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Global existence of positive periodic solutions of a general differential equation with neutral type
Advances in Difference Equations volume 2021, Article number: 254 (2021)
Abstract
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 agestructure 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 \(u(t,\tau )=u(0,\tau t)\) and \(u(t,\infty )=0\). Function w is a weighted average of the total population with weight function \(\rho (a)\geq 0\). 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 \(a=\tau \) and
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 \(a=\tau \)
where δ is the delta function. Then it can be verified that \(N(t)\) satisfies
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 models
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 [2–6]). Several studies have also been carried out on model (2) with time periodic coefficients (see [7, 8]), on discrete Nicholson’s blowflies model (see [9–12]), and on diffusive Nicholson’s blowflies model (see [13–17]). Mackey–Glass equations (3) and (4) and Lasota–Wazewska model (5) have been studied in [18–23]. 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 \(f(s)=pse^{\alpha s}\) [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 \([0, \infty )\) with \(f(\infty )=0\) and satisfies one of the following assumptions:

(H1)
\(f(0)>0\) and \(f(s)\) is strictly monotone decreasing when \(s\in (0,\infty )\).

(H2)
\(f(0)=0\), and \(\frac{f(s)}{s}\) is strictly monotone decreasing when \(s\in (0,\infty )\).
It is easy to see that the function f in (2) or (3) satisfies (H2) and the function f in (4) or (5) satisfies (H1).
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.
Theorem 2.1
If \(\Gamma =\Gamma _{1}\) or \(\Gamma =\Gamma _{2}\), then Γ is a positively invariant set for Eq. (1).
Proof
Assume that \(\theta \in [\tau , 0]\) and \(\phi \in \Gamma \). Then, for \(t\in (0, \tau ]\), we have \(N(t\tau )=\phi (t\tau )\geq 0\). Since function f is nonnegative in \([0, \infty )\), we can obtain
If \(\Gamma =\Gamma _{1}\), one has
and
Therefore, by mathematical induction, \(\Gamma _{1}\) is a positively invariant set for Eq. (1).
If \(\Gamma =\Gamma _{2}\), then the solution \(N(t)\) of Eq. (1) satisfies that \(D^{}N(t)\) exists,
and
Therefore, by mathematical induction, \(\Gamma _{2}\) 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 Γ with either \(\Gamma =\Gamma _{1}\) or \(\Gamma =\Gamma _{2}\), and formulate our results accordingly.
Theorem 2.2
Assume that \(r>0\), \(0< c<1\), and f is a nonnegative and \(C^{2}\) function in \([0, \infty )\) with \(f(\infty )=0\).

(i)
If (H1) holds, then Eq. (1) has a unique positive equilibrium \(N_{0}\).

(ii)
If (H2) and \(f'(0)>r(1c)\) hold, then Eq. (1) has a unique positive equilibrium \(N_{0}\).
Proof
Obviously, the positive equilibrium \(N_{0}\) of Eq. (1) satisfies
If (H1) holds, then we have \(g(0)=f(0)>0\), \(g(\infty )=\infty \). Since \(g'(s)=r(1c)+f'(s)<0\), there exists a unique \(N_{0}\) in the interval \((0,\infty )\) such that \(g(N_{0})=0\). If (H2) holds, we have \(\lim_{s\rightarrow 0^{+}}\frac{f(s)}{s}=f'(0)>r(1c)\) and \(\lim_{s\rightarrow \infty }\frac{f(s)}{s}=0\). There exists a unique value of \(N_{0}\) in the interval \((0,\infty )\) satisfying \(\frac{f(N_{0})}{N_{0}}=r(1c)\) due to the monotonicity of \(\frac{f(s)}{s}\), which implies that \(g(s)=s (r(1c)+\frac{f(s)}{s} )\) has a unique zero \(N_{0}\) in the interval \((0,\infty )\). The proof is completed. □
Remark 2.3
From the proof of Theorem 2.2, one has that under assumption (H2), if \(f'(0)>r(1c)\) does not hold, Eq. (1) has no positive equilibrium. Therefore, in the remainder of this paper, we replace assumption (H2) by
 (H2)′:

