Global existence of positive periodic solutions of a general differential equation with neutral type

In this paper, the dynamics of a general differential equation with neutral type are investigated. Under certain assumptions, the stability of positive equilibrium and the existence of Hopf bifurcation are obtained by analyzing the distribution of eigenvalues. And global existence of positive periodic solutions is established by using the global Hopf bifurcation result of Krawcewicz et al. Finally, by taking neutral Nicholson’s blowflies model and neutral Mackey–Glass model as two examples, some numerical simulations are carried out to illustrate the analytical results.


Introduction
The standard age-structure model is as follows: with the initial condition u(0, a) = u 0 (a). Here, u(t, a) is the age distribution at time t, which satisfies u(t, τ ) = u(0, τt) and u(t, ∞) = 0. Function w is a weighted average of the total population with weight function ρ(a) ≥ 0. Functions b and μ denote the birth and death rates, respectively, which all depend on the age and the average w.
Let τ be the critical age that separates adults and juveniles. Then the total population of the mature individuals is Assume that the death rate μ is a constant step function where H is the Heaviside function with jump at a = τ and ρ(a) = H τ (a), which implies w(t) = N(t) and the birth rate only depends on the age a and the population of the mature individuals N . Further assume that the juveniles do not reproduce and the birth rate b has a delta peak at a = τ b(a, N) = g(N)H τ (a) + cδ τ (a), where δ is the delta function. Then it can be verified that N(t) satisfieṡ where f (s) = sg(s). For more details on the derivation of Eq. (1), we refer the reader to [1]. Model (1) is a neutral differential equation, and the corresponding delay equation (c = 0) describes many famous models for population growth, which include Nicholson's blowflies model the Mackey-Glass modelṡ and the Lasota-Wazewska model Nicholson's blowflies model (2) has been extensively studied in the literature, where its results mainly concern the global attractivity of positive equilibrium and oscillatory behaviors of solutions (see [2][3][4][5][6]). Several studies have also been carried out on model (2) with time periodic coefficients (see [7,8]), on discrete Nicholson's blowflies model (see [9][10][11][12]), and on diffusive Nicholson's blowflies model (see [13][14][15][16][17]). Mackey-Glass equations (3) and (4) and Lasota-Wazewska model (5) have been studied in [18][19][20][21][22][23]. A majority of results deal with the global attractivity of the positive equilibrium and the Hopf bifurcation problem. Meanwhile, there are several papers on the complex dynamics of model (3). For example, Mackey and Glass [24] and Namajūnas et al. [25] studied the chaotic behavior, while Losson et al. [26] investigated the multistability. (Further, we refer to [27] and [28] for biological models, involving chemotaxis and nonlinear diffusive mechanism, formulated by the introduction of reactions coupling growth and death impacts. ) Wei investigated the global Hopf bifurcation of Eq. (1) with c = 0 [29], and Li et al. investigated the global Hopf bifurcation of Eq. (1) with f (s) = pse -αs [30]. The purpose of the present paper is to study the global Hopf bifurcation of the general neutral differential equation (1). Here, r > 0, 0 < c < 1, and f is a nonnegative and C 2 function in [0, ∞) with f (∞) = 0 and satisfies one of the following assumptions: (H1) f (0) > 0 and f (s) is strictly monotone decreasing when s ∈ (0, ∞).
The rest of the paper is organized as follows. In Sect. 2, we investigate the existence of positive equilibrium and analyze the distribution of the roots of the characteristic equation to give various conditions on the stability of positive equilibrium and the existence of Hopf bifurcation. In Sect. 3, we establish the extended existence of bifurcation periodic solutions by using the global Hopf bifurcation result of Krawcewicz et al. [31]. In Sect. 4, we carry out some numerical simulations to support the analytical results.

