GLOBAL HOPF BIFURCATION OF A POPULATION MODEL WITH STAGE STRUCTURE AND STRONG ALLEE EFFECT

. This paper is devoted to the study of a single-species population model with stage structure and strong Allee eﬀect. By taking τ as a bifurcation parameter, we study the Hopf bifurcation and global existence of periodic solu- tions using Wu’s theory on global Hopf bifurcation for FDEs and the Bendix-son criterion for higher dimensional ODEs proposed by Li and Muldowney. Some numerical simulations are presented to illustrate our analytic results using MATLAB and DDE-BIFTOOL. In addition, interesting phenomenon can be observed such as two kinds of bistability.


Introduction. The well-known Mackey-Glass equatioṅ
was proposed by Mackey and Glass [14] to describe a physiological control system in 1977. Since then, a fair amount of work [23,22,11] has been devote to it, such as the study the Hopf bifurcation and global Hopf bifurcation [23,22], chaos, peak-adding and period-doubling bifurcations [11].
Notice that when δ = 0 and n = 1, the equation (1) reduced to the classical Mackey-Glass equation. Thus equation (1) can be regarded as a modification of the classical Mackey-Glass equation whose dynamical behavior is more complicated.
In [15], Morozov et al. indicated that the growth of adult population in system (1) has the property of strong Allee effect when m > n > 1, where the 'Allee effect' named after Allee [1]. Literatures [10,21] as well as the references therein showed that the phenomenon of the growth rate per capita achieves its peak at a positive density is called an Allee effect. Furthermore, a strong Allee effect refers to the phenomenon that the growth rate per capita is negative when the population is small, and the weak Allee effect means that the growth rate per capita is smaller than the maximum but still positive for small population. Therefore, it is strong Allee effect if m > n > 1 (see Figure 1(a)), and not-strong Allee effect if n ≤ 1 (see Figure 1(b)). We would like to mention that the situation in Figure 1(b)(classical Mackey-Glass case) is not Allee effect since the growth rate per capita r(y) always decreases as y increases. A large amount of simulations has been done in [15] to reveal the rich phenomenon of strong Allee effect (n = 2, m = 4) model, such as long-term transients, peakadding bifurcations, and period-doubling bifurcations. Morozov et al. have done a lot of work to exhibit them in numerical simulation by taking A = αe −δτ as a constant.
In this paper, we will concentrate on (1) with stage structure and general strong Allee effect m > n > 1 to study the stability, Hopf bifurcation, and global Hopf bifurcation.
The rest of the paper is organized as follows: in Section 2, we investigate some properties of the system, such as positivity and boundedness of the solutions, analysis of equilibria and their stabilities. In Section 3, we study the Hopf bifurcation, particularly in Section 3.1 the occurrence of Hopf bifurcation, and in Section 3.2 the direction and stability of the Hopf bifurcation. A global Hopf bifurcation is established in Section 4. Finally, some simulations are carried out for illustrating the analytic results.

2.
Preliminaries. In this section, we shall investigate the positivity and boundedness of the solutions and the equilibria of system (1) with nonnegative initial conditions. For any τ > 0, denote C = C([−τ, 0], R) as the Banach space of continuous functions on [−τ, 0] to R with the norm From a biological point of view, it is reasonable to take initial conditions from the nonnegative cone C + = C([−τ, 0], R + ) where R + = [0, ∞). For the initial conditions ϕ ∈ C + and ϕ(θ) ≡ 0 for θ ∈ [−τ, 0], (2) system (1) admits a unique solution for each continuous initial condition by the fundamental theory in Hale's book [8], and then, the following conclusion holds.
Lemma 2.1. Each solution of (1) with initial condition (2) is nonnegative and ultimately uniformly bounded in C + .
For the ultimately uniformly boundedness, using the fact that we havė Dm .
This completes the proof.
When ξ > 1 and τ < τ 0 , system (1) has three fixed points: y 0 , y 1 , and y 2 , where both y 1 and y 2 change with τ . Furthermore, we would like to point out that the following statement holds.
The linearization of equation (1) around y i (i = 0, 1, 2) iṡ Notice that b 0 = 0, and b 1 , b 2 are functions of τ since y 1 , y 2 changs over τ . The characteristic equation associated with (5) is The following conclusions hold.
Compared to the stability of trivial equilibrium y 0 = 0, we are more concerned about the complex behavior of positive equilibria y 1 and y 2 . In the following sections, we always assume that ξ > 1 and τ < τ 0 , which stands for the existence of positive equilibria.
3. Hopf bifurcation analysis. In this section, we are going to investigate the existence of Hopf bifurcation around y 1 and y 2 by taking the time delay τ as a bifurcation parameter. Without loss of generality, we denote y 1 or y 2 as y * . Then the characteristic equation (7) around y * becomes where b * stands for b 1 or b 2 , which are in the denotation of (6).

