Bifurcation analysis of fish-algae-nutrient interactions in aquatic ecosystems

The overgrowth of algae in lakes often stems from an influx of nutrients from various sources, such as run-off from agricultural areas, anthropogenic and industrial drainage. Phosphorus and nitrogen play a crucial role as catalysts for algae growth, driving their rapid proliferation and leading to the formation of algal blooms. Both herbivorous and carnivorous fish play vital roles in the aquatic food web, and their presence can significantly affect the dynamics of algae within the aquatic ecosystem. Thus, a mathematical model is proposed to investigate the influence of fish on algae-nutrient interactions. For the model formulation, herbivorous fish are considered to depend on algae as their primary food source, while carnivorous fish rely on herbivorous fish for their survival and growth. Our analytical results confirm the existence of one parametric bifurcation, including saddle-node and Hopf bifurcations. Additionally, when the model is transformed into discrete-time intervals, it undergoes a Neimark-Sacker bifurcation. The existence of one parametric bifurcation is shown by considering the maximum uptake rate of nutrients by algae as a bifurcation parameter. Numerical simulations further demonstrate that the proposed model system exhibits two-parametric bifurcations, such as cusp, Bogdanov-Takens, generalized Hopf, Chenciner, and zero-Hopf bifurcations. The basin of stability is used to assess how the initial conditions and parameter values influence the bistability of the proposed mathematical model. This comprehensive analysis of algae-nutrient-fish interactions provides valuable insights into the complex dynamics of aquatic ecosystems, offering a foundation for better understanding and potentially managing algal blooms in aquatic ecosystem.