Dynamics of a fractional three-species food chain system with mixed functional responses and fear effect

Abstract

In this paper, a novel fractional three-species food chain system with Ivlev-type and ratio-dependent functional responses is proposed. Also the fear effect of the middle predator on the prey is taken into account. First the boundedness of the solutions and the existence of positive equilibrium point are investigated for the non-delayed system, and then the delay-induced Hopf bifurcation is dealt with for the delayed system via feedback control method. Some sufficient Hopf bifurcation conditions are obtained based on Hopf bifurcation theorem and stability theorem for fractional dynamical systems. A simulation example is given to support our theoretical results, and especially the effect of the feedback gains and the level of fear on the stability and bifurcation behavior are discussed by illustrations.