Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-22T04:04:21.885Z Has data issue: false hasContentIssue false
  • English
  • Français

Large deviations for the stochastic predator–prey model with nonlinear functional response

Part of: Stochastic analysis Nonlinear dynamics Limit theorems

Published online by Cambridge University Press:  22 June 2017

M. Suvinthra  and
K. Balachandran
M. Suvinthra*
Affiliation:
Bharathiar University
K. Balachandran*
Affiliation:
Bharathiar University
*
* Postal address: Department of Mathematics, Bharathiar University, Coimbatore 641046, India.
* Postal address: Department of Mathematics, Bharathiar University, Coimbatore 641046, India.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we consider a diffusive stochastic predator–prey model with a nonlinear functional response and the randomness is assumed to be of Gaussian nature. A large deviation principle is established for solution processes of the considered model by implementing the weak convergence technique.

Keywords

Predator–prey modelpopulation dynamicslarge deviationstochastic partial differential equation

MSC classification

Primary: 92D25: Population dynamics (general) 60H15: Stochastic partial differential equations
Secondary: 70K50: Bifurcations and instability 60F10: Large deviations
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 2 , June 2017 , pp. 507 - 521
Copyright
Copyright © Applied Probability Trust 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Adams, R. A. and Fournier, J. J. F. (2003). Sobolev Spaces, 2nd edn. Elsevier/Academic Press, Amsterdam. Google Scholar
[2] Arratia, R. and Gordon, L. (1989). Tutorial on large deviations for the binomial distribution. Bull. Math. Biol. 51, 125–131. CrossRefGoogle ScholarPubMed
[3] Bertini, L. et al. (2007). Large deviations of the empirical current in interacting particle systems. Theory Prob. Appl. 51, 2–27. CrossRefGoogle Scholar
[4] Bessam, D. (2015). Large deviations in a Gaussian setting: the role of the Cameron-Martin space. In From Particle Systems to Partial Differential Equations II (Springer Proc. Math. Statist. 129), eds P. Goncalves and A. J. Soares, Springer, pp. 135–152. CrossRefGoogle Scholar
[5] Bressloff, P. C. and Newby, J. M. (2014). Path integrals and large deviations in stochastic hybrid systems. Phys. Rev. E 89, 042701. CrossRefGoogle ScholarPubMed
[6] Budhiraja, A. and Dupuis, P. (2000). A variational representation for positive functionals of infinite dimensional Brownian motion. Prob. Math. Statist. 20, 39–61. Google Scholar
[7] Chen, L. and Jungel, A. (2004). Analysis of a multidimensional parabolic population model with strong cross-diffusion. SIAM J. Math. Anal. 36, 301–322. CrossRefGoogle Scholar
[8] Chow, P.-L. (2011). Explosive solutions of stochastic reaction-diffusion equations in mean L p -norm. J. Differential Equat. 250, 2567–2580. CrossRefGoogle Scholar
[9] Da Prato, G. and Zabczyk, J. (1992). Stochastic Equations in Infinite Dimensions. Cambridge University Press. CrossRefGoogle Scholar
[10] Dembo, A. and Zeitouni, O. (2010). Large Deviations Techniques and Applications. Springer, New York. CrossRefGoogle Scholar
[11] Dupuis, P. and Ellis, R. S. (1997). A Weak Convergence Approach to the Theory of Large Deviations. John Wiley, New York. CrossRefGoogle Scholar
[12] Florens-Landais, D. and Pham, C. H. (1999). Large deviations in estimation of an Ornstein-Uhlenbeck model. J. Appl. Prob. 36, 60–77. CrossRefGoogle Scholar
[13] Freidlin, M. (1991). Coupled reaction-diffusion equations. Ann. Prob. 19, 29–57. CrossRefGoogle Scholar
[14] Gross, L. (1965). Potential theory on Hilbert space. J. Funct. Anal. 1, 123–181. CrossRefGoogle Scholar
[15] Khaminskii, R. Z., Klebaner, F. C. and Liptser, R. (2003). Some results on the Lotka-Volterra model and its small random perturbations. Acta Appl. Math. 78, 201–206. CrossRefGoogle Scholar
[16] Klebaner, F. C., Lim, A. and Liptser, R. (2007). FCLT and MDP for stochastic Lotka-Volterra model. Acta Appl. Math. 97, 53–68. CrossRefGoogle Scholar
[17] Klebaner, F. C. and Liptser, R. (2001). Asymptotic analysis and extinction in a stochastic Lotka-Volterra model. Ann. Appl. Prob. 11, 1263–1291. CrossRefGoogle Scholar
[18] Kratz, P., Pardoux, E. and Kepgnou, B. S. (2015). Numerical methods in the context of compartmental models in epidemiology. ESAIM Proc. Surveys 48, 169–189. CrossRefGoogle Scholar
[19] Li, A.-W. (2011). Impact of noise on pattern formation in a predator–prey model. Nonlinear Dynamics 66, 689–694. CrossRefGoogle Scholar
[20] Manna, U. and Mohan, M. T. (2013). Large deviations for the shell model of turbulence perturbed by Lévy noise. Commun. Stoch. Anal. 7, 39–63. Google Scholar
[21] Mo, C. and Luo, J. (2013). Large deviations for stochastic differential delay equations. Nonlinear Anal. 80, 202–210. CrossRefGoogle Scholar
[22] Pakdaman, K., Thieuller, M. and Wainrib, G. (2010). Diffusion approximation of birth-death processes: comparison in terms of large deviations and exit points. Statist. Prob. Lett. 80, 1121–1127. CrossRefGoogle Scholar
[23] Peszat, S. and Zabczyk, J. (2007). Stochastic Partial Differential Equations with Lévy Noise. Cambridge University Press. CrossRefGoogle Scholar
[24] Sambath, M. and Balachandran, K. (2013). Spatiotemporal dynamics of a predator-prey model incorporating a prey refuge. J. Appl. Anal. Comput. 3, 71–80. Google Scholar
[25] Sambath, M., Balachandran, K. and Suvinthra, M. (2016). Stability and Hopf bifurcation of a diffusive predator-prey model with hyperbolic mortality. Complexity 21, 34–43. CrossRefGoogle Scholar
[26] Setayeshgar, L. (2014). Large deviations for a stochastic Burgers' equation. Commun. Stoch. Anal. 8, 141–154. Google Scholar
[27] Shangerganesh, L. and Balachandran, K. (2011). Existence and uniqueness of solutions of predator-prey type model with mixed boundary conditions. Acta Appl. Math. 116, 71–86. CrossRefGoogle Scholar
[28] Sivakumar, M., Sambath, M. and Balachandran, K. (2015). Stability and Hopf bifurcation analysis of a diffusive predator-prey model with Smith growth. Internat. J. Biomath. 8, 18 pp. CrossRefGoogle Scholar
[29] Sritharan, S. S. and Sundar, P. (2006). Large deviations for two-dimensional Navier-Stokes equations with multiplicative noise. Stoch. Process. Appl. 116, 1636–1659. CrossRefGoogle Scholar
[30] Suvinthra, M., Sritharan, S. S. and Balachandran, K. (2015). Large deviations for stochastic tidal dynamics equation. Commun. Stoch. Anal. 9, 477–502. Google Scholar
[31] Varadhan, S. R. S. (2008). Large deviations. Ann. Prob. 36, 397–419. CrossRefGoogle Scholar
[32] Weber, J. K., Jack, R. L., Schwantes, C. R. and Pande, V. S. (2014). Dynamical phase transitions reveal amyloid-like states on protein folding landscapes. Biophys. J. 107, 974–982. CrossRefGoogle ScholarPubMed