Abstract
We propose a patch type model for mosquitoes that have aquatic larvae inhabiting ponds. Partial differential equations (PDEs) model the larvae on each of several disconnected patches representing the ponds, with conditions varying in each patch, coupled via the adults in the air. From the PDEs a scalar delay differential equation, with multiple delays, for the total adult mosquito population is derived. The various delays represent the larval development times in the patches. The coefficients contain all the relevant information about the sizes and geometry of the individual patches inhabited by the larvae, the boundary conditions applicable to those patches and the diffusivity of the larvae in each patch. For patches of general shapes and sizes, and without the need to specify the criteria by which an adult mosquito selects an oviposition patch, the modern theory of monotone dynamical systems and persistence theory enables a complete determination of the conditions for the mosquito population to go extinct or to persist. More detailed biological insights are obtained for the case when the patches are squares of various sizes, which allows a detailed discussion of the effects of scale, and for two particular criteria by which mosquitoes might select patches for oviposition, being (i) selection based solely on patch area, and (ii) selection based both on area and expected larval survival probability for each patch. In some parameter regimes, counterintuitive phenomena are predicted.
