No CrossRef data available.
Article contents
Multiplicative-noise Can Suppress Chaotic Oscillation inLotka-Volterra Type Competitive Model
Published online by Cambridge University Press: 12 December 2012
Abstract
Recently, Wang and Xiao studied a four-dimensional competitive Lotka-Volterra systemwithin a deterministic environment in [11]. Withthe help of numerical example they showed the existence of a chaotic attractor through theperiod doubling route. In this paper, we are interested to study the dynamics of the samemodel in presence of environmental driving forces. To incorporate the environmentaldriving force into the deterministic system, we perturb the growth rates of each speciesby white noise terms. Then we prove that the unique positive global solution exists forthe noise added system and the general p-th order moment of it is boundedfor p ≥ 1, which ensures that the solution is stochasticallybounded. It is also shown that the solution of the stochastic system is stochasticallypermanent under some simple conditions. Finally , we demonstrate the noise inducedoscillation for the concerned model with the help of numerical example..
Information
- Type
- Research Article
- Information
- Mathematical Modelling of Natural Phenomena , Volume 7 , Issue 6: Biological oscillations , 2012 , pp. 23 - 46
- Copyright
- © EDP Sciences, 2012
References
Références
Spagnolo, B., Valenti, D., Fiasconaro, A.. Noise in ecosystems: A short review. Math. Bios. Eng., 1 (2004) 185–211. Google ScholarPubMed
R.M. May. Stability and complexity in Model Ecosystems. Princeton University Press, Princeton, NJ, USA, 1973.
J.D. Murray, Mathematical Biology, Berlin, Springer-Verlag, 1989.
E. Renshaw. Modelling Biological Populations in Space and Time. Cambridge University Press, Cambridge, 1993.
P. Turchin. Complex Population Dynamics: A theoretical/empirical synthesis. Princeton, NJ: Princeton University Press, 2003.
Smale, S.. On the differential equations of species in competition. J. Math. Biol., 3 (1976) 5–7. CrossRefGoogle Scholar
Ellner, S., Turchin, P.. Chaos in noisy world: new methods and evidence from time-series analysis. Am. Natur., 145 (1995) 343–375. CrossRefGoogle Scholar
Volterra, V.. Variazione e fluttuazione del numero d’individui in specie animali convinenti. Mem. Accad. Nazionale Lincei, 2 (1926) 31–113. Google Scholar
A.J. Lotka . Elements of Mathematical Biology. New York, Dover, 1958.
Vano, J.A., Wildenberg, J.C., Anderson, M.B., Noel, J.K., Sprott, J.C.. Chaos in low-dimensional Lotka-Volterra Models of competition. Nonlinearity, 19 (2006) 2391–2404. CrossRefGoogle Scholar
Wang, R., Xiao, D.. Bifurcations and chaotic dynamics in a 4-dimensional competitive Lotka-Volterra system. Nonlin Dyn, 59 (2010) 411–422. CrossRefGoogle Scholar
Zeeman, E.C., Zeeman, M.L.. An n-dimensional competitive Lotka-Volterra system is generically determined by the edges of its carrying simplex. Nonlinearity , 15 (2002) 2019–2032. CrossRefGoogle Scholar
Hirsch, M.W.. Systems of differential equations which are competitive or cooperative III: Competing species. Nonlinearity 1 (1988) 51–71. CrossRefGoogle Scholar
Hirsch, M.W.. Systems of differential equations which are competitive or cooperative V: Convergence in 3-dimensional systems. J. Differ. Equ., 80 (1989) 94–106. CrossRefGoogle Scholar
Hirsch, M.W.. Systems of differential equations which are competitive or cooperative IV: Structural stability in three dimensional systems. SIAM J. Math. Anal., 21 (1990) 1225–1234. CrossRefGoogle Scholar
Zeeman, M.L.. Hopf bifurcations in competitive three dimensional Lotka-Volterra systems. Dyn. Stab. Sys., 8 (1993) 189–217. Google Scholar
Gyllenberg, M., Yan, P., Wang, Y.. A 3D competitive Lotka-Volterra system with three limit cycles: A falsification of a conjecture by Hofbauer and So. Appl. Math. Lett., 19 (2006) 1–7. CrossRefGoogle Scholar
