markov chains. Stoch Process Appl 6(3):223–240MathSciNetCrossRefzbMATHGoogle Scholar
- Kuske R, Gordillo LF, Greenwood P (2007) Sustained oscillations via coherence resonance in sir. J Theor Biol 245(3):459–469MathSciNetCrossRefGoogle Scholar
- Mata MA, Greenwood P, Tyson R (2018) The relative contribution of direct and environmental transmission routes in stochastic avian flu epidemic recurrence: an approximate analysis. Bull Math Biol. https://doi.org/10.1007/s11538-018-0414-6
- May RM, Anderson RM et al (1979) Population biology of infectious diseases: part ii. Nature 280(5722):455–461CrossRefGoogle Scholar
- McKane AJ, Newman TJ (2005) Predator-prey cycles from resonant amplification of demographic stochasticity. Phys Rev Lett 94(21):218102CrossRefGoogle Scholar
- Milstein GN (1994) Numerical integration of stochastic differential equations, vol 313. Springer, BerlinzbMATHGoogle Scholar
- Nisbet RM, Gurney W (2003) Modelling fluctuating populations: reprint of first edition (1982). Blackburn Press, CaldwellGoogle Scholar
- Rozhnova G, Nunes A (2010) Stochastic effects in a seasonally forced epidemic model. Phys Rev E (3) 82(4):041906, 12. https://doi.org/10.1103/PhysRevE.82.041906
**(ISSN 1539-3755)**MathSciNetCrossRefGoogle Scholar - Stein EM, Shakarchi R (2011) Fourier analysis: an introduction, vol 1. Princeton University Press, PrincetonzbMATHGoogle Scholar
- van Kampen N (1992) Stochastic processes in physics and chemistry. Elsevier, AmsterdamzbMATHGoogle Scholar