Global Stability and Bifurcation Analysis of a Delayed Predator–prey System with Prey Immigration

A delayed predator–prey system with a constant rate immigration is considered. Local and global stability of the equilibria are studied, a fixed point bifurcation appears near the boundary equilibrium and Hopf bifurcation occurs near the positive equilibrium when the time delay passes some critical values. We also show the existence of the global Hopf bifurcation, and the properties of the fixed point bifurcation and the stability and direction of the Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem.


Introduction
Since the Lotka-Volterra model was first prosed in 1920s, it has been studied in various models.Furthermore, many ecological concepts such as diffusion, functional responses and time delays have been added to the Lotka-Volterra equations to gain more accurate description and better understanding [1,11,[15][16][17][18].In [19] the author studied a Rosenzweig-MacArthur model first.In this model the prey has a logistic growth and the predator has a Holling II functional response.In [12-14, 20, 27] the global stability are discussed.There are also many researches on the limit cycle of Rosenweig-MacArthur model [6,21,26,28].Brauer et al. studied the stability of predator-prey systems with constant rate harvesting and stocking in [2][3][4][5].Sugie et al. discussed the existence and uniqueness of limit cycles in predator-prey systems with a constant immigration in [23].
In this paper, we study the delayed Rosenweig-MacArthur model with a constant rate immigration, which has the following form: Corresponding author.Email: zhuganghit@163.comwhere a, b, d, k, r, µ are all positive constants, and the meaning of them are the same as those in [23], and τ ≥ 0 is the constant delay due to the gestation of the predator.The initial conditions for system (1.1) take the form where (φ 1 (θ), φ 2 (θ)) ∈ C([−τ, 0], R 2 +0 ), the Banach space of continuous functions mapping the interval [−τ, 0] into R 2 +0 , where R 2 +0 = {(x 1 , x 2 ) : x i ≥ 0, i = 1, 2}.Solving the algebraic equation we get that the system (1.1) has equilibria and Combining the biological meaning, we take then the equilibrium E 0 (x 0 , 0) always exists, which means that the predator will extinct.
It is easy to see that if µ < d, then system (1.1) has no positive equilibrium and E 0 (x 0 , 0) is the unique equilibrium of (1.1).
The rest of the present paper is organized as follows: in Section 2, we show the positiveness and boundedness of the solutions of (1.1).In Section 3, we analyze the local stability of E 0 and E * , and the existence of Hopf bifurcation at E * .In Section 4, we study the global stability of the equilibria E 0 and E * .In Section 5, we determine the properties of the bifurcating periodic solution and discuss the existence of the global Hopf bifurcation.In Section 6, some numerical simulations are carried out to illustrate the analytic results.

Positiveness and boundedness of the solutions
In this section, we study the positiveness and boundedness of the solutions of system (1.1).Theorem 2.1.All the solutions of system (1.1) through initial conditions (1.2) are positive for t ≥ 0.
Proof.Solving the following ordinary differential equation we can get the following solution Obviously the solution is positive for all t > 0, furthermore, by the comparison theorem, we know that the solution x(t) of system (1.1) is positive.
Proof.Consider the following ordinary differential equation then all the solutions of equation (2.1) with positive initial conditions are positive, and Since x 0 > k and all the solutions of (2.1) are positive, we have V(t) ≤ 0 and V(t) = 0 if and only if x = x 0 , so the equilibrium x 0 of equation (2.1) is global asymptotically stable.
Basing on the comparison theorem, we know that the solution x(t) of system (1.1) with initial conditions (1.2) satisfies lim sup t→∞ x(t) ≤ x 0 , then for > 0, we have x(t) ≤ x 0 + when t is sufficiently big.Let which implies This completes the proof.