Stability and Hopf bifurcation.
From the consequence of the distribution of zeros of a transcendental function given by Ruan and Wei [19], we know that as τ varies, the sum of the orders of the roots of (7) in the open right half plane can change only if a zero appears on and crosses the imaginary axis.
We would like to mention that b * changes with τ since y * is a function of τ . In order to investigate the critical values of τ when there exists a pair of purely imaginary roots of (8), we are going to use the method introduced by Beretta and Kuang [2]. Rewrite equation (8) as We mention that P , Q have real coefficients. This ensures that if λ = iω, for some real ω, is a root of (8), then λ = −iω is a root of (8) as well. Let λ = iω (ω > 0) be a purely imaginary root of equation (8), then Separating the real and imaginary parts yields Squaring both sides of (9) and summing the two equations, we obtain

PENGMIAO HAO, XUECHEN WANG AND JUNJIE WEI
Make the following hypothesis: Then F (ω, τ ) = 0 has a unique positive real root given by Collecting all τ which satisfies (P1) that Before applying the geometry criterion in [2] to (8), a sequence of conditions on P and Q are required to be verified. This is accomplished by the following proposition.
Proof. (a) We know from the expression of b i in (6) and hypothesis (P1). That is, λ = 0 is not a root of (8).
The conclusion is valid because F (ω, τ ) is a linear polynomial in ω 2 and the fact that b * is a continuous function of τ .
Then, θ(τ ) is well and uniquely defined for all τ ∈ I, and we have ω(τ )τ = θ(τ )+2nπ obviously. Therefore, iω * , ω * = ω(τ * ) > 0, is a purely imaginary root of (8) if and only if τ * is a zero of the function S n (τ ) defined by Obviously, S n (0 + ) < 0. In the following, we will investigate S n (τ ) when τ → τ 1 − according to two cases. When y * = y 1 , we have lim We also know that θ(τ ) = 0, 2π on I in terms of (b) in Proposition 1, and S n (τ ) are continuous and differentiable on I from Lemma 2.1 in [2]. The following result in [2] can be used to verify the occurrence of Hopf bifurcations when τ = τ * .
Furthermore, S n (τ ) − S n+1 (τ ) = 2π ω(τ ) > 0 due to the positivity of ω(τ ). Therefore, we see S n (τ ) > S n+1 (τ ) with n ∈ N for all τ ∈ I. That is to say, if S 0 (τ ) has no zeros in I, then S n (τ ) have no zeros in I for all n ∈ N, and if the function S n (τ ) has a positive zero τ ∈ I for some n * ∈ N, there exists at least one zero satisfying S n (τ * ) = 0 and dS n (τ * ) dτ = 0 with n ≤ n * .
Define the set of possible Hopf bifurcation values by from the decreasing property of S n w.r.t. n, we know the set J is finite, so denote the minimum and maximum element to be τ min and τ max respectively. We first state the stability of y 1 and y 2 when τ = 0 in the following.
Theorem 3.2. y 1 is unstable and y 2 is stable when τ = 0.
Proof. When τ = 0, from (7) we have Thus it follows that y 1 is unstable and y 2 is stable when τ = 0.

