We divide our population at time t into four compartments: eggs E(t), immature I(t), host-seeking H(t), and gravid G(t), where each compartment describes the size of that population at time t. The immature compartment is composed of the larval and pupal stages; these two stages were lumped together due to lack of data to distinguish between them. Adult female mosquitoes enter a reproductive cycle of mating, blood-feeding, and ovipositing. We consider mosquitoes who are mating and blood-feeding as host-seeking H, while mosquitoes who are seeking to oviposit are gravid G. Thus, the H and G compartments only contain female mosquitoes. As temperature and precipitation have a significant effect on the mosquito population, we will choose functional forms for some of the transfer and transition rates in our model to accurately describe the dynamics of the mosquito population at an average trapping site in Knox County, Tennessee for the summer of 2013.

Seen in our diagram below, Fig 1, we assume that γ is the rate at which gravid mosquitoes are ovipositing at time t, hence γG is the rate of ovipositing eggs. We assume that when a gravid mosquito oviposits, there are j eggs laid on average. Thus, jγG is on average the rate of the number of eggs laid by a gravid mosquito per day. With the development rate of eggs, k, and the immature development rate of immatures, g, we have that kE is the transition of eggs into immature and gI is the transition rate of immatures into adults. We assume that there is a constant sex ratio in the population, where ρ is the proportion of females in the population. Thus, ρgI is the transition rate of female immatures to host-seeking H, as we are only considering female mosquitoes in the host-seeking and gravid compartments. Note that (1 − ρ)gI is the rate of males leaving the I compartment, as the females transition to the H compartment. We have φH as the rate at which host-seeking mosquitoes finish mating/blood-feeding and become gravid G. Also, γG is the rate that gravid mosquitoes finish resting/ovipositing and become host-seeking H. We denote the death rates as λE, λl, λH, and λG for eggs, immature, host-seeking, and gravid compartments, correspondingly.

We assume that the oviposition rate, j, of eggs depends on accumulated precipitation P, since mosquitos need standing water to lay eggs and to have eggs hatch. The development rate of eggs, k, depends on degree days DD, and the immature development rate, g, depends on both accumulated precipitation, P, and degree days, DD. The structure of the functional forms of j, k and γ will be chosen from the corresponding biological mechanisms and our environmental and mosquito data. Below is our system of differential equations for our population model associated with our diagram, Fig 1.

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