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Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA


Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania


Analysis of Stochastic Diffusive Predator PreyModel with Hyperbolic Mortality Rate

Discontinuity, Nonlinearity, and Complexity 5(3) (2016) 223--237 | DOI:10.5890/DNC.2016.09.003

M. Suvinthra$^{1}$, K. Balachandran$^{1}$, M. Sambath$^{2}$

$^{1}$ Department of Mathematics, Bharathiar University, Coimbatore 641 046, India

$^{2}$ Department of Mathematics, Periyar University, Salem 636 011, India

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In this work, we establish a Freidlin-Wentzell type large deviation principle for a diffusive predator-prey modelwith hyperbolic mortality rate perturbed by multiplicative type Gaussian noise. We implement the variational representation developed by Budhiraja and Dupuis to establish the large deviation principle for the solution processes.


The first author would like to thank the Department of Science and Technology (DST), Government of India, New Delhi for their financial support under the INSPIRE Fellowship Scheme. The work of the second author is supported by Defence Research and Development Organization (DRDO), Government of India. The work of the third author is supported by University Grants Commission (UGC), New Delhi, Government of India under the Special Assistance Programme (SAP - 1).


  1. [1]  Chen, L. and Jungel, A. (2006), Analysis of a parabolic cross-diffusion population model without self-diffusion, Journal of Differential Equations, 224, 39-59.
  2. [2]  Bendahmane, M. (2010),Weak and classical solutions to predator-prey system with cross-diffusion, Nonlinear Analysis, 73, 2489-2503.
  3. [3]  Shangerganesh, L. and Balachandran, K. (2011), Existence and uniqueness of solutions of predator-prey type model with mixed boundary conditions, Acta ApplicandaeMathematicae, 116, 71-86.
  4. [4]  Leonetti, M., Boedec, G. and Jaeger, M. (2013), Breathing instability in biological cells, patterns of membrane proteins, Discontinuity, Nonlinearity, and Complexity, 2, 75-84.
  5. [5]  Sambath, M. and Balachandran, K. (2013), Spatiotemporal dynamics of a predator-prey model incorporating a prey refuge, Journal of Applied Analysis and Computation, 3, 71-80.
  6. [6]  Erjaee, G.H., Ostadzad, M.H., Okuguchi K. and Rahimi, E. (2013), Fractional differential equations system for commercial fishing under predator-prey interaction, Journal of Applied Nonlinear Dynamics, 2, 409-417.
  7. [7]  Sambath, M., Gnanavel, S. and Balachandran, K. (2013), Stability and Hopf bifurcation of a diffusive predator-prey model with predator saturation and competition, Applicable Analysis, 92, 2439-2456.
  8. [8]  Sivakumar, M., Sambath, M. and Balachandran, K. (2015), Stability and Hopf bifurcation analysis of a diffusive predator-prey model with Smith growth, International Journal of Biomathematics, 8, 1550013.
  9. [9]  Khaminskii, R.Z., Klebaner, F.C. and Liptser, R. (2003), Some results on the Lotka-Volterra model and its small random perturbations, Acta Applicandae Mathematicae, 78, 201-206.
  10. [10]  Li, A-W. (2011), Impact of noise on pattern formation in a predator-prey model, Nonlinear Dynamics, 66, 689-694.
  11. [11]  Dupuis, P. and Ellis, R.S. (1997), A Weak Convergence Approach to the Theory of Large Deviations, Wiley- Interscience: New York.
  12. [12]  Dembo, A. and Zeitouni, O. (2007), Large Deviations Techniques and Applications, Springer, New York.
  13. [13]  Varadhan, S.R.S. (2008), Large deviations, The Annals of Probability, 36, 397-419.
  14. [14]  Freidlin,M.I. andWentzell, A.D. (1970), On small random perturbations of dynamical systems, Russian Mathematical Surveys, 25, 1-55.
  15. [15]  Budhiraja, A. and Dupuis, P. (2000), A variational representation for positive functionals of infinite dimensional Brownian motion, Probability and Mathematical Statistics, 20, 39-61.
  16. [16]  Arratia, R. and Gordon, L. (1989), Tutorial on large deviations for the binomial distribution, Bulletin of Mathematical Biology, 51, 125-131.
  17. [17]  Florens-Landais, D. and Pham, C.H. (1999), Large deviations in estimation of an Ornstein-Uhlenbeck model, Journal of Applied Probability, 36, 60-77.
  18. [18]  Champagnat, N., Ferriere, R. and Meleard, S. (2006), Unifying evolutionary dynamics: From individual stochastic processes to macroscopic models, Theoretical Population Biology, 69, 297-321.
  19. [19]  Klebaner, F.C., Lim, A. and Liptser, R. (2007), FCLT and MDP for stochastic Lotka-Volterra model, Acta Applicandae Mathematicae, 97, 53-68.
  20. [20]  Bertini, L., De Sole, A., Gabrielli, D., Jona-Lasinio, G. and Landim, C. (2007), Large deviations of the empirical current in interacting particle systems, Theory of Probability and its Applications, 51, 2-27.
  21. [21]  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, Biophysical Journal, 107, 974-982.
  22. [22]  Zint, N., Baake, E. and den Hollander, F. (2008), How T-cells use large deviations to recognize foreign antigens, Journal of Mathematical Biology, 57, 841-861.
  23. [23]  Pakdaman, K., Thieuller, M. and Wainrib, G. (2010), Diffusion approximation of birth-death processes: Comparison in terms of large deviations and exit points, Statistics and Probability Letters, 80, 1121-1127.
  24. [24]  Klebaner, F.C. and Liptser, R. (2001), Asymptotic analysis and extinction in a stochastic Lotka-Volterra model, The Annals of Applied Probability, 11, 1263-1291.
  25. [25]  Bressloff, P.C. and Newby, J.M. (2014), Path integrals and large deviations in stochastic hybrid systems, Physical Review E, 89, 042701.
  26. [26]  Kratz, P., Pardoux, E. and Kepgnou, B.S. (2015), Numerical methods in the context of compartmental models in epidemiology, in N. Champagnat, T. Lelievre and A. Nouy (Eds.) ESAIM: Proceedings and Surveys, 48, 169-189.
  27. [27]  Sambath, M., Balachandran, K. and Suvinthra, M. (2015), Stability and Hopf bifurcation of a diffusive predator-prey model with hyperbolic mortality, Complexity, DOI: 10.1002/cplx.21708 .
  28. [28]  Adams, R.A. and Fournier, J.J.F. (2003), Sobolev Spaces, Academic Press, Amsterdam.
  29. [29]  Sritharan, S.S. and Sundar, P. (2006), Large deviations for two dimensionalNavier-Stokes equations with multiplicative noise, Stochastic Processes and their Applications, 116, 1636-1659.
  30. [30]  Budhiraja, A. and Dupuis, P. (2008), Large deviations for infinite dimensional stochastic dynamical systems, The Annals of Probability, 36, 1390-1420.
  31. [31]  Da Prato, G. and Zabczyk, J. (1992), Stochastic Equations in Infinite Dimensions, Cambridge University Press: Cambridge.