Stability analysis for the phytoplankton-zooplankton model with depletion of dissolved oxygen and strong Allee effects

The photosynthetic activity of phytoplankton in the seas is responsible for an estimated 50-80 % of the world's oxygen generation. Both phytoplankton and zooplankton require some of this synthesized oxygen for cellular respiration. This study aims to better understand how the oxygen-phytoplankton dynamics are altered due to the Allee effect in phytoplankton development, particularly when considering the time-dependent oxygen generation rate. The dynamic analysis of the model is dedicated to finding the possible equilibrium points. The analysis reveals that three equilibrium points can be obtained. The stability study demonstrates that one of the equilibrium points is always stable. The remaining equilibrium points are stable under specific conditions. We also identify bifurcations originating from these equilibrium points, including transcritical, pitchfork, and Hopf bifurcation. We derive conditions for stable limit cycles (supercritical Hopf bifurcation) and, in some cases, establish the non-existence of solutions. Numerical simulations are performed to validate our theoretical findings. Furthermore, it is noted that the Allee threshold for the phytoplankton population (k(0)) significantly influences the overall dynamics of the system. When k(0) < 0.001, the population of plankton is at risk of extinction. On the other hand, when 0.001 < k(0) < 0.01, the population of zooplankton is at risk of extinction. When 0.01 < k(0) < 2, the solution reaches a stable condition of coexistence. Conversely, when k(0) >= 2.1, the solution exhibits periodic attractor behaviour.