Stability and local Hopf bifurcation analysis
In this section, we consider the stability of positive equilibrium and the existence of local Hopf bifurcation by taking time delay τ as a bifurcation parameter.
For neutral differential equations, positive initial conditions can lead to positive solutions is not a trivial matter. Here, we provide some positively invariant sets for Eq. (1). Without loss of generality, assume that the initial time for Eq. (1) is zero and define Then we have the following result. Proof Assume that θ ∈ [-τ , 0] and φ ∈ . Then, for t ∈ (0, τ ], we have Therefore, by mathematical induction, 1 is a positively invariant set for Eq. (1). If = 2 , then the solution N(t) of Eq. (1) satisfies that D -N(t) exists, Therefore, by mathematical induction, 2 is a positively invariant set for Eq. (1).
In the remainder of the paper, we investigate the dynamics of Eq. (1) in a positively invariant region with either = 1 or = 2 , and formulate our results accordingly. (1) has no positive equilibrium. Therefore, in the remainder of this paper, we replace assumption (H2) by Hence, the corresponding characteristic equation is Lemma 2.4 Assume that either (H1) or (H2) holds. When τ = 0, the root of Eq. (6) is negative.
Proof When τ = 0, the root of Eq. (6) is Clearly, we have f (N 0 ) < 0 under assumption (H1). Assume that (H2) holds. From the monotonicity of f (s) s , we can obtain that It follows that The proof is completed.
Lemma 2.4 implies that the positive equilibrium N 0 of Eq. (1) is asymptotically stable when τ = 0. Thus, with the increase of τ , a stability change at N 0 can only happen when there are characteristic roots crossing the imaginary axis to the right. In addition, from the proof of Lemma 2.4, one can see that f (N 0 ) < r(1c) under assumption (H1) or (H2) . This leads to λ = 0 is not a root of Eq. (6) under assumption (H1) or (H2) . Now, let λ = iω 0 (ω 0 > 0) be a root of Eq. (6). Substituting it into Eq. (6) and separating the real and imaginary parts, we have Hence, Since we have f (N 0 ) < r(1c) under assumption (H1) or (H2) , ω 0 > 0 makes sense if and only if Furthermore, from Eq. (7), one can obtain that Therefore, when f (N 0 ) < -r(1 + c), Eq. (6) has a pair of imaginary roots ±iω 0 if and only if τ = τ j , where From the discussions above, we have the following result. (ii) If f (N 0 ) < -r(1 + c), then Eq. (6) has a pair of imaginary roots ±iω 0 when τ = τ j , where τ j is defined by Eq. (9).

Lemma 2.6
Assume that either (H1) or (H2) holds and f Proof Differentiating Eq. (6) with respect to λ, we have Thus, The proof is completed.
By applying Lemmas 2.4, 2.5, and 2.6, one can easily obtain the following stability properties of the positive steady state N 0 of Eq. (1).

Global Hopf bifurcation analysis
In this section, we investigate the global existence of positive periodic solutions of Eq. (1) by using the global Hopf bifurcation theorem [31].
Then, for any fixed t, we get Replacing t with tτ in the above inequality, we have Hence, for any integer m, Let m → ∞ and we can obtain that By using a similar method, one also has In addition, from the definitions of T 0 and T 0 , we know that Now we can distinguish two cases. Case (i): (H1) holds. From Eq. (11), Eq. (12), and Eq. (13), it is easy to see that and Case Then, by applying Eq. (11) and Eq. (13), we have In particular, from Eq. (12), we have It follows that Thus, combining with Eq. (13), we can obtain that Now, we need to show that u(T 0τ ) > 0. In fact, if u(T 0τ ) = 0, we can get u(T 0 ) = 0 and On the other hand, since u(t) is a nonconstant periodic solution of Eq. (1), we have f (u(tτ )) ≡ 0. This leads to Without loss of generality, assume that there exists t * > 0 such that u(t * )cu(t *τ ) > 0. Then we can obtain that, for t ≥ t * , From the periodicity of u(t), we have u(t) > cu(tτ ) ≥ 0 for any t ∈ [0, ∞), which implies that all nonconstant periodic solutions of Eq. (1) are positive. Hence, we get u(T 0τ ) > 0. Then, from Eq. (16), one has By using the fact that f (s) s is strictly monotone decreasing when s ∈ (0, ∞), we can obtain Note that from Eq. (15) Therefore, we can define such that .
Clearly, f m only depends on function f and parameters r and c. Moreover, we have f m > 0, because f (s) s is strictly monotone decreasing and tends to zero as s tends to infinity. The proof is completed. and Then we know that from Lemma 3.1 all of the positive periodic solutions of Eq. (1) belong to the region G. Then the conclusion follows from the fact that the first order autonomous ODE has no nonconstant periodic solutions.  Then u, v ∈ G and (u(t), v(t)) is a periodic solution to the following system of ordinary differential equations: That is, Hence, If (H1) holds, we have f (s) < 0 for all s ∈ G. Thus, ∂P ∂u + ∂Q ∂v < 0. If (H2) holds, for all s ∈ G, we have Then, from cf (G 1 ) < r(1c 2 )G 1 , we also can conclude that ∂P ∂u + ∂Q ∂v < 0. The classical Bendixson criterion implies that Eq. (20) has no nonconstant periodic solution in G.
Then we have the following result.