Hofbauer, J., So, J.W.H.. Multiple limit cycles for three dimensional Lotka-Volterra equations. Appl. Math. Lett., 7 (1994) 65–70. CrossRefGoogle Scholar
Lu, Z., luo, Y.. Three limit cycles for a three-dimensional Lotka-Volterra competitive system with a heteroclinic cycle. Comput. Math. Appl., 46 (2003) 231–238. CrossRefGoogle Scholar
May, R.M., Leonard, W.J.. Nonlinear aspects of competition between three species. SIAM J. Appl. Math., 29 (1975) 243–253. CrossRefGoogle Scholar
Driessche, P.V.D., Zeeman, M.L.. Three-dimensional competitive Lotka-Volterra systems with no periodic orbits. SIAM J. Appl. Math., 58 (1998) 227–234. Google Scholar
Xiao, D., Li, W.. Limit cycles for the competitive three dimensional Lotka-Volterra systems. J. Differ. Equ., 164 (2000) 1–15. CrossRefGoogle Scholar
Arneodo, A., Coullet, P., Tresser, C.. Occurrence of strange attractors in three-dimensional Volterra equations. Phys. Lett. A, 79 (1980) 259–263. CrossRefGoogle Scholar
Arneodo, A., Coullet, P., Peyraud, J., Tresser, C.. Strange attractors in Volterra equations for species in competition. J. Math. Biol., 14 (1982) 153–157. CrossRefGoogle Scholar
Hastings, A., Powell, T.. Chaos in a three-species food chain. Ecology, 72 (1991), 896–903. CrossRefGoogle Scholar
Klebanoff, A., Hastings, A.. Chaos in three species food chains. J. Math. Biol., 32 (1994), 427–451. CrossRefGoogle Scholar
McCann, K., Yodzis, P.. Biological conditions for chaos in a three–species food chain. Ecology, 75 (1994), 561–564. CrossRefGoogle Scholar
Vayenas, D. V., Pavlou, S.. Chaotic dynamics of a microbial system of coupled food chains. Ecol. Model., 136 (2001), 285–295. CrossRefGoogle Scholar
Abbas, S., Bahuguna, D., Banerjee, M.. Effect of stochastic perturbation on a two species competitive model. Non. Anal. Hyb. Syst., 3 (2009) 195–206. CrossRefGoogle Scholar
S.L. Pimm. The balance of nature ? Ecological issue in the conservation of species and communities. University of Chicago Press, Chicago, 1991.
Steele, J.H.. A comparison of terrestrial and marine ecological systems. Nature, 313 (1985) 355–358. CrossRefGoogle Scholar
Mandal, P.S., Banerjee, M.. Stochastic persistence and stationary distribution in a Holling-Tanner type prey-predator model. Physica A, 391 (2012) 1216–1233. CrossRefGoogle Scholar
Nyccka, D., Ellner, S., Mccaffrey, D., Gallant, A.R.. Finding chaos in noisy systems. J. Roy. Statist. Soc. B., 54 (1992) 399–426. Google Scholar
Bandyopadhyay, M.. Effect of environmental fluctuation on a detritus based Ecosystem. J. Appl. Math. Comput., 26 (2008) 433–450. CrossRefGoogle Scholar
Valenti, D., Flasconaro, A., Spagnolo, B.. Stochastic resonance and noise delayed extinction in a model of two competing species. Physica. A., 331 (2004) 477–486. CrossRefGoogle Scholar
Agudov, N.V., Spagnolo, B.. Noise-enhanced stability of periodically driven metastable states. Phys. Rev. E., 64 (2001) 035102(R). CrossRefGoogle ScholarPubMed
Broeck, C.V.D., Parrondo, J.M.R., Toral, R., Kawai, R.. nonequilibrium phase transitions induced by multiplicative noise. Phys. Rev. E., 55 (1997) 4084–4094. CrossRefGoogle Scholar
Berry, H.. Nonequilibrium phase transition in a self-activated biological network. Phys. Rev. E., 67 (2003) 031907. CrossRefGoogle Scholar
Li, J., Hanggi, P.. Spatially periodic stochastic system with infinite globally coupled oscillators. Phys. Rev. E., 64 (2001) 011106. CrossRefGoogle ScholarPubMed
Arditi, R., Ginzburg, L.R.. Coupling in predator-prey dynamics: ratio-dependence. J. Theor. Biol., 139 (1989) 311–326. CrossRefGoogle Scholar
M. Bandyopadhyay, R. Bhattacharya, C.G. Chakrabarti. A nonlinear two species oscillatory system: Bifurcation and stability analysis. Int. J. Math. Sci., (2003) 1981–1991.
