Phytoplankton are single celled organisms capable of phytosynthesis, and are present in all the major oceans and lakes in the world. Phytoplankton contribute to 50% of the total primary production on Earth, and are the dominating primary producer in most aquatic ecosystems. This thesis is based on the 1D deterministic model by Jäger et. al. (2010) which models phytoplankton dynamics in freshwater lakes, where phytoplankton growth is limited by the availability of light and phosphorus. The original model is here extended to two dimensions to include a horizontal dimension as well as a vertical dimension, in order to simulate phytoplankton dynamics under varying lake bottom topographies. The model was solved numerically using a grid transform and a finite volume method in MATLAB. Using the same parameter settings as the 1D case studied by Jäger et. al. (2010), an initial study of plankton dynamics was done by varying the horizontal and vertical diffusion coefficients independently.