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

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.