Impact of Fear and Prey Refuge Parameters in a Fuzzy Prey–Predator Model with Group Defense

Prey–predator interactions are perhaps the most ordinarily noticed phenomena in the environment. In this article, we have proposed a three-species prey–predator model incorporating three important factors, namely, prey refuge, group defense, and the growth rate of two prey species which is reduced for the amount of fear of the predator species. All the biological parameters of our system have been presented as fuzzy numbers to make them more realistic. The system has been studied analytically and numerically in the fuzzy sense. Model analyses such as positivity, boundedness, and permanence of the system are investigated. Stability analysis at all equilibrium points of the system has been studied. Hopf bifurcation analysis around the positive interior equilibrium point has been discussed. All the numerical simulations of the system are presented with suitable tables and graphical diagrams by using MATHEMATICA and MATLAB. Numerically, we have seen that the fear effect and prey refuge parameter can stabilize the system from chaos to a stable region. The effect of fear and prey refuge on stability has been analyzed in the numerical section. Stable focus and limit cycle analysis are investigated in crisp as well as fuzzy environment. The system undergoes Hopf bifurcation at the positive equilibrium point when the fear parameter k and refuge parameter m cross the threshold value in crisp as well as fuzzy environment.