Stability and Hopf Bifurcation in a Modified Holling-Tanner Predator-Prey System with Multiple Delays

and Applied Analysis 3 transformed into the following nondimensional form: dx dt x 1 − x t − τ1 − xy a1 bx c1y , dy dt y [ δ − β t − τ2 x t − τ2 ] , 1.4 where a1 a/K, c1 cr/α, δ s/r, β sh/α are the non-dimensional parameters and they are positive. The main purpose of this paper is to consider the effect of multiple delays on system 1.4 . The local stability of the positive equilibrium and the existence of Hopf bifurcation are investigated. By employing normal form and center manifold theory, the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are determined. Finally, some numerical simulations are also included to illustrate the theoretical analysis. 2. Local Stability and the Existence of Hopf Bifurcation In this section, we study the local stability of each of feasible equilibria and the existence of Hopf bifurcation at the positive equilibrium. Obviously, system 1.4 has a unique boundary equilibrium E1 1, 0 and a unique positive equilibrium E∗ x∗, y∗ , where x∗ −[ a1 − b β 1 − c1 δ] √[ a1 − b β 1 − c1 δ]2 4a1β(bβ c1δ) 2 ( bβ c1δ ) , y∗ δ β x∗. 2.1 The Jacobian matrix of system 1.4 at E1 takes the form J E1 ⎛ ⎜⎝−e − 1 a1 b 0 δ ⎞ ⎟⎠. 2.2 The characteristic equation of system 1.4 at E1 is of the form λ − δ ( λ e−λτ1 ) 0. 2.3 Clearly, the boundary equilibrium E1 1, 0 is unable. 4 Abstract and Applied Analysis Next, we discuss the existence of Hopf bifurcation at the positive equilibrium E x∗, y∗ . Let x t z1 t x∗, y t z2 t y∗, and still denote z1 t and z2 t by x t and y t , respectively, then system 1.4 becomes dx dt a11x t a12y t b11x t − τ1 ∑ i j k≥2 f ijk 1 x yx t − τ1 , dy dt c21x t − τ2 c22y t − τ2 ∑ i j k≥2 f ijk 2 y x t − τ2 y t − τ2 , 2.4