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.
Highlights
1 Introduction The aim of this paper is investigating the dynamic behavior of the following Lotka– Volterra predato–prey system incorporating both Michaelis–Menten-type harvesting and fear effect of the prey: du =
We have discussed the dynamics of a prey–predator system where the prey is provided with fear effect and harvesting at Michaelis–Menten-type rate
For system (1.7) without the fear effect, we obtained that the trivial equilibrium was always a saddle, the axial equilibrium was unique, and the number of the interior equilibria depended on the expression η + α qE mm2 rr1
Summary
B1x c1y d1 + y2 dy dt = y(a2 – b2y) – q2Emy. The author showed that if the system admits a unique positive equilibrium, it is globally asymptotically stable. For the limited harvesting case, the author showed that the system admits a unique globally stable positive equilibrium. Chandra, and Banerjee [10] incorporated the Michaelis–Menten-type harvesting to the predator–prey model, which has led to the following model: dx x dt = r0x They showed that the system has at most two interior equilibria and can have saddle-node, transcritical, Hopf–Andronov, and Bogdanov–Takens bifurcations. Remark 2.1 As was pointed out by Wang, Zanette, and Zou [1], for system (1.2), the positive equilibrium is locally asymptotically stable as long as it exists (see Theorem 3.1 in [1]). Remark 3.1 Theorem A shows that for system (1.2), the positive equilibrium is globally asymptotically stable as long as it exists. If the species in a system without harvesting can coexist in a stable state, Theorem 3.1(2) shows that for small enough catching ability q, the system can still coexist in a stable state
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