A Three-Species Food Chain System with Two Types of Functional Responses

and Applied Analysis 3 The main object of this paper is to investigate the dynamic properties and behaviors of system 1.2 . In this context, the paper is organized as follows. In Section 2, we show the dissipativeness of system 1.2 and find a necessary condition for the mid-predator to survive. The local stabilities of the equilibrium points of system 1.2 are examined in Section 3, and the conditions for persistence of system 1.2 are found out according to the existence of limit cycles in Section 4 and some numerical examples are given to substantiate our theoretical results. Moreover, in Section 5, we provide numerical evidence of the existence of chaotic phenomena by illustrating bifurcation diagrams of system 1.2 and by calculating the largest Lyapunov exponent. Finally, conclusions are given in Section 6. 2. Dissipativeness Obviously, the right-hand sides of system 1.2 are continuous and have continuous partial derivatives on the state space R3 { x, y, z T | x ≥ 0, y ≥ 0, z ≥ 0}. In fact, they are Lipschitzian on R3 and then the solution of system 1.2 with nonnegative initial condition exists and is unique, as the solution of system 1.2 initiating in the nonnegative octant is bounded. Moreover, from 19 , it is easy to show that R3 is an invariant domain of system 1.2 . A system is said to be dissipative if all population initiating in R3 are uniformly limited by their environment 20 . Thus, the dissipativeness of system 1.2 is carried out in the following theorem. Theorem 2.1. System 1.2 is dissipative. Proof. From the first equation of system 1.2 , we get dx t /dt ≤ x t a − bx t . So, by comparison theorem, x t ≤ a/ b Ce−at for all t ≥ 0, where C a− bx0 /x0, which implies that x t ≤ a/b for sufficiently large t. Define V t c2/c1 x t y t c3/c4 z t . Then dV t /dt ≤ a a 1 c2/bc1 − mV t , where m min{1, d1, d2}. So, by comparison theorem, we obtain that V t ≤ a a 1 c2/bc1m − a a 1 c2/bc1m e−mt for t ≥ 0. Thus, c2/c1 x t y t c3/c4 z t ≤ a a 1 c2/bc1m for sufficiently large t, which means that all species are uniformly bounded for any initial value in R3 . Therefore, system 1.2 is dissipative. The following proposition provides a necessary condition for survival of the midpredator in system 1.2 . Proposition 2.2. A necessary condition for the mid-predator species y to survive is d1 < ac2 bα1 a . 2.1 Proof. From the second equation of system 1.2 , we get dy t dt −d1y t c2x t y t α1 x t − c3y t z t α2 y t βz t ≤ −d1y t c2x t y t α1 x t ≤ y t ( −d1 ac2 bα1 a ) ( By Theorem 2.1 ) . 2.2 Then we have y t ≤ y0e, where A −d1 ac2/ bα1 a . Thus, for A < 0, limt→∞y t 0. Hence, 2.1 is a necessary condition for the survival of the mid-predator y. 4 Abstract and Applied Analysis 3. Stability Analysis In order to investigate the stability of the equilibrium points of system 1.2 , first, we consider the following two-dimensional dynamical system: dx t dt x t a − bx t − c1x t y t α1 x t , dy t dt −d1y t c2x t y t α1 x t . 