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