\(f(0)=0\), \(f'(0)>r(1c)\), and \(\frac{f(s)}{s}\) is strictly monotone decreasing when \(s\in (0,\infty )\).
The linearization of Eq. (1) at \(N=N_{0}\) is given by
Hence, the corresponding characteristic equation is
Lemma 2.4
Assume that either (H1) or (H2)′ holds. When \(\tau =0\), the root of Eq. (6) is negative.
Proof
When \(\tau =0\), the root of Eq. (6) is
Clearly, we have \(f'(N_{0})<0\) under assumption (H1). Assume that (H2)′ holds. From the monotonicity of \(\frac{f(s)}{s}\), 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 \(\tau =0\). 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 \(\lambda =0\) is not a root of Eq. (6) under assumption (H1) or (H2)′.
Now, let \(\lambda =i\omega _{0}\) (\(\omega _{0}>0\)) 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)′, \(\omega _{0}>0\) 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 \(\pm i\omega _{0}\) if and only if \(\tau =\tau _{j}\), where
From the discussions above, we have the following result.
Lemma 2.5
Assume that either (H1) or (H2)′ holds.

(i)
If \(f'(N_{0})\geq r(1+c)\), then Eq. (6) has no imaginary root for any \(\tau >0\).

(ii)
If \(f'(N_{0})<r(1+c)\), then Eq. (6) has a pair of imaginary roots \(\pm i\omega _{0}\) when \(\tau =\tau _{j}\), where \(\tau _{j}\) is defined by Eq. (9).
Lemma 2.6
Assume that either (H1) or (H2)′ holds and \(f'(N_{0})<r(1+c)\) is satisfied. Let \(\lambda (\tau )=\alpha (\tau )+i\beta (\tau )\) be the root of Eq. (6) near \(\tau =\tau _{j}\) satisfying \(\alpha (\tau _{j})=0\) and \(\beta (\tau _{j})=\omega _{0}\). Then we have
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).
Theorem 2.7
Assume that \(r>0\), \(0< c<1\), f is a nonnegative and \(C^{2}\) function in \([0, \infty )\) with \(f(\infty )=0\) and either (H1) or (H2)′ holds.

(i)
If \(f'(N_{0})\geq r(1+c)\), then for Eq. (1), \(N=N_{0}\) is locally asymptotically stable for all \(\tau \geq 0\).

(ii)
If \(f'(N_{0})<r(1+c)\), then for Eq. (1), \(N=N_{0}\) is locally asymptotically stable for \(\tau \in [0,\tau _{0})\) and unstable for \(\tau >\tau _{0}\). Furthermore, system (1) undergoes a Hopf bifurcation at \(N=N_{0}\) when \(\tau =\tau _{j}\), \(j=0,1,2,\ldots \) .
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].
Lemma 3.1
Assume that either (H1) or (H2)′ holds. Then all periodic solutions of Eq. (1) are uniformly bounded.
Proof
By Theorem 2.1, we know that all periodic solutions of Eq. (1) in Γ are bounded below by 0. Let \(u(t)\) be a nonconstant periodic solution of Eq. (1). Then there exist \(T^{0}\) and \(T_{0}\) such that
Then, for any fixed t, we get
Replacing t with \(t\tau \) in the above inequality, we have
Hence, for any integer m,
Let \(m\rightarrow \infty \) 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 (ii): (H2)′ holds.
From \(f(0)=f(\infty )=0\) and the continuity of function f, we can define
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 \(u(T_{0}\tau )>0\). In fact, if \(u(T_{0}\tau )=0\), 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 \(f(u(t\tau ))\not \equiv 0\). This leads to
Without loss of generality, assume that there exists \(t_{*}>0\) such that \(u(t_{*})cu(t_{*}\tau )>0\). Then we can obtain that, for \(t\geq t_{*}\),
From the periodicity of \(u(t)\), we have \(u(t)>cu(t\tau )\geq 0\) for any \(t\in [0, \infty )\), which implies that all nonconstant periodic solutions of Eq. (1) are positive. Hence, we get \(u(T_{0}\tau )>0\). Then, from Eq. (16), one has
By using the fact that \(\frac{f(s)}{s}\) is strictly monotone decreasing when \(s\in (0,\infty )\), 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 \(\frac{f(s)}{s}\) is strictly monotone decreasing and tends to zero as s tends to infinity. The proof is completed. □
Denote \(G=[G_{1}, G_{2}]\) with
and
Then we know that from Lemma 3.1 all of the positive periodic solutions of Eq. (1) belong to the region G.
Lemma 3.2
Equation (1) has no nonconstant periodic solution when \(\tau =0\).
Proof
Obviously, when \(\tau =0\), Eq. (1) becomes
Then the conclusion follows from the fact that the first order autonomous ODE has no nonconstant periodic solutions. □
Remark 3.3
The proof of Lemma 3.2 is also able to show that Eq. (1) has no nonconstant periodic solution of period τ.
Lemma 3.4

