Impact of predator incited fear and prey refuge in a fractional order prey predator model

In this article, a predator-prey model has been evolved in the form of a system of fractional order differential equations incorporating two important factors, namely, fear factor and prey refuge factor. Here, the fractional calculus has been taken into consideration to investigate the dynamical behaviour of the solutions of the proposed model system as the changes in life cycle of prey species are of memory bound. Biological validation and well-posedness such as positivity and boundedness of solutions of the model system have been proved analytically. Stability analysis of all the feasible equilibrium points of the model system has been performed in a systematic way. Some important dynamical features of the model system (such as transition of stability of the system) have been demonstrated through rigorous numerical simulation. It is observed that our proposed model system experiences Hopf-bifurcation around the interior equilibrium point with respect to both the parameters f and m1, which are linked with amount of predator induced fear and rate of prey refuge, respectively. The system dynamics is more likely to be stable in the framework of fractional order derivative in comparison to integer-order derivative. The high amount of predator induced fear f and prey refuge rate m1 are independently capable to make the system dynamics to be stable in integer order model system. On the other hand, the dynamics of the model system shifts towards the stability from its unstable behaviour when we continuously reduce the order of the model system; especially under the scenario of low level of predator induced fear and prey refuge rate. Thus, our comprehensive mathematical findings reveal the fact that fading memory can play a contributory role towards stable coexistence of the predator-prey system whereas strong memory of the species deteriorates the stable coexistence of the model system.