Dynamics of a Mosquito Egg-Larvae Model with Seasonality

Abstract

We propose a two stages mosquito egg-larvae model with seasonality as a simplification of a four stages one. For the simplified model we characterize the dynamics in terms of the vectorial reproduction number, [Formula: see text], obtaining extinction if [Formula: see text] and convergence to a unique positive periodic orbit if [Formula: see text]. We illustrate each case with an example, by providing general conditions on the periodic coefficients for its occurrence. These examples are further developed using numerical simulations where the periodic parameters satisfy the conditions obtained. In the [Formula: see text] case, real climatic data is used for inferring the parameter behaviour. For the four stage system, using alternative oviposition rate functions, we present a result which generalizes others given for models with delays and even with diffusion to the case in which competition between the larvae is introduced. The analytical study of our initial four stages system when [Formula: see text] remains open, since we were not able to prove that in this case the system is dissipative.

Keywords: Monotone systems; Mosquito life cycle; Periodic attractor; Systems with seasonality; Vectorial reproduction number.

Fig. 1

Average maximum temperature and average precipitation per month in a Tropical region (Brasilia) and in a temperate region (Valencia). Reference years 1981–2010 in both cases (Color Figure Online)

Fig. 2

Time dependence of parameters in scenarios 1 (left) and 2 (right). Functions have been normalized to compare purely their periodic behaviours (Color Figure Online)

Fig. 3

Evolution of the egg and larva population in the first scenario plotted against time (left) and its evolution in the phase space (right) (Color Figure Online)

Fig. 4

Evolution of the egg and larva population in the second scenario plotted against time (left) and its evolution in the phase space (right) (Color Figure Online)

Fig. 5

Phase space of solutions to system (1) for different initial conditions in scenarios 1 (left) and 2 (right) (Color Figure Online)

Fig. 6

Solution of the linear version of system (1), system (9), in the Example 2 setting plotted against time (left). Phase space of the solution (right) (Color Figure Online)