(i)
If (H1) holds, then Eq. (1) has no nonconstant periodic solution of period 2τ.

(ii)
If (H2)′ and \(cf(G_{1})< r(1c^{2})G_{1}\) hold, then Eq. (1) has no nonconstant periodic solution of period 2τ.
Proof
Let \(u(t)\) be a nonconstant periodic solution to Eq. (1) of period 2τ and \(v(t)=u(t\tau )\). Then \(u, v\in G\) 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 \(s\in G\). Thus, \(\frac{\partial P}{\partial u}+\frac{\partial Q}{\partial v}<0\). If (H2)′ holds, for all \(s\in G\), we have
Then, from \(cf(G_{1})< r(1c^{2})G_{1}\), we also can conclude that \(\frac{\partial P}{\partial u}+\frac{\partial Q}{\partial v}<0\). The classical Bendixson criterion implies that Eq. (20) has no nonconstant periodic solution in G. Therefore, Eq. (1) has no nonconstant periodic solution of period 2τ. The proof is completed. □
Denote
Then we have the following result.
Lemma 3.5
Assume that either (H1) or (H2)′ holds. If
then Eq. (1) has no nonconstant periodic solution of period 4τ.
Proof
Let \(u(t)\) be a nonconstant periodic solution to Eq. (1) of period 4 and set \(v_{i}(t)=u(t(i1)\tau )\), \(i=1,2,3,4\). Then \(V=(v_{1}(t), v_{2}(t), v_{3}(t), v_{4}(t))\) is a nonconstant periodic solution to the following system of ordinary differential equations:
That is,
whose orbits belong to \(\tilde{G}:=\{V\in \mathbb{R}^{4}\mid v_{i}\in G, i=1,2,3,4\}\). Next, we will employ a general Bendixson criterion in higher dimensions developed in [32] to exclude nonconstant periodic solutions of Eq. (23) in region G, which will guarantee that there are no 4τperiodic solutions to Eq. (1). The Jacobian matrix \(J(V)\) of Eq. (23), for \(V\in \mathbb{R}^{4}\), is
The second additive compound matrix \(J^{[2]}(V)\) of \(J(V)\) is
where S is a \(6\times 6\) matrix, whose rows \(S_{i}\) are given in the following row vectors:
Choose \(l_{\infty }\) norm in \(\mathbb{R}^{6}\), namely, \(x=\max_{1\leq i\leq 6} x_{i}\). Let A be the diagonal matrix given by
Then the Lozinskii measure of \(AJ^{[2]}(V)A^{1}\) is
where
Obviously, from Eq. (23), we have that \(\mu (AJ^{[2]}(V)A^{1})<0\) for \(V\in \tilde{G}\), which is a Bendixson condition that rules out nonconstant periodic orbits of Eq. (23) in G̃ [32]. Therefore, Eq. (1) has no nonconstant periodic solution of period 4τ. The proof is completed. □
Now, we are in a position to state the following global Hopf bifurcation results.
Theorem 3.6
Assume that \(r>0\), \(0< c<1\), and f is a nonnegative and \(C^{2}\) function in \([0, \infty )\) with \(f(\infty )=0\).

(i)
If either (H1) or (H2)′ holds and \(f'(N_{0})<r(1+c)\) is satisfied, then Eq. (1) has at least one positive periodic solution for \(\tau >\tau _{1}\).

(ii)
If (H1), \(r(1+c^{2})< cf'(N_{0})<rc(1+c)\), and Eq. (22) hold, then Eq. (1) has at least one positive periodic solution for \(\tau >\tau _{0}\) and two positive periodic solutions for \(\tau >\tau _{1}\).

