Dynamical analysis of a prey–predator model with Beddington–DeAngelis type function response incorporating a prey refuge

This paper discusses a prey–predator model with reserved area. The feeding rate of consumers (predators) per consumer (i.e., functional response) is considered to be Beddington–DeAngelis type. The Beddington–DeAngelis functional response is similar to the Holling type II functional response but contains an extra term describing mutual interference by predators. We investigate the role of reserved region and degree of mutual interference among predators in the dynamics of system. We obtain different conditions that affect the persistence of the system. We also discuss local and global asymptotic stability behavior of various equilibrium solutions to understand the dynamics of the model system. The global asymptotic stability of positive interior equilibrium solution is established using suitable Lyapunov functional. It is found that the Hopf bifurcation occurs when the parameter corresponding to reserved region (i.e., m) crosses some critical value. Our result indicates that the predator species exist so long as prey reserve value (m) does not cross a threshold value and after this value the predator species extinct. To mimic the real-world scenario, we also solve the inverse problem of estimation of model parameter (m) using the sampled data of the system. The results can also be interpreted in different contexts such as resource conservation, pest management and bio-economics of a renewable resource. At the end, we perform some numerical simulations to illustrate our analytical findings.