3.2.
Stability and direction of the Hopf bifurcation. When conditions in Theorem 3.3(ii) hold, there are small amplitude periodic solutions bifurcating at τ ∈ J. In this section, we shall study the direction, stability, and the period of the bifurcating periodic solution. The way to do this is the combination of the normal form method and center manifold theory in [9]. Without loss of generality, let τ * be any critical value such that equation (8) has a pair of purely imaginary roots ±iω * , and system (1) undergoes Hopf bifurcation at y 2 . Then, by setting τ = τ * + µ, µ = 0 is the Hopf bifurcation value of (1). Letỹ(t) = y(τ t) − y 2 to normalize the delay and move y 2 to the origin. Then (1) becomesẏ where b 2 defined in (6), d 2 and e 2 defined in the following Notice that b 2 , d 2 , and e 2 depend on the parameter τ , since y 2 is continuous function of τ . and By the Riesz representation theorem, there exists a bounded variation function η(θ, µ) for θ ∈ [−1, 0], such that In fact, η(θ, µ) can be chosen as Define operators A(µ) and R(µ) as Then system (13) is equivalent to the following operator equatioṅ

PENGMIAO HAO, XUECHEN WANG AND JUNJIE WEI
where y t (θ) = y(t + θ) for θ ∈ [−1, 0]. For ψ ∈ C 1 ([0, 1], R), define an operator and a bilinear inner form where η(θ) = η(θ, 0). Then A(0) and A * are adjoint operators. As shown in Section 3.1, we know that ±iω * τ * are eigenvalues of A(0), thus, they are also eigenvalues of A * . It can be verified that the vectors are the eigenvectors of A(0) and A * corresponding to the eigenvalues iω * τ * and −iω * τ * , respectively. When choosing Following the algorithms provided in Hassard [9] and using a computation process similar to that in [17,18,6], we obtain the following coefficients: So far, g 20 , g 11 , g 02 , g 21 can be calculated exactly. According to Hassard et al. [9], Furthermore, we can compute the following quantities: which determine the properties of bifurcating periodic solutions. From the discussion in Section 3.1, we have the following results immediately. Assume that the conditions in (ii) of Theorem 3.3 hold. Then µ 2 , β 2 , T 2 determine the direction, stability, and period of the corresponding Hopf bifurcation, respectively: (i) The direction of Hopf bifurcation of system (1) at y 2 when τ = τ * is backward (forward) if µ 2 < 0 (µ 2 > 0), that is, there exists a bifurcating periodic solution for τ < τ * (τ > τ * ). (ii) The bifurcating periodic solution on the center manifold is unstable (stable) if β 2 > 0 (β 2 < 0). Particularly, the stability of the bifurcating periodic solutions of (1) is same as that of bifurcating periodic solutions on the center manifold when τ * = τ min and τ * = τ max . (iii) The period of the bifurcating periodic solution decreases (increases) if T 2 < 0 (T 2 > 0).

Global existence of periodic solutions. In Section 3.1, Theorem 3.3(ii)
shows that periodic solutions can bifurcate from y * when τ passes through certain critical values. But will the periodic solutions last for a long interval, this is what we concerned in this section. The method we do this is the global Hopf bifurcation theory proposed by Wu [24]. We first make some notations as in Wu [24] and verify that the assumptions (A1)-(A4) in [24] hold. Copy (1) as follows for conveniencė and rewrite (14) as a general functional differential equation with two parameters τ and T in the following forṁ where is completely continuous with x t ∈ X = C ([−τ, 0] , R + ) and x t (θ) = x(t + θ) for θ ∈ [−τ, 0]. Restricting F on R + × R + × R + , we havê
In the following, define a closed subset Σ(F ) of X × R + × R + by (15) .
Literature [24] gives a definition of a stationary solution to be a center: if ∆ (x0,τ0,T0) im 2π T0 = 0 for some positive integer m. Moreover, the center is said to be isolated if it is the only center in some neighborhood of (x 0 , τ 0 , T 0 ) and it has only finitely many purely imaginary characteristic values of the form im 2π T0 (m is an integer). The analysis of eigenvalues in Section 2 shows that there is no purely imaginary roots for ∆ (y0,τ,T ) (λ) = 0.
Lemma 4.1. All positive periodic solutions of system (15) are uniformly bounded, that is, the projection of C y * , τ j , 2π ωj onto X is bounded. Theorem 2.3 shows that y 1 = 0 is globally asymptotically stable when τ > τ 0 . This implies the following statement holds.
Then system (14) has no periodic solutions of period 4τ .
Proof. Let x(t) be a periodic solution to (14) with period 4τ . Set 2τ ), and x 4 (t) = x(t − 3τ ). Then x = (x 1 , x 2 , x 3 , x 4 ) is a periodic solutions of the system of ordinary differential equationsẋ PENGMIAO HAO, XUECHEN WANG AND JUNJIE WEİ To rule out the 4-periodic solution to (14), it suffices to prove the nonexistence of nonconstant periodic solutions of (16). We use a general Bendixsons criterion in higher dimensions developed by Li and Muldowney [12]. For x ∈ R 4 , the Jacobian matrix J = ∂f ∂x of (16) is Calculate the second additive compound matrix J [2] follow the algorithm in Li and Muldowney [13] obtains Theorem 4.1 in Muldowney [16] shows that µ ∂f [2] ∂x < 0, or µ − ∂f [2] ∂x < 0 holds for all x ∈ R 4 implies that system (16) has no nonconstant periodic solutions, where the Lozinskil measure µ(A) is defined in [3] to be the right-hand derivative To apply this conclusion, we choose a vector norm in R 6 the same as in [22] that Then the matrix norm of A induced by previous vector norm is Apply this into µ J [2] we have µ J [2] = lim where i ∈ {1, 2, 3, 4} and (j, k) ∈ {(1, 3), (2, 4)}. It then follows that µ J [2] < 0 if and only if α|b ′ i |e −δτ < √ 2D.
Proof. Let x(t) be a periodic solution to (14) of period 2τ , then x 1 (t) = x(t) and x 2 (t) = x(t−τ ) are periodic solutions of the system of ordinary differential equationṡ Let (P (x 1 , x 2 ), Q(x 1 , x 2 )) denote the vector field of (17), then Thus no nonconstant periodic solutions of (17) exists due to the classical Bendixsons negative criterion, i.e. there is no periodic solutions with period 2τ of (14). This completes the proof.

