We investigated a nonlinear model of the interaction between nutrients and plankton, which was addressed using a pair of reaction-advection-diffusion equations. Based on numerical analysis, we studied a model without diffusion and sinking terms, and we found that the phytoplankton density (a stable state) increased with the increase of nutrient density. We analyzed the model using a linear analysis technique and found that the sinking of phytoplankton could affect the system. If the sinking velocity exceeded a certain critical value, the stable state became unstable and the wavelength of phytoplankton increased with the increase of sinking velocity. Furthermore, band patterns were also produced by our model, which was affected by the diffusion and sinking of phytoplankton. Thus, the change in the diffusion and sinking of phytoplankton led to different spatial distributions of phytoplankton. All of these results are expected to be useful in the study of plankton dynamics in aquatic ecosystems. 1. Introduction Plankton play an important role in the ecology of the ocean and climate because of their participation in the global carbon cycle at the base of the food chain . In certain environmental conditions, lakes, reservoir, and marine waters may experience plankton or algal blooms [2, 3]. However, the local and global impacts of plankton blooms on water quality, carbon cycling, and climate may be damaging. If nutrient source is abundant, and some conditions are satisfied, blooms may become long-term events that affect ecosystems. Plankton blooms can change the types of species present at the base of the aquatic food web and affect human health. Thus, the study of plankton dynamics is currently of major interest. In the past years, there were many researches on the model between nutrient and phytoplankton and zooplankton [4–6]. A larger number of researchers have attempted to model the relationship between nutrient and phytoplankton and zooplankton, to investigate the dynamics in plankton model. Truscott and Brindley  presented a model for the evolution of phytoplankton and zooplankton populations which resembles models for the behavior of excitable media. Luo  investigated phytoplankton-zooplankton dynamics in periodic environments, where eutrophication was considered. El Saadi and Bah  modeled phytoplankton aggregation using numerical treatment and explored the asymptotic behavior of the model. Banerjee and Venturino  studied a phytoplankton-toxic phytoplankton-zooplankton model and found that the toxic phytoplankton does not drive the
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