Mathematical analysis of a spatio‐temporal model for the population ecology of anopheles mosquito

Abstract

We propose a novel model for the population dynamics of mosquitoes by considering the dispersal states of female mosquitoes of oviposition's cycle and spatial variations. From the modeling perspective, a general functional form of eggs oviposition rate is used including the Malthusian, the Verhlust‐Pearl logistic, the Hassell, and the Maynard‐Smith‐Slatkin functions. From the theoretical and numerical perspectives, the study is done in two steps using the more realistic birth Maynard‐Smith‐Slatkin function. First, we consider an ordinary differential equations model and show that the mosquito‐free equilibrium (MFE) is globally asymptotically stable, whenever the basic offspring number is less than unity. Using a fluctuation argument, we prove that the unique mosquito‐persistent equilibrium (MPE) is globally attractive, whenever exceeds the unity. Moreover, the temporal model undergoes a Hopf bifurcation in the absence of density‐dependent mortality in the aquatic stage of mosquitoes. Second, the temporal model is extended into an advection‐reaction‐diffusion model in order to account for the movement of mosquitoes and their spatial source of heterogeneity. We establish the uniform persistence and the existence of at least one positive steady state whenever the spatial basic offspring number is greater than unity. Finally, for the case study of malaria vector agent (Anopheles mosquito), we construct a nonstandard finite difference scheme that is dynamically consistent with the features of the continuous model to illustrate our results, including the spatial heterogeneity of mosquito resources.