3.1 It is well known that the Kolmogorov theorem is applicable in two-dimensional dynamical system and guarantees the existence of either a stable equilibrium point or stable limit cycle behavior in the positive quadrant of phase space of the system, provided certain conditions are satisfied cf. 20, 21 . Such conditions ensure that the parametric values are biologically relevant. Now, it is observed that subsystem 3.1 is a Kolmogorov system under the condition 0 < α1d1 c2 − d1 < a b . 3.2 From now on, we assume that subsystem 3.1 satisfies the condition 3.2 . By applying the local stability analysis to a Kolmogorov system 3.1 we have the following results 11 . I The equilibrium point E00 0, 0 always exists and is a saddle point. II The equilibrium point E01 a/b, 0 always exists and is a saddle point. III The positive equilibrium point E02 x̃, ỹ exists, where x̃ α1d1 c2 − d1 , ỹ a − bx̃ α1 x̃ c1 , 3.3 and it is a locally asymptotically stable point if the following condition holds: d1 > c2 a − bα1 a bα1 . 3.4 Moreover, the solution to system 3.1 approaches to a stable limit cycle if d1 < c2 a − bα1 / a bα1 . Now, we will study the dynamic behavior of the solution of system 1.2 . First, we think over the stability of equilibrium points of system 1.2 . In fact, there are at most four nonnegative equilibrium points of system 1.2 . The existence conditions of them are mentioned as follows. Abstract and Applied Analysis 5 I The trivial equilibrium point E0 0, 0, 0 and one species equilibrium point E1 a/b, 0, 0 always exist. However, the predators die out in the absence of the prey. Thus the equilibrium points 0, yc, 0 and 0, 0, zc with yc, zc > 0 do not exist. II Two-species equilibrium point E2 x̃, ỹ, 0 exists in the interior of positive quadrant of xy plane under the Kolmogorov condition 3.2 , where x̃ and ỹ are given in 3.3 . On the other hand, the absence of the mid-predator causes no equilibrium point in the xz plane. Moreover, if there exists no prey, then neither y nor z can survive, which means that there is no equilibrium point in the yz plane. III The positive equilibrium point E3 x∗, y∗, z∗ exists in the interior of the first octant if and only if there exists a positive solution to the following algebraic nonlinear simultaneous equations:and Applied Analysis 5 I The trivial equilibrium point E0 0, 0, 0 and one species equilibrium point E1 a/b, 0, 0 always exist. However, the predators die out in the absence of the prey. Thus the equilibrium points 0, yc, 0 and 0, 0, zc with yc, zc > 0 do not exist. II Two-species equilibrium point E2 x̃, ỹ, 0 exists in the interior of positive quadrant of xy plane under the Kolmogorov condition 3.2 , where x̃ and ỹ are given in 3.3 . On the other hand, the absence of the mid-predator causes no equilibrium point in the xz plane. Moreover, if there exists no prey, then neither y nor z can survive, which means that there is no equilibrium point in the yz plane. III The positive equilibrium point E3 x∗, y∗, z∗ exists in the interior of the first octant if and only if there exists a positive solution to the following algebraic nonlinear simultaneous equations: f1 ( x, y, z ) a − bx − c1y α1 x 0, f2 ( x, y, z ) −d1 c2x α1 x − c3z α2 y βz 0, f3 ( x, y, z ) −d2 c4y α2 y βz 0. 3.5 By applying elementary calculation to 3.5 , we obtain that y∗ a − bx∗ α1 x∗ c1 , z∗ c4 − d2 y∗ − d2α2 d2β , 3.6 where x∗ is a positive solution of the quadratic equation px2 qx r 0, p −bc2c4β bc4d1β bc3c4 − bc3d2, q ac2c4β bc4d1α1β − ac4d1β ac3d2 bc3c4α1 − ac3c4 − bc3d2α1, r −ac4d1α1β c1c3d2α2 − ac3c4α1 ac3d2α1. 