Sea lice infection is one of the major threats in the marine fishery, especially for farmed salmon. In this work, we propose two mathematical models, one is for the growth of sea lice with a three-stage structure: non-infectious immature, infectious immature and adults where the level of non-infectious immature development depends on the size of the adult population; another is for the control of sea lice with the predator-prey interaction between cleaner fish and sea lice. By using mathematical techniques and an appropriate change of variables, we first describe the nonlinear dynamics by a system of partial differential equations, then transform it into a system of delay differential equations with constant delay. We provide two important indexes: the adult reproduction number Rs for sea lice and the net reproductive number of cleaner fish Rf ,  address the global dynamics  relating to Rs and Rf theoretically, including the global/local stability of the equilibria, uniformly persistence and possible Hopf bifurcation. Numerical simulations are provided to confirm the theoretical results.

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