This reveals
Lemma 4.5. The projection of C y * , τ j , 2π ωj onto T is bounded.
Up to now, form Lemmas 4.1, 4.2, and 4.5 we know the connected component C y * , τ j , 2π ωj is bounded, which exclude case (a) of Theorem 3.3 in Wu [24], and hence (b) is satisfied. Follow the description in [20] and the fact of the nonexistence of non-constant 1-periodic solution, we know each local periodic solution bifurcated from the Hopf bifurcation exactly connects another corresponding periodic solution. That is the existence of periodic solution in a wide range which is the so called global Hopf bifurcation.
The previous discussion shows that for system (1) with the data (18): 1. The direction of the Hopf bifurcation at y 2 is forward when τ = τ 0 , and backward when τ = τ 1 . 2. The bifurcating periodic solutions bifurcated from y 1 are unstable, and the bifurcating periodic solutions bifurcated from y 2 are stable.
This reveals that conditions in Theorem 4.6 hold and the connected components C y 1 ,τ 0 , 2π ω0 and C y 2 , τ 0 , 2π ω0 are separated. In the following, we carry out numerical simulation to show the Hopf bifurcation branch in connected component C y 2 , τ 0 , 2π ω0 connecting τ 0 = 2.8 and τ 1 = 14.6 (see Figure 8) using DDE-BIFTOOL developed by Engelborghs et al. [4,5]. But it is a little pity that we didn't draw the Hopf bifurcation branch in connected component C y 1 ,τ 0 , 2π ω0 . We should point out that it may be difficult to separate C y 1 ,τ 0 , 2π ω0 from C y 2 , τ 0 , 2π ω0 by period when there are too many zeros ofS n (τ ) and S n (τ ). In the following, we add the periodic solutions obtained by global Hopf bifurcation bifurcated from y 1 connecting τ 0 = 2.8 and τ 1 = 14.6 on Figure 2. Then we get their stability in the following Figure 9. As we can see, when τ ∈ (0, τ 0 ) ∪ (τ 1 , τ 0 ), there are two stable equilibria 0 and y 2 . When τ ∈ (τ 0 , τ 1 ), besides a stable equilibria 0, there is a stable periodic solutions bifurcated from y 2 . This is known as bistable.
Numerical simulations in this section and theoretical analysis in previous sections shows that the introduction of the stage-structure term e −δτ may change the number of possible Hopf bifurcation points from infinite to finite. Furthermore, the strong Allee effect model (1) with m > n > 1 may have two positive equilibria, which is more complicated than the classical Mackey-Glass equation, who has only one positive equilibrium. In fact, we speculate that the long-term transients phenomenon observed by Figure 4 and Figure 6 in [15] might be the result of the unstable periodic solutions bifurcated from y 1 due to Hopf bifurcation.