Dynamics of a predator-prey model with generalized Holling type functional response and mutual interference

Mutual interference and prey refuge are important drivers of predator-prey dynamics. The "exponent" or degree of mutual interference has been under much debate in theoretical ecology. In the present work, we investigate the interplay of the mutual interference exponent, and prey refuge, on the behavior of a predator-prey model with a generalized Holling type functional response - considering in particular the "non-smooth" case. This model can also be used to model an infectious disease where a susceptible population, moves to an infected class, after being infected by the disease. We investigate dynamical properties of the system and derive conditions for the occurrence of saddle-node, transcritical and Hopf-bifurcations. A sufficient condition for finite time extinction of the prey species has also been derived. In addition, we investigate the effect of a prey refuge on the population dynamics of the model and derive conditions such that the prey refuge would yield persistence of the population. We provide additional verification of our analytical results via numerical simulations. Our findings are in accordance with classical experimental results in ecology (Gause, 1934), that show that extinction of predator and prey populations is possible in a finite time period - but that bringing in refuge can effectively yield persistence.