### Abstract

A delay ordinary deterministic differential equation model for the population dynamics of the malaria vector is rigorously analysed subject to two forms of the vector birth rate function: the Verhulst-Pearl logistic growth function and the Maynard-Smith-Slatkin function. It is shown that, for any birth rate function satisfying some assumptions, the trivial equilibrium of the model is globally-asymptotically stable if the associated vectorial reproduction number is less than unity. Further, the model has a non-trivial equilibrium which is locally-asymptotically stable under a certain condition. The non-trivial equilibrium bifurcates into a limit cycle via a Hopf bifurcation. It is shown, by numerical simulations, that the amplitude of oscillating solutions increases with increasing maturation delay. Numerical simulations suggest that the Maynard-Smith-Slatkin function is more suitable for modelling the vector population dynamics than the Verhulst-Pearl logistic growth model, since the former is associated with increased sustained oscillations, which (in our view) is a desirable ecological feature, since it guarantees the persistence of the vector in the ecosystem.

Original language | English (US) |
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Pages (from-to) | 3286-3313 |

Number of pages | 28 |

Journal | Applied Mathematics and Computation |

Volume | 217 |

Issue number | 7 |

DOIs | |

State | Published - Dec 1 2010 |

Externally published | Yes |

### Keywords

- Birth rate function
- Hopf bifurcation
- Maturation delay
- Vector population dynamics

### ASJC Scopus subject areas

- Computational Mathematics
- Applied Mathematics

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## Cite this

*Applied Mathematics and Computation*,

*217*(7), 3286-3313. https://doi.org/10.1016/j.amc.2010.08.062