Multiplicative-noise Can Suppress Chaotic Oscillation in Lotka-Volterra Type Competitive Model

P.S. Mandal¹ ; M. Banerjee¹

¹ Department of Mathematics and Statistics Indian Institute of Technology, Kanpur Kanpur - 208016, INDIA

Mathematical modelling of natural phenomena, Tome 7 (2012) no. 6, pp. 23-46.

Résumé

Recently, Wang and Xiao studied a four-dimensional competitive Lotka-Volterra system within a deterministic environment in [11]. With the help of numerical example they showed the existence of a chaotic attractor through the period doubling route. In this paper, we are interested to study the dynamics of the same model in presence of environmental driving forces. To incorporate the environmental driving force into the deterministic system, we perturb the growth rates of each species by white noise terms. Then we prove that the unique positive global solution exists for the noise added system and the general p-th order moment of it is bounded for p ≥ 1, which ensures that the solution is stochastically bounded. It is also shown that the solution of the stochastic system is stochastically permanent under some simple conditions. Finally , we demonstrate the noise induced oscillation for the concerned model with the help of numerical example..

Détail

DOI : 10.1051/mmnp/20127602

Bibliographie

[1] B. Spagnolo, D. Valenti, A. Fiasconaro Math. Bios. Eng. 2004 185 211

[2] R.M. May. Stability and complexity in Model Ecosystems. Princeton University Press, Princeton, NJ, USA, 1973.

[3] J.D. Murray, Mathematical Biology, Berlin, Springer-Verlag, 1989.

[4] E. Renshaw. Modelling Biological Populations in Space and Time. Cambridge University Press, Cambridge, 1993.

[5] P. Turchin. Complex Population Dynamics: A theoretical/empirical synthesis. Princeton, NJ: Princeton University Press, 2003.

[6] S. Smale J. Math. Biol. 1976 5 7

[7] S. Ellner, P. Turchin Am. Natur. 1995 343 375

[8] V. Volterra Mem. Accad. Nazionale Lincei 1926 31 113

[9] A.J. Lotka . Elements of Mathematical Biology. New York, Dover, 1958.

[10] J.A. Vano, J.C. Wildenberg, M.B. Anderson, J.K. Noel, J.C. Sprott Nonlinearity 2006 2391 2404

[11] R. Wang, D. Xiao Nonlin Dyn 2010 411 422

[12] E.C. Zeeman, M.L. Zeeman Nonlinearity 2002 2019 2032

[13] M.W. Hirsch Nonlinearity 1988 51 71

[14] M.W. Hirsch J. Differ. Equ. 1989 94 106

[15] M.W. Hirsch SIAM J. Math. Anal. 1990 1225 1234

[16] M.L. Zeeman Dyn. Stab. Sys. 1993 189 217

[17] M. Gyllenberg, P. Yan, Y. Wang Appl. Math. Lett. 2006 1 7

[18] J. Hofbauer, J.W.H. So Appl. Math. Lett. 1994 65 70

[19] Z. Lu, Y. Luo Comput. Math. Appl. 2003 231 238

[20] R.M. May, W.J. Leonard SIAM J. Appl. Math. 1975 243 253

[21] P.V.D. Driessche, M.L. Zeeman SIAM J. Appl. Math. 1998 227 234

[22] D. Xiao, W. Li J. Differ. Equ. 2000 1 15

[23] A. Arneodo, P. Coullet, C. Tresser Phys. Lett. A 1980 259 263

[24] A. Arneodo, P. Coullet, J. Peyraud, C. Tresser J. Math. Biol. 1982 153 157

[25] A. Hastings, T. Powell Ecology 1991 896 903

[26] A. Klebanoff, A. Hastings J. Math. Biol. 1994 427 451

[27] K. Mccann, P. Yodzis Ecology 1994 561 564

[28] D. V. Vayenas, S. Pavlou Ecol. Model. 2001 285 295

[29] S. Abbas, D. Bahuguna, M. Banerjee Non. Anal. Hyb. Syst. 2009 195 206

[30] S.L. Pimm. The balance of nature ? Ecological issue in the conservation of species and communities. University of Chicago Press, Chicago, 1991.

[31] J.H. Steele Nature 1985 355 358

[32] P.S. Mandal, M. Banerjee Physica A 2012 1216 1233

[33] D. Nyccka, S. Ellner, D. Mccaffrey, A.R. Gallant J. Roy. Statist. Soc. B. 1992 399 426

[34] M. Bandyopadhyay J. Appl. Math. Comput. 2008 433 450

[35] D. Valenti, A. Flasconaro, B. Spagnolo Physica. A. 2004 477 486

[36] N.V. Agudov, B. Spagnolo Phys. Rev. E. 2001

[37] C.V.D. Broeck, J.M.R. Parrondo, R. Toral, R. Kawai Phys. Rev. E. 1997 4084 4094

[38] H. Berry Phys. Rev. E. 2003 031907

[39] J. Li, P. Hanggi Phys. Rev. E. 2001 011106

[40] R. Arditi, L.R. Ginzburg J. Theor. Biol. 1989 311 326

[41] M. Bandyopadhyay, R. Bhattacharya, C.G. Chakrabarti. A nonlinear two species oscillatory system: Bifurcation and stability analysis. Int. J. Math. Sci., (2003) 1981–1991.

[42] C.S. Elton. The pattern of Animal Communities. London, Methuen, 1966.

[43] W. Horsthemke, R. Lefever. Noise Induced Transitions. Springer-Verlag, Berlin, 1984.

[44] C.W. Gardiner. Handbook of Stochastic Methods. Springer-Verlag, New York, 1983.

[45] T.C.Gard. Introduction to Stochastic Differential Equations. Marcel Decker, New York, 1987.

[46] I.I. Gikhman, A.V. Skorokhod. The Theory of Stochastic Process-I. Berlin, Springer, 1979.

[47] X. Mao, G. Marion, E. Renshaw Stoc. Proc. Appl. 2002 95 110

[48] D.J. Higham SIAM Rev. 2001 525 546

[49] V.B. Kolmanovskii, L.E. Shaikhet Appl. Math. Lett. 2002 355 360

[50] V.B. Kolmanovskii, L.E. Shaikhet Math. Comp. Model. 2002 691 716

[51] L. Arnold. Stochastic Differential Equations: Theory and Applications. Wiley, New York, 1972.

[52] X. Mao. Stochastic Differential Equations and Applications. Horwood, New York, 1997.

[53] R.Z. Khasminskii. Stochastic Stability of Differential Equations. Sijthoff and Noordhoff, Alphen a/d Rijn, 1981.

[54] X. Mao. Stability of Stochastic Differential Equations with respect to Semimartingales. Longman Scientific and Technical, New York, 1991.

[55] X. Mao. Exponential Stability of Stochastic Differential Equations. Marcel Dekker, New York, 1994.

[56] N. Dalal, D. Greenhalgh, X. Mao J. Math. Anal. Appl. 2008 1084 1101

[57] E. Allen. Modeling With Itô Stochastic Differential Equations. Dordrecht, The Netherlands, 2007.

[58] V. Hutson, J.S. Pym. Applications of Functional Analysis and Operator Theory. Academic Press, London, 1980.

[59] A.N. Kolmogorov, S. V. Fomin. Introductory Real Analysis. Dover Publications, Inc., New York, 1970.

[60] M. Liu, K. Wang, Q. Wu Bull. Math. Biol. 2011 1969 2012

[61] P.S. Mandal, M. Banerjee. Deterministic Chaos vs. Stochastic Fluctuation in an Eco-epidemic Model. Math. Model. Nat. Phenom., (In press), (2012).