##
**Deterministic and stochastic analysis of a nonlinear prey-predator system.**
*(English)*
Zbl 1054.92044

From the introduction: The object of the present paper is to develop a stochastic dynamic model of a nonlinear predator-prey ecosystem and to examine the stability and extinction of the system under random perturbations. We also want to make a critical comparative analysis of stability of the model ecosystem under both deterministic and stochastic perturbations. For this we have first considered a deterministic analysis of stability about the nontrivial steady-state and bifurcations.

The second part of the work consists of stochastic dynamic modeling of the prey-predator system by addition of random perturbations leading to nonlinear stochastic differential equations. The system of nonlinear stochastic differential equations has then been reduced to a system of deterministic moment equations by the method of statistical linearization. The solution of the system of moment equations leads to an expression of nonequilibrium fluctuation and criteria of stability of the stationary states of the original prey-predator systems. A comparative study of the stability of the system under both deterministic and stochastic perturbations leads to the characteristic behavior of the stochastic model of the system.

The second part of the work consists of stochastic dynamic modeling of the prey-predator system by addition of random perturbations leading to nonlinear stochastic differential equations. The system of nonlinear stochastic differential equations has then been reduced to a system of deterministic moment equations by the method of statistical linearization. The solution of the system of moment equations leads to an expression of nonequilibrium fluctuation and criteria of stability of the stationary states of the original prey-predator systems. A comparative study of the stability of the system under both deterministic and stochastic perturbations leads to the characteristic behavior of the stochastic model of the system.

### MSC:

92D40 | Ecology |

37N25 | Dynamical systems in biology |

60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |

92D25 | Population dynamics (general) |

34C23 | Bifurcation theory for ordinary differential equations |

### Keywords:

Prey-predator system; stability; Hopf-bifurcation; white noise; non-equilibrium fluctuation; stochastic stability
PDF
BibTeX
XML
Cite

\textit{M. Bandyopadhyay} and \textit{C. G. Chakrabarti}, J. Biol. Syst. 11, No. 2, 161--172 (2003; Zbl 1054.92044)

Full Text:
DOI

### References:

[1] | Arnold L., Stochastic Differential Equations: Theory and Applications (1974) · Zbl 0278.60039 |

[2] | DOI: 10.1007/BF02459962 · Zbl 0608.92024 |

[3] | Banerjee S., Bull. Cal. Math. Soc. 88 pp 235– |

[4] | Banerjee S., Syst. Anal. Mod. Simul. 30 pp 1– |

[5] | Birkhoff G., Ordinary Differential Equations (1982) |

[6] | DOI: 10.1137/0146043 · Zbl 0608.92016 |

[7] | DOI: 10.1007/978-3-662-02377-8 |

[8] | DOI: 10.1007/978-3-642-88338-5 |

[9] | Hassard B. D., Theory and Application of Hopf-bifurcation (1981) · Zbl 0474.34002 |

[10] | Horsthemke W., Noise Induced Transitions (1984) · Zbl 0529.60085 |

[11] | Jumarie G., Journal of Franklin Institute: Engineering and Applied Mathematics |

[12] | May R. M., Stability and Complexity in Model Ecosystems (1974) |

[13] | DOI: 10.1007/b98869 · Zbl 1006.92002 |

[14] | Nicolis G., Self-organization in Nonequilibrium Systems (1977) · Zbl 0363.93005 |

[15] | Nisbet R. M., Modeling Fluctuating Populations (1982) · Zbl 0593.92013 |

[16] | DOI: 10.1890/0012-9658(2000)081[2767:LEPTEF]2.0.CO;2 |

[17] | DOI: 10.1086/282272 |

[18] | Samanta G., J. Math. Phys. Sc. 25 pp 399– |

[19] | DOI: 10.1115/1.3119486 |

[20] | Svirezhev Y. M., Stability of Biological Community (1983) |

[21] | Turelli M., Mathematical Ecology (1986) |

[22] | DOI: 10.1007/BF01009680 · Zbl 0587.60047 |

[23] | Van Kampen N. G., Stochastic Processes in Physics and Chemistry (1981) · Zbl 0511.60038 |

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.