Local stability of the boundary equilibrium
Linearizing system (1.1) near the boundary equilibrium E 0 (x 0 , 0), we get and the characteristic equation 2), so in the following we study the second factor of equation (3.2).
We denote If condition µx 0 /(a + x 0 ) > d holds, then it is easy to show that Hence f (λ) = 0 has at least one positive real root.Therefore, equilibrium E 0 (x 0 , 0) is unstable.
If condition µx 0 /(a + x 0 ) < d holds, we need to consider the effect of the delay τ.
When τ = 0, we get which implies that the equilibrium E 0 (x 0 , 0) is locally asymptotically stable when τ = 0.If λ = iω (ω > 0) be a root of (3.2) when τ > 0, substituting λ = iω into (3.2) and separating the real and the imaginary parts, we have and furthermore, which implies that Eq. (3.2) has no purely imaginary root.Then by theorem in [22], we know that all roots of Eq. (3.2) have negative real part, and the equilibrium E 0 (x 0 , 0) is locally asymptotically stable for all τ ≥ 0.
If condition µx 0 /(a + x 0 ) = d holds, then λ = 0 is a simple root of (3.2).So to determine the stability of E 0 , we need to compute the restriction of system (1.1) on the center manifold.
Here we use the center manifold theorem by [24], and the normal form method from [7,8] Let Λ = {0} and B = 0, clearly the non-resonance conditions relative to Λ are satisfied.Therefore there exists a 1-dimensional ODE which governs the dynamics of system (1.1) near E 0 .
For convenience, we denote d 0 = µx 0 /(a + x 0 ).Firstly, we re-scale the time delay by t → (t/τ) to normalize the delay and let d = d 0 + ε, then ε = 0 is the critical value for the fixed point bifurcation, so system (1.1) can be written in the form: Clearly, the phase space for Eq.(3.3) where then by the Riesz representation theorem, we obtain Furthermore, we choose Then Eq. (3.3) can be rewritten in the form: where u = (u 1 , u 2 ), and Let operator A 0 , which satifies be the infinitesimal generator for the semigroup defined by the solutions of the following equation d dt where L 0 = L(0), and ), here R 2 * denotes the space of row vectors, we consider the adjoint bilinear form on C * × C defined by For the eigenvalue of A 0 , Λ = 0, we use the formal adjoint theory for FDEs to decompose the phase space C by Λ = {0}.Let P be the center space of equation (3.4), the generalized eigenspace for A 0 associated with the eigenvalue zero, and P * the center space of the adjoint equation of (3.4), then the phase space C can be decomposed by Λ = {0} as C = P ⊕ Q, where In fact, letting Thus, we can choose Similarly, let us choose and we can verify that they are the bases of P and P * , respectively, satisfying (Ψ, Φ) = 1.Thus the dual bases satisfy Φ = ΦB and Ψ = −BΨ, where B = 0. Taking the enlarged phase space we obtain the abstract ODE with the form: where and A is an extension of the infinitesimal generator A 0 , defined by A : and X 0 (θ) is given by The definition of the continuous projection allows us to decompose the enlarged space by Λ as BC = C ⊕ Ker π.Since π commutes with A in C 1 , and using the decomposition u t = Φx + y, the abstract ODE (3.5) is therefore decomposed as the system where A Q 1 is the restriction of A as an operator from Q 1 to the Banach space Ker π.And by the expressions of Φ(θ), we get By Taylor's theorem, we expand the nonlinear terms in Eq. (3.6) at (x, y, ε) = (0, 0, 0) as Then the ordinary differential equation for the flow of Eq. (3.6) on the center manifold which is given in normal form up to second order terms by letting y = 0 has the form and it is easy to check that .

Local stability of the positive equilibrium and the existence of Hopf bifurcation
Linearizing system (1.1) near the positive equilibrium E * (x * , y * ), we get and the characteristic equation where When τ = 0, Eq. (3.8) becomes notice that µx * a + x * = d and p 0 + q 0 = ady * (a + x * ) 2 > 0, so we know that both roots of Eq. (3.9) have negative real parts when p 1 + q 1 > 0.
Furthermore, since dλ dτ we can get that Sgn Re dλ dτ which implies that the transversal condition holds.
In conclusion, we have the following results.

