Modelling hiding behaviour in a predator-prey system by both integer order and fractional order derivatives

Most of the preys are well aware of sensing predation risk. Consequently, to escape from predators they usually adopt several defense mechanisms, specially refuge themselves to become invulnerable. In view of this, a mathematical model has been formulated incorporating prey refuge, where it is assumed that prey refuge is a function of predators availability in the system. It is shown that the model system is well-posed. It has been found that the hiding level and consumption rate of predators have a suitable interrelation between them. Both the parameters act as Hopf bifurcation parameters, but they play opposite role in case of stabilization of the system dynamics. Also, hiding level plays crucial role in maintaining the mean density of both the populations. Furthermore, as hiding behaviour of prey is not instantaneous, so a time delay, namely hiding delay has been introduced to make the model system more realistic and it is observed that the delay parameter destabilizes the system. Modelling approach through fractional calculus has been further deployed to study how the process of forgetting life history influences the dynamical intricacy of the population level dynamics. All the analytical findings have been testified by proper numerical performances.