Complex Algal Dynamics and Optimal Control with Algicidal Activity and Reabsorption of Algal Cell Contents

Algaecides utilizing bacteriolytic algae are considered as a promising approach for algae control. These bacteria inhibit the continuous reproduction of algae cells in various ways, including lysing the cells, which leads to the release of cellular contents and affects the levels of nitrogen and phosphorus in the environment. In this paper, we establish a novel mathematical model with algicidal activities and the reabsorption of algal cell contents. The model exhibits complex dynamical phenomena: (i) backward and forward bifurcations; (ii) transcritical bifurcation and saddle-node bifurcation discussed via Sotomayor's theorem; (iii) Hopf bifurcation; (iv) the codimension 2 bifurcations, exemplified by the Bogdanov-Takens bifurcation, via the methodologies of normal form theory and the center manifold theorem. We also obtain an explicit formula for the ultimate lower bound of algal bloom. Sensitivity analysis of the basic ecological reproductive indices $R_0$ is conducted, and the optimal control problem is formulated by integrating environmental factors and physical algal control methods. The analysis indicates that using algicidal bacteria to lyse algal cells can result in two scenarios: algicidal dominance and nutrient supplementation dominance. The former effectively curbs the sustained reproduction of algal cells and is more effective than physical algal control methods.