Dynamic analysis of a biocontrol of sea lice by age-structured model

Due to the responsibility for many illnesses on farmed salmon and inflicting huge financial losses, sea lice treatment has become one of the highest priorities in aquaculture studies. From the viewpoint of dynamics, we consider three stages in sea louse life cycle and study the predator–prey interaction between cleaner fish and sea lice. Through mathematical analysis, we provide two important indexes: the adult reproduction number Rs\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {R}}_s$$\end{document} for sea lice and the net reproductive number of cleaner fish Rf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {R}}_f$$\end{document}, which address the global dynamics relating to Rs\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {R}}_s$$\end{document} and Rf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {R}}_f$$\end{document} theoretically, including the global/local stability of the equilibria, uniformly persistence and possible Hopf bifurcation. Numerical simulation is provided to show the oscillation behavior in the system.