Series solution of the multispecies Lotka-Volterra equations by means of the homotopy analysis method.

*(English)*Zbl 1160.34302##### MSC:

34A25 | Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc. |

92D25 | Population dynamics (general) |

##### References:

[1] | A. J. Lotka, Elements of Physical Biology, Williams & Wilkins, Baltimore, Md, USA, 1925. · JFM 51.0416.06 |

[2] | V. Volterra, “Variazioni e fluttuazioni del numero d’individui in specie animali conviventi,” Memorie della Reale Accademia dei Lincei, vol. 2, pp. 31-113, 1926. · JFM 52.0450.06 |

[3] | R. M. May and W. J. Leonard, “Nonlinear aspects of competition between three species,” SIAM Journal on Applied Mathematics, vol. 29, no. 2, pp. 243-253, 1975. · Zbl 0314.92008 |

[4] | E. C. Pielou, An Introduction to Mathematical Ecology, Wiley-Interscience, New York, NY, USA, 1969. · Zbl 0259.92001 |

[5] | V. W. Noonburg, “A neural network modeled by an adaptive Lotka-Volterra system,” SIAM Journal on Applied Mathematics, vol. 49, no. 6, pp. 1779-1792, 1989. · Zbl 0684.92008 |

[6] | Z. Noszticzius, E. Noszticzius, and Z. A. Schelly, “Use of ion-selective electrodes for monitoring oscillating reactions. 2. Potential response of bromide- iodide-selective electrodes in slow corrosive processes. Disproportionation of bromous and iodous acids. A Lotka-Volterra model for the halate driven oscillators,” Journal of Physical Chemistry, vol. 87, no. 3, pp. 510-524, 1983. |

[7] | K.-I. Tainaka, “Stationary pattern of vortices or strings in biological systems: lattice version of the Lotka-Volterra model,” Physical Review Letters, vol. 63, no. 24, pp. 2688-2691, 1989. |

[8] | A. M. Stuart and A. R. Humphries, Dynamical Systems and Numerical Analysis, vol. 2 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, UK, 1996. · Zbl 1080.35528 |

[9] | A. H. Nayfeh, Perturbation Methods, Wiley Classics Library, John Wiley & Sons, New York, NY, USA, 2000. · Zbl 1082.74523 |

[10] | S. J. Liao, The proposed homotopy analysis techniques for the solution of nonlinear problems, Ph.D. dissertation, Shanghai Jiao Tong University, Shanghai, China, 1992. |

[11] | S. J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, vol. 2 of CRC Series: Modern Mechanics and Mathematics, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2004. · Zbl 1137.90669 |

[12] | S. J. Liao, “An approximate solution technique not depending on small parameters: a special example,” International Journal of Non-Linear Mechanics, vol. 30, no. 3, pp. 371-380, 1995. · Zbl 0837.76073 |

[13] | S. J. Liao, “A kind of approximate solution technique which does not depend upon small parameters-II: an application in fluid mechanics,” International Journal of Non-Linear Mechanics, vol. 32, no. 5, pp. 815-822, 1997. · Zbl 1031.76542 |

[14] | S. J. Liao, “An explicit, totally analytic approximate solution for Blasius’ viscous flow problems,” International Journal of Non-Linear Mechanics, vol. 34, no. 4, pp. 759-778, 1999. · Zbl 1342.74180 |

[15] | S. J. Liao, “On the homotopy analysis method for nonlinear problems,” Applied Mathematics and Computation, vol. 147, no. 2, pp. 499-513, 2004. · Zbl 1086.35005 |

[16] | S. J. Liao and I. Pop, “Explicit analytic solution for similarity boundary layer equations,” International Journal of Heat and Mass Transfer, vol. 47, no. 1, pp. 75-85, 2004. · Zbl 1137.90669 |

[17] | S. J. Liao, “Comparison between the homotopy analysis method and homotopy perturbation method,” Applied Mathematics and Computation, vol. 169, no. 2, pp. 1186-1194, 2005. · Zbl 1082.65534 |

[18] | S. J. Liao, “A new branch of solutions of boundary-layer flows over an impermeable stretched plate,” International Journal of Heat and Mass Transfer, vol. 48, no. 12, pp. 2529-2539, 2005. · Zbl 1137.90012 |

