### Abstract

This paper develops a stochastic population size model for the black-margined pecan aphid. Prajneshu [Prajneshu, A nonlinear statistical model for aphid population growth. J. Indian Soc. Agric. Statist. 51 (1998), p. 73] proposes a novel nonlinear deterministic model for aphid abundance. The per capita death rate in his model is proportional to the cumulative population size, and the solution is a symmetric analytical function. This paper fits Prajneshu's deterministic model to data. An analogous stochastic model, in which both the current and the cumulative aphid counts are state variables, is then proposed. The bivariate solution of the model, with parameter values suggested by the data, is obtained by solving a large system of Kolmogorov equations. Differential equations are derived for the first and second order cumulants, and moment closure approximations are obtained for the means and variances by solving the set of only five equations. These approximations, which are simple for ecologists to calculate, are shown to give accurate predictions of the two endpoints of applied interest, namely (1) the peak aphid count and (2) the final cumulative aphid count.

Original language | English |
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Pages (from-to) | 148-168 |

Number of pages | 21 |

Journal | Mathematical Biosciences |

Volume | 198 |

Issue number | 2 |

DOIs | |

State | Published - Dec 2005 |

### Keywords

- Birth-death processes
- Cumulant truncation
- Normal approximation

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## Cite this

*Mathematical Biosciences*,

*198*(2), 148-168. https://doi.org/10.1016/j.mbs.2005.07.009