Dynamics of a PDE model with size-structure: Characterizing the growth and flocculation effects of unicellular algae

The commercial production of algae is of great practical significance to the reduction of environmental pollution, the development of bioenergy and nutritional food. Heterotrophic culture is one of the most important methods in commercial production of algae, and flocculation is usually used to collect algae cells in the reactor. The size of algae cells has a direct impact on the nutrient absorption and the flocculation effect of algae cells. In this paper, a size-structured PDE model considering the size of algae cells is established, which depicts the evolutionary relationship of the growth of algae cells, the nutrient absorption and flocculation effect. In the construction of the model, it is assumed that the number of newborn algae cells is determined by nutrition and the weighted total number of algal cells. The numerical observation shows that the model has complex dynamics, such as the appearances of the backward bifurcation of equilibrium states and bistability. The main theoretical results are as follows: (1) The well-posedness of the solution is obtained by using the fixed point theorem. (2) Under the condition of threshold quantity R0>1, the existence of the coexistence equilibrium state of algae is analyzed. Then, by using the theory of contraction semigroups, the sufficient condition for the local stability of the coexistence equilibrium state of algae is obtained. (3) Sufficient condition for the global stability of the algae-free equilibrium state is given.