(iii)
If (H2)′, \(r(1+c^{2})< cf'(N_{0})<rc(1+c)\), \(cf(G_{1})< r(1c^{2})G_{1}\), and Eq. (22) hold, then Eq. (1) has at least one positive periodic solution for \(\tau >\tau _{0}\) and two positive periodic solutions for \(\tau >\tau _{1}\).
Proof
First, from the definition of \(\tau _{j}\) in Eq. (9), we know that \(\tau _{j}<\tau _{j+1}\). Define
and denote by \(C(N_{0}, \tau _{j}, \frac{2\pi }{\omega _{0}})\) the connected component of \((N_{0}, \tau _{j}, \frac{2\pi }{\omega _{0}})\) in Σ, where \(\omega _{0}\) is defined in Eq. (8). Note that, under assumption (H1) or (H2)′, the stationary points \((N_{0}, \tau _{j}, \frac{2\pi }{\omega _{0}})\) of Eq. (1) are nonsingular and they are isolated centers (see [31]). Thus, Lemma 2.6 implies that there exist \(\epsilon >0\), \(\delta >0\), and a smooth curve \(\lambda : (\tau _{j}\delta , \tau _{j}+\delta )\rightarrow \mathbb{C}\) such that
for all \(\tau \in [\tau _{j}\delta , \tau _{j}+\delta ]\), where Δ is defined as (6), and
Let
It is easy to see that on \([\tau _{j}\delta , \tau _{j}+\delta ]\times \bar{\Omega }_{\epsilon }\), \(\Delta _{(N_{0}, \tau , T)}(u+\frac{2\pi i}{p})=0\) if and only if \(\tau =\tau _{j}\), \(u=0\), and \(p= \frac{2\pi }{\omega _{0}}\). Moreover, put
Then, for any \(j=0,1,2,\ldots \) , the crossing number is
It follows that
Therefore, we conclude that the connected component is nonempty and unbounded from the local and global Hopf bifurcation theorem given by Krawcewicz, Wu, and Xia (see [31]).
By Lemma 3.1, the projection of \(C(N_{0}, \tau _{j}, \frac{2\pi }{\omega _{0}})\) onto the yspace is bounded. Meanwhile, the projection of \(C(N_{0}, \tau _{j}, \frac{2\pi }{\omega _{0}})\) onto τspace is bounded below from Lemma 3.2.
From Eq. (9), we know that
which implies
By Remark 3.3, for \(j\geq 1\), one has that \(\frac{\tau }{j+1}< T<\frac{\tau }{j}\) if \((y, \tau , T)\in C(N_{0}, \tau _{j}, \frac{2\pi }{\omega _{0}})\). This fact shows that the projection of \(C(N_{0}, \tau _{j}, \frac{2\pi }{\omega _{0}})\) onto the Tspace is bounded if τ is bounded. Thus, in order for \(C(N_{0}, \tau _{j}, \frac{2\pi }{\omega _{0}})\) to be unbounded, its projection onto the τspace must be unbounded. In other words, the projection of \(C(N_{0}, \tau _{j}, \frac{2\pi }{\omega _{0}})\) onto the τspace includes \([\tau _{j}, \infty )\). The proof of (i) is completed.
If either (H1) or (H2)′ holds and \(r(1+c^{2})< cf'(N_{0})<rc(1+c)\) is satisfied, then from Eq. (7), Eq. (8), and Eq. (9), we have
which implies
By Lemmas 3.4 and 3.5, if (H1) and Eq. (22) hold, we have \(2\tau < T<4\tau \) when \((y, \tau , T)\in C(N_{0}, \tau _{0}, \frac{2\pi }{\omega _{0}})\). It follows that the projection of \(C(N_{0}, \tau _{0}, \frac{2\pi }{\omega _{0}})\) onto the Tspace is bounded if τ is bounded. Thus, the projection onto τspace of \(C(N_{0}, \tau _{0}, \frac{2\pi }{\omega _{0}})\) must be unbounded. In other words, the projection of \(C(N_{0}, \tau _{j}, \frac{2\pi }{\omega _{0}})\) onto the τspace includes \([\tau _{0}, \infty )\). The proof of (ii) is completed.
Similarly, if (H2)′, \(cf(G_{1})< r(1c^{2})G_{1}\) and Eq. (22) hold, we also have \(2\tau < T<4\tau \) when \((y, \tau , T)\in C(N_{0}, \tau _{0}, \frac{2\pi }{\omega _{0}})\). Therefore, the projection onto the τspace of \(C(N_{0}, \tau _{0}, \frac{2\pi }{\omega _{0}})\) must include \([\tau _{0}, \infty )\). The proof of (iii) is completed. □
