Dynamics of three-species food chain model with two delays of fear

In this paper, we study the dynamics of a three-species food chain model with two delays duo to fear. First we discuss the permanence of this delayed model, then we discuss dynamics under six cases: (1) τ 1 = τ 2 = 0 , (2) τ 1 > 0 , τ 2 = 0 , (3) τ 2 > 0 , τ 1 = 0 , (4) τ 1 = τ 2 > 0 , (5) τ 1 ∈ ( 0 , τ 10 ) , τ 2 > 0 , (6) τ 2 ∈ ( 0 , τ 20 ) , τ 1 > 0 . We shall study the Hopf bifurcation about case (5) by center manifold theorem and normal form. At last we shall give simulations which reveal the influence of delays and fear on the dynamics of this predator–prey system.