Influence of prey refuge and fear effect on fractional order delayed predator-prey model

Abstract We propose a fractional-order predator-prey model with maturation delay in prey species and Beddington-DeAngelis functional response. We have also incorporated a prey refuge and fear effect. There is a research gap as very few analyses have examined the impacts of fractional order, prey refuge, and fear on population fluctuation in the presence of time delays. In this analysis, the stability behaviour of the coexisting equilibrium is explored, taking delay as the bifurcating parameter for the system. Our primary focus is to check whether time delay, fractional order, prey refuge, and fear effect can stabilize a destabilized system or destabilize a stable system when the parameters are varied. We observe that (i) fractional order has a destabilizing effect, (ii) time delay has a destabilizing effect for fixed memory, (iii) prey refuge has a stabilizing effect; however, switching of instability may also occur, and (iv) depending on the other parameter set fear factor may have stabilizing effect, destabilizing effect, and switching of instability. All the theoretical analyses are confirmed by extensive numerical simulation.