Stability and Bifurcation Analysis of Hassell–Varley Prey–Predator System with Fear Effect

• Published in: International journal of applied and computational mathematics Vol. 6; no. 5
• Main Authors: Kumar, Vikas, Kumari, Nitu
• Format: Journal Article
• Language: English
• Published: New Delhi Springer India 01.10.2020; Springer Nature B.V
• Subjects: Applications of Mathematics; Applied mathematics; Computational mathematics; Computational Science and Engineering; Dynamic stability; Fear; Hopf bifurcation; Limit cycle oscillations; Mathematical and Computational Physics; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Nuclear Energy; Operations Research/Decision Theory; Original Paper; Predator-prey simulation; Predators; Stability analysis; Theoretical; Stable focus; Permanence; Hopf bifurcation; Limit cycle
• ISSN: 2349-5103; 2199-5796
• DOI: 10.1007/s40819-020-00899-y

In this work, we study a prey–predator system with Hassell–Varley functional response, which is a predator-dependent functional response. Our goal is to study the dynamics of the system in the presence of fear of predation risk. Hassell–Varley functional response is used when the predator population forms a fixed number of tight groups like for foraging purposes. Those predators can be terrestrial and aquatic. We assume that the birth rate of prey population reduces due to fear of the predator. Here, we have discussed the existence of biologically relevant equilibria and derived the condition for local asymptotic stability. Further, permanence and persistence are discussed to understand the dynamics of the system. The system undergoes Hopf bifurcation for both: the fear parameter f and interference coefficient γ . Stability and the direction of Hopf bifurcation are also discussed. We observe that limit cycle oscillation can be prevented by increasing the cost of fear of the predator population. Numerical simulation is performed to substantiate our analytical findings.