Two examples and simulations
In this section, we carry out some numerical simulations for Eq. (1).
Example 1: neutral Nicholson’s blowflies model
Choose \(f(s)=pse^{\alpha s}\) and consider the following neutral Nicholson’s blowflies model:
We would like to mention that the authors of reference [30] have investigated the existence of global Hopf bifurcation of Eq. (24). However, the global Hopf bifurcation conditions given in [30] depend on the existence of a positive real number, which is denoted by β in [30], and are very difficult to verify. In this subsection, we take the parameter values as follows:
and show that this set of parameters satisfies the global Hopf bifurcation conditions in this paper.
Firstly, we have that \(f'(0)=p>r(1c)\). Thus, (H2)′ holds and Eq. (24) has a unique positive equilibrium \(N_{0}=0.7675\). By calculation, we have \(f'(N_{0})=0.5862<r(1+c)=0.55\). Then, from Theorem 2.7(ii), \(N=N_{0}\) is locally asymptotically stable for \(\tau \in [0,\tau _{0})\) (see Fig. 1(a)) and unstable for \(\tau >\tau _{0}\), and system (24) undergoes a local Hopf bifurcation at \(N=N_{0}\) when \(\tau =\tau _{j}\), \(j=0,1,2,\ldots \) , where
Next, we check the conditions of the global Hopf bifurcation (Theorem 3.6(iii)). Obviously, \(r(1+c^{2})< cf'(N_{0})\) is satisfied. From the definitions given by Eq. (14), Eq. (17), Eq. (18), Eq. (19), and Eq. (21), we have
It follows that
and
Therefore, the conditions of Theorem 3.6(iii) are all satisfied, and Eq. (24) has at least one periodic solution for \(\tau >\tau _{0}\) (see Fig. 1(b)) and two periodic solutions for \(\tau >\tau _{1}\) (see Fig. 1(c)).
Example 2: neutral Mackey–Glass model
Choose \(f(s)=\frac{bs}{1+s^{3}}\) and consider the following neutral Mackey–Glass model:
with the parameters \(r=0.5\), \(c=0.1\), \(b=1.8\).
Obviously, we have that \(f'(0)=b>r(1c)\). It follows that (H2)′ holds and Eq. (25) has a unique positive equilibrium \(N_{0}=1.4422\). Moreover, we can obtain that \(f'(N_{0})=0.5625<r(1+c)=0.55\). Then, from Theorem 2.7(ii), \(N=N_{0}\) is locally asymptotically stable for \(\tau \in [0,\tau _{0})\) (see Fig. 2(a)) and unstable for \(\tau >\tau _{0}\), and system (25) undergoes a local Hopf bifurcation at \(N=N_{0}\) when \(\tau =\tau _{j}\), \(j=0,1,2,\ldots \) , where
Now, we check the conditions of the global Hopf bifurcation (Theorem 3.6(iii)). Similarly, we can obtain that \(r(1+c^{2})< cf'(N_{0})\) and
It follows that
and
Therefore, the conditions of Theorem 3.6(iii) are all satisfied, and Eq. (25) has at least one periodic solution for \(\tau >\tau _{0}\) (see Fig. 2(b)) and two periodic solutions for \(\tau >\tau _{1}\) (see Fig. 2(c)).
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The authors are very grateful to the referees for their valuable suggestions.
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This work is supported by the Fundamental Research Funds for the Central Universities (No. 2572020BC09).
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Liu, M., Cao, J. & Xu, X. Global existence of positive periodic solutions of a general differential equation with neutral type. Adv Differ Equ 2021, 254 (2021). https://doi.org/10.1186/s1366202103295y
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Keywords
 Differential equation
 Neutral
 Stability
 Global Hopf bifurcation