Global stability analysis
In this section, we investigate the global stability of equilibria E 0 and E * .
Proof.From Section 1, we know that system (1.1) has no positive equilibrium when R 0 < 1, and E 0 (x 0 , 0) is the unique equilibrium.Let where c 1 = x 0 /(ad), then we get the derivative of V 11 (t) along solutions of system (1.1) From the positiveness of x(t) and x 0 > k, combined with the condition we have V1 (t) ≤ 0, and V1 (t) = 0 ⇔ (x(t), y(t)) = (x 0 , 0).By LaSalle's invariant set principle we know that E 0 is global asymptotically stable.
Proof.System (1.1) has a positive equilibrium where c 2 = x * /(ad), then we get the derivative of V 21 (t) along solutions of system (1.1) From the positiveness of x(t), combining the condition, we have V2 (t) ≤ 0, and V2 (t) = 0 if and only if x(t) = x * , y(t) = y * .By LaSalle's invariant set principle we know that E * (x * , y * ) is global asymptotically stable.

Hopf bifurcation analysis
In Section 3, we found that under some conditions the system undergoes a Hopf bifurcation when τ passes through some critical value.In this section we study some properties of the Hopf bifurcation and the global existence of the periodic solutions.For convenience, we assume that the condition for Hopf bifurcation (H 1 ) p 1 + q 1 > 0, p 0 − q 0 < 0 is always satisfied in this section.

Properties of bifurcating periodic solutions
In this part, we will study the properties of the bifurcating periodic solutions such as the orbital stability and the direction of Hopf bifurcation.The method we used is based on the normal form method and the center manifold theory introduced by Hassard et al. [10].
Re-scale the time by t → (t/τ) to normalize the delay, and let τ = τ 0 + ε, ε ∈ R, then we can rewrite system (1.1) in the following form where Clearly, the phase space is C = C([−1, 0], R 2 ).From the analysis above we know that ε = 0 is the Hopf bifurcation value for system (5.1).

Global existence of periodic solutions
In this subsection, we shall study the global existence of periodic solutions bifurcating from the point (E * , τ j ), j = 0, 1, 2, . . .for system (1.1) by a global Hopf bifurcation theorem by Wu [25].
For simplification of notations, setting z t = (x t , y t ), we may rewrite systems (1.1) as the following functional differential equation: , and p is the period of the solution of the above equation.
Proof.To the contrary, suppose that system (1.1) has a τ-periodic solution, then the following system of ordinary differential equations also has a periodic solution (5.3) As to the existence of the limit cycle of this ODE system, Sugie and his coworkers have obtained some results in [23].Based on Theorem 2.3 in [23], we know that the ODE system has no limit cycles when condition (H 2 ) holds, which completes the proof.
Basing on the analysis in Section 4, we can get Therefore, we know that the equilibrium E * is asymptotically stable when τ ∈ [0, 0.2727), which is shown in Figure 6.1, here we choose τ = 0.25 and the initial value is taken as (x 0 , y 0 ) = (0.1, 10.5).Furthermore, we know that the equilibrium E * is unstable when τ > 0.2727 and the Hopf bifurcation is forward and the bifurcating periodic solutions are orbitally asymptotically stable, which is shown in Figure 6.2, here we choose τ = 0.3 and the initial value is taken as (x 0 , y 0 ) = (0.1, 10.5).Finally, by the analysis in Subsection 5.2, we know that the bifurcating periodic solutions exist for all τ ∈ (0.2727, +∞), which is shown in Figure 6.3.

Conclusion
Sugie et al. studied the existence of the limit cycles of system (5.3) in [23].They showed that when condition (H 2 ) holds, the coexistence equilibrium E * (x * , y * ) is globally asymptotically stable and system (5.3) has no limit cycle.In this paper, we consider the same system with a constant delay τ due to the gestation of predator.We find that even if the condition (H 2 ) is satisfied, combining with the condition (H 1 ), the equilibrium E * (x * , y * ) loses its stability and an orbitally asymptotically stable periodic solution arises from the Hopf bifurcation when the delay τ passes through some critical value τ 0 , and the bifurcating periodic solution always exists for all τ ∈ (τ 0 , +∞).This shows the important influence of the time delay τ on the system.
888,which implies that the conditions for the Hopf bifurcation are satisfied, and by the previous algorithm we can get