C.S. Elton. The pattern of Animal Communities. London, Methuen, 1966.
W. Horsthemke, R. Lefever. Noise Induced Transitions. Springer-Verlag, Berlin, 1984.
C.W. Gardiner. Handbook of Stochastic Methods. Springer-Verlag, New York, 1983.
T.C.Gard. Introduction to Stochastic Differential Equations. Marcel Decker, New York, 1987.
I.I. Gikhman, A.V. Skorokhod. The Theory of Stochastic Process-I. Berlin, Springer, 1979.
Mao, X., Marion, G., Renshaw, E.. Environmental Brownian noise suppresses explosions in population dynamics. Stoc. Proc. Appl., 97 (2002) 95–110. CrossRefGoogle Scholar
Higham, D.J.. An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev., 43 (2001) 525–546. CrossRefGoogle Scholar
Kolmanovskii, V.B., Shaikhet, L.E.. Some peculiarities of the general method of Lyapunov functionals construction. Appl. Math. Lett., 15 (2002) 355–360. CrossRefGoogle Scholar
Kolmanovskii, V.B., Shaikhet, L.E.. Construction of Lyapunov functionals for stochastic hereditary systems: a survey of some recent results. Math. Comp. Model., 36 (2002) 691–716. CrossRefGoogle Scholar
L. Arnold. Stochastic Differential Equations: Theory and Applications. Wiley, New York, 1972.
X. Mao. Stochastic Differential Equations and Applications. Horwood, New York, 1997.
R.Z. Khasminskii. Stochastic Stability of Differential Equations. Sijthoff and Noordhoff, Alphen a/d Rijn, 1981.
X. Mao. Stability of Stochastic Differential Equations with respect to Semimartingales. Longman Scientific and Technical, New York, 1991.
X. Mao. Exponential Stability of Stochastic Differential Equations. Marcel Dekker, New York, 1994.
Dalal, N., Greenhalgh, D., Mao, X.. A stochastic model for internal HIV dynamics. J. Math. Anal. Appl., 341 (2008) 1084–1101. CrossRefGoogle Scholar
E. Allen. Modeling With Itô Stochastic Differential Equations. Dordrecht, The Netherlands, 2007.
V. Hutson, J.S. Pym. Applications of Functional Analysis and Operator Theory. Academic Press, London, 1980.
A.N. Kolmogorov, S. V. Fomin. Introductory Real Analysis. Dover Publications, Inc., New York, 1970.
Liu, M., Wang, K., Wu, Q.. Survival analysis of stochastic competitive models in a polluted environment and stochastic competitive exclusion principle. Bull. Math. Biol., 73 (2011) 1969–2012. CrossRefGoogle Scholar
P.S. Mandal, M. Banerjee. Deterministic Chaos vs. Stochastic Fluctuation in an Eco-epidemic Model. Math. Model. Nat. Phenom., (In press), (2012).