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, R0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R_0$$\\end{document}, obtaining extinction if R0≤1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R_0\\le 1$$\\end{document} and convergence to a unique positive periodic orbit if R0>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R_0>1$$\\end{document}. 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 R0>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R_0>1$$\\end{document} 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 R0≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R_0\\ge 1$$\\end{document} remains open, since we were not able to prove that in this case the system is dissipative.