3.7 Therefore, sufficient conditions for the existence of the positive equilibrium point in the interior of the first octant are easily obtained as follows: q2 − 4pr ≥ 0, 0 < x∗ < a b , 0 < α2d2 c4 − d2 < y∗. 3.8 Now, in order to investigate the stabilities of the equilibrium points, we consider the variational matrix V x, y, z of system 1.2 . Thus, we get the matrix V ( x, y, z ) ⎛ ⎜⎜ ⎝ v11 v12 v13 v21 v22 v23 v31 v32 v33 ⎞ ⎟ ⎠, 3.9 6 Abstract and Applied Analysis where v11 x ∂f1 ∂x f1 a − 2bx − c1α1y α1 x 2 , v12 x ∂f1 ∂y − c1x α1 x , v13 x ∂f1 ∂z 0, v21 y ∂f2 ∂x c2α1y α1 x 2 , v22 y ∂f2 ∂y f2 −d1 c2x α1 x − c3z ( α2 βz ) ( α2 y βz )2 , v23 y ∂f2 ∂z − c3y ( α2 y ) ( α2 y βz )2 , v31 z ∂f3 ∂x 0, v32 z ∂f3 ∂y c4z ( α2 βz ) ( α2 y βz )2 , v33 z ∂f3 ∂z f3 −d2 c4y ( α2 y ) ( α2 y βz )2 , 3.10 and f1, f2, and f3 are in 3.5 . Using the variational matrix V x, y, z , the local stability of system 1.2 near the equilibrium points are obtained as follows. I The trivial equilibrium point E0 is a hyperbolic saddle point. In fact, near E0 0, 0, 0 the prey population is increasing, while both of the predators populations are decreasing. II The equilibrium point E1 a/b, 0, 0 is locally stable if d1 > ac2/ bα1 a . However, under the Kolmogorov condition 3.2 , that is, d1 < ac2/ bα1 a , the point E1 is a saddle point with locally stable manifold in xz plane and with locally unstable manifold in y-direction. III Clearly, the equilibrium point E2 x̃, ỹ, 0 has the same stability behavior as E02 x̃, ỹ in the interior of positive coordinate xy plane. However, the stability of the point E2 is determined by the positive direction orthogonal to the xy plane, that is, z-direction, depending on whether the eigenvalue λ̃3 −d2 c4ỹ/ α2 ỹ is negative or positive, respectively. IV Let V ∗ v∗ i,j be the variational matrix at the equilibrium point E3 x ∗, y∗, z∗ . Then we have v∗ 11 x ∗ −b c1y/ α1 x∗ 2 , v∗ 22 c3yz/ α2 y∗ βz∗ 2 > 0 and v∗ 33 −c4βyz/ α2 y∗ βz∗ 2 < 0 because f1 x∗, y∗, z∗ f2 x∗, y∗, z∗ f3 x∗, y∗, z∗ 0 and v∗ 12 < 0, v ∗ 21 > 0, v ∗ 23 < 0, and v ∗ 32 > 0. Moreover, the characteristic equation of V ∗ is λ3 Aλ2 Bλ C 0, where A − ( v∗ 11 v ∗ 22 v ∗ 33 ) , B v∗ 11v ∗ 33 v ∗ 22v ∗ 33 v ∗ 11v ∗ 22 − v∗ 12v ∗ 21 − v ∗ 23v ∗ 32, C ( v∗ 12v ∗ 21 − v ∗ 11v ∗ 22 ) v∗ 33 v ∗ 11v ∗ 23v ∗ 32. 3.11 By the Routh-Hurwitz criterion 20 , E3 x∗, y∗, z∗ is locally asymptotically stable if and only if A, C, and AB − C are positive. Abstract and Applied Analysis 7 A sufficient condition for the local stability of E3 is given in the following theorem. Theorem 3.1. Suppose that the positive equilibrium point E3 x∗, y∗, z∗ exists in the interior of the positive octant. Then E3 is locally asymptotically stable if the following conditions hold: bx∗ > y∗ ( c1x ∗ φ2 c3z ∗ φ2 ) , 3.12and Applied Analysis 7 A sufficient condition for the local stability of E3 is given in the following theorem. Theorem 3.1. Suppose that the positive equilibrium point E3 x∗, y∗, z∗ exists in the interior of the positive octant. Then E3 is locally asymptotically stable if the following conditions hold: bx∗ > y∗ ( c1x ∗ φ2 c3z ∗ φ2 ) , 3.12 bc3z ∗φ3 < c1 ( c2α1φ 2 c3yzφ ) ,