[19] | S. J. Liao and Y. Tan, “A general approach to obtain series solutions of nonlinear differential equations,” Studies in Applied Mathematics, vol. 119, no. 4, pp. 297-354, 2007. |

[20] | S. J. Liao, “Notes on the homotopy analysis method: some definitions and theorems,” Communications in Nonlinear Science and Numerical Simulation. In press. · Zbl 1221.65126 |

[21] | M. Ayub, A. Rasheed, and T. Hayat, “Exact flow of a third grade fluid past a porous plate using homotopy analysis method,” International Journal of Engineering Science, vol. 41, no. 18, pp. 2091-2103, 2003. · Zbl 1211.76076 |

[22] | T. Hayat, M. Khan, and S. Asghar, “Homotopy analysis of MHD flows of an Oldroyd 8-constant fluid,” Acta Mechanica, vol. 168, no. 3-4, pp. 213-232, 2004. · Zbl 1128.76378 |

[23] | T. Hayat and M. Khan, “Homotopy solutions for a generalized second-grade fluid past a porous plate,” Nonlinear Dynamics, vol. 42, no. 4, pp. 395-405, 2005. · Zbl 1094.76005 |

[24] | M. Sajid and T. Hayat, “Comparison of HAM and HPM methods in nonlinear heat conduction and convection equations,” Nonlinear Analysis: Real World Applications. In press. · Zbl 1156.76436 |

[25] | I. Hashim, O. Abdulaziz, and S. Momani, “Homotopy analysis method for fractional IVPs,” Communications in Nonlinear Science and Numerical Simulation. In press. · Zbl 1221.65277 |

[26] | M. S. H. Chowdhury, I. Hashim, and O. Abdulaziz, “Comparison of homotopy analysis method and homotopy-perturbation method for purely nonlinear fin-type problems,” Communications in Nonlinear Science and Numerical Simulation. In press. · Zbl 1221.80021 |

[27] | Y. Tan and S. Abbasbandy, “Homotopy analysis method for quadratic Riccati differential equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 13, no. 3, pp. 539-546, 2008. · Zbl 1137.62012 |

[28] | S. Abbasbandy, “The application of homotopy analysis method to nonlinear equations arising in heat transfer,” Physics Letters A, vol. 360, no. 1, pp. 109-113, 2006. · Zbl 1236.80010 |

[29] | S. Abbasbandy, “The application of homotopy analysis method to solve a generalized Hirota-Satsuma coupled KdV equation,” Physics Letters A, vol. 361, no. 6, pp. 478-483, 2007. · Zbl 1137.65442 |

[30] | A. S. Bataineh, M. S. M. Noorani, and I. Hashim, “Solving systems of ODEs by homotopy analysis method,” Communications in Nonlinear Science and Numerical Simulation, vol. 13, no. 10, pp. 2060-2070, 2008. · Zbl 1221.65194 |

[31] | A. S. Bataineh, M. S. M. Noorani, and I. Hashim, “Solutions of time-dependent Emden-fowler type equations by homotopy analysis method,” Physics Letters A, vol. 371, no. 1-2, pp. 72-82, 2007. · Zbl 1209.65104 |

[32] | A. S. Bataineh, M. S. M. Noorani, and I. Hashim, “Approximate solutions of singular two-point BVPs by modified homotopy analysis method,” Physics Letters A, vol. 372, no. 22, pp. 4062-4066, 2008. · Zbl 1220.34026 |

[33] | A. S. Bataineh, M. S. M. Noorani, and I. Hashim, “Approximate analytical solutions of systems of PDEs by homotopy analysis method,” Computers & Mathematics with Applications, vol. 55, no. 12, pp. 2913-2923, 2008. · Zbl 1142.65423 |

[34] | A. S. Bataineh, M. S. M. Noorani, and I. Hashim, “Modified homotopy analysis method for solving systems of second-order BVPs,” Communications in Nonlinear Science and Numerical Simulation. In press. · Zbl 1221.65196 |

[35] | A. S. Bataineh, M. S. M. Noorani, and I. Hashim, “On a new reliable modification of homotopy analysis method,” Communications in Nonlinear Science and Numerical Simulation. In press. · Zbl 1221.65195 |

[36] | J.-H. He, “Homotopy perturbation method: a new nonlinear analytical technique,” Applied Mathematics and Computation, vol. 135, no. 1, pp. 73-79, 2003. · Zbl 1030.34013 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.