Mathematical assessment of the role of non-linear birth and maturation delay in the population dynamics of the malaria vector

• Published in: Applied mathematics and computation Vol. 217; no. 7; pp. 3286 - 3313
• Main Authors: Ngwa, Gideon A., Niger, Ashrafi M., Gumel, Abba B.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 01.12.2010; Elsevier
• Subjects: Birth; Birth rate function; Computer simulation; Delay; Dynamic tests; Exact sciences and technology; Hopf bifurcation; Mathematical analysis; Mathematical models; Mathematics; Maturation delay; Nonlinear dynamics; Numerical analysis; Numerical analysis. Scientific computation; Ordinary differential equations; Sciences and techniques of general use; Vector population dynamics; Vectors (mathematics); Vector population dynamics; Birth rate function; Hopf bifurcation; Maturation delay; Differential equation; Delay equation; Numerical analysis; Applied mathematics; Ecosystem; Oscillation; Numerical simulation; Population dynamics; Dynamic model; Limit cycle
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2010.08.062

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.