THE QUALITATIVE BEHAVIOR OF A PLANKTON-FISH INTERACTION MODEL WITH FOOD LIMITED GROWTH RATE AND NON-CONSTANT FISH HARVESTING

The manuscript aims to investigate the qualitative analysis of a plankton-fish interaction with food limited growth rate of plankton population and non-constant harvesting of fish population. The ecological feasibility of population densities of both plankton and fish in terms of positivity and boundedness of solutions is shown. The conditions for the existence of various equilibrium points and their stability are derived thoroughly. This study mainly focuses on how the harvesting affects equilibrium points, their stability, periodic solutions and bifurcations in the proposed system. It is shown that the system exhibits saddle-node bifurcation in the form of a collision of two interior equilibrium points. Existence conditions for the occurrence of Hopf-bifurcation around interior equilibrium points are discussed. Lyapunov coefficients are examined to check the stability properties of these periodic solutions. We have also plotted the bifurcation diagrams for saddle-node, transcritical and Hopf bifurcations. A detailed algorithm for the occurrence of Bogdanov-Takens bifurcation is derived and finally some numerical simulations are also carried out to validate the theoretical results. This work suggests that the harvesting of fish population can change the dynamics of the system, which may be useful for the ecological management.