Impact of Michaelis–Menten type harvesting in a Lotka–Volterra predator–prey system incorporating fear effect

Abstract

Abstract We propose and study a Lotka–Volterra predator–prey system incorporating both Michaelis–Menten-type prey harvesting and fear effect. By qualitative analysis of the eigenvalues of the Jacobian matrix we study the stability of equilibrium states. By applying the differential inequality theory we obtain sufficient conditions that ensure the global attractivity of the trivial equilibrium. By applying Dulac criterion we obtain sufficient conditions that ensure the global asymptotic stability of the positive equilibrium. Our study indicates that the catchability coefficient plays a crucial role on the dynamic behavior of the system; for example, the catchability coefficient is the Hopf bifurcation parameter. Furthermore, for our model in which harvesting is of Michaelis–Menten type, the catchability coefficient is within a certain range; increasing the capture rate does not change the final number of prey population, but reduces the predator population. Meanwhile, the fear effect of the prey species has no influence on the dynamic behavior of the system, but it can affect the time when the number of prey species reaches stability. Numeric simulations support our findings.

Keywords

• Lotka–Volterra predator–prey model
• Michaelis–Menten-type harvesting
• Stability
• Fear effect