Dynamics of a nonlocal phytoplankton species with nonlinear boundary conditions

In this paper, a nonlocal reaction–diffusion–advection model with nonlinear boundary conditions arising from phytoplankton species is investigated, where the species depends solely on light for its metabolism. Firstly, we prove that a single phytoplankton model with nonlinear boundary conditions is a strongly monotone dynamical system with respect to a non-standard cone related to the cumulative distribution functions, and further determine the global dynamics of single phytoplankton species in terms of boundary reaction functions. Secondly, we obtain the existence and uniqueness of positive steady states in terms of the critical death rate of the phytoplankton species by using the Lyapunov–Schmidt reduction method. Thirdly, we analyze the effects of advection rate, diffusion rate and water column depth on critical death rate under some conditions. Finally, we investigate the asymptotic behaviors of positive steady state for large advection rate.