The qualitative behavior of a plankton-fish interaction model with food limited growth rate and non-constant fish harvesting

Abstract

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.

High codimension bifurcations analysis in a predator-prey system with Michaelis-Menten type predator harvesting

“…To solve this problem, harvesting terms were added to the predator-prey model. Depending on the type of harvesting [19,20,42], there are three types of harvesting: constant-yield harvesting [28,30,39], constant-effort harvesting [1,6], and nonlinear type harvesting [2,10,11,13,37,45], such as Michaelis-Menten type harvesting [16]. Nonlinear type harvesting has a functional relationship of Holling II type with the current number of harvested biological populations and human effort, which has a certain saturation effect on the current number of biological populations and human effort.…”