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Bifurcation analysis and chaotic behavior of a discretetime delayed genetic oscillator model
Advances in Difference Equations volume 2017, Article number: 3 (2017)
Abstract
In this paper, a genetic oscillator model with time delay is discretized by the Euler method. The discrete oscillator model is discussed by using NeimarkSacker bifurcation theory. The direction and the stability of the NeimarkSacker bifurcation has been studied using the center manifold theorem and normal form theory. Numerical simulations illustrate the theoretical results.
Introduction
In recent years, the dynamics of genetic oscillators has been investigated in the past decades [1–12]. Goldbeter shows that the genetic oscillators have a crucial impact on many aspects of cell physiology activities [3, 4]. Mathematical modeling has been playing key roles for understanding the dynamics of genetic oscillators [1–4]. The evolution of genetic oscillators can be described by ordinary differential equations or difference equations. Oscillation phenomena are common in nature, especially in those systems that have rhythm behavior. It has been demonstrated that one of the most common causes of the oscillation phenomenon is the existence of certain bifurcations [5–16]. In continuoustime systems, oscillations appear mainly due to Hopf bifurcations [8–11]. In discretetime systems, however, the essential cause is NeimarkSacker bifurcation or perioddoubling bifurcation [13, 14].
Recently, increasing attention has been paid to discrete time models. The reasons are as follows: The numerical simulations of continuoustime models are obtained by the discretizing models. It is common practice to discretize the continuoustime model for experimental or computational purposes. The discretetime model inherits the dynamic characteristics of the continuoustime model, and it also retains functional similarity to the continuoustime system and any physical or biological reality that the continuoustime model has [17]. At last, the discretetime models have rich dynamical behaviors as compared to continuoustime models. We can get more accurate numerical simulations results from discrete time models. Herein, we will consider an oscillator model described by difference equations.
In this paper, we will investigate stability and NeimarkSacker bifurcation of a discrete genetic oscillator model with delay. The genetic oscillator model [4] can be described by the differential equations:
where \(X(t)\) and \(Y(t)\) denote the concentration of mRNA and protein concentration at time t. \(k_{1}\) and \(k_{sy}\) represent the synthetic rate of mRNA and protein. S is the concentration of a transcription factor. \(K_{d}\) is the dissociation constant for binding of \(Y(t)\). P is Hill coefficient. \(k_{dx}\) is the degradation rate of mRNA. \(E_{T}\) is the total concentration of enzyme; its turnover rate is \(k_{2}\) and its Michaelis constant is \(K_{m}\). τ is the time delay required for transcription and translation.
This paper is organized as follows. In Section 2, the stability of the positive equilibrium and the existence of a NeimarkSacker bifurcation are discussed. In Section 3, the direction and the stability of bifurcating periodic solutions can be determined by using the center manifold theorem and normal form theory. In Section 4, a simulation example is applied to verify the theoretical results. The last section contains conclusions.
Stability analysis
In order to be concise, let \(P = 2\), \(n_{1} = k_{1}S\), \(n_{2} = K_{d}^{2}\), \(n_{3} =k_{dx}\), \(n_{4} = k_{sy}\), \(n_{5} = k_{2}E_{T}\), \(n_{6} = K_{m}\), then equation (1.1) can be rewritten as
Let \(\tilde{x}(t) = X(\tau t)\), \(\tilde{y}(t) = Y(\tau t)\), then equation (2.1) can be rewritten as
Applying the forward Euler scheme to equation (2.2), we obtain a model as follows:
Here \(h = 1 / d\) is the step size, d is a positive integer.
In the following sections, we will study the dynamics behavior of the system (2.3) including stability and the bifurcation phenomenon. We will use some lemmas to illuminate existence conditions of a unique positive root of equation (2.4).
We denote \((x,y)\) as the steady state and \(( x^{ *},y^{ *} ) \) as the positive equilibrium point of equation (2.3). Let \((x, y)\) be a steady state of equation (2.3), then we have
From equation (2.4), we obtain
where \(M = \frac{n_{2}n_{3}n_{5} n_{1}n_{2}n_{4}}{n_{3}n_{5}}\), \(N = \frac{n_{1}n_{2}n_{4}n_{6}}{n_{3}n_{5}}\).
Let \(f(y) = y^{3} + My  N\).
Taking the derivative of \(f(y)\), we have
Lemma 2.1
System (2.3) has a unique positive equilibrium point \(( x^{*},y^{*}) \) if one of the following assumptions holds:

(i)
\(0 < n_{1}n_{4} / n_{3}n_{5} < 1\);

(ii)
\(n_{1}n_{4} / n_{3}n_{5} > 1\) and \(f(r_{1}) < 0\);

(iii)
\(n_{1}n_{4} / n_{3}n_{5} > 1\), \(f(r_{1}) > 0\), and \(f(r_{2}) > 0\),
where \(f(y) = y^{3} + My  N\), \(r_{1}\) and \(r_{2}\) are roots of \(f'(y) =0\).
Proof
As \(f(y) = y^{3} + My  N\). Since \(f(0) =  N <0\), \(f( + \infty ) = + \infty \), it is clear that equation (2.5) has at least one positive solution.
We will consider three cases as follows:
If (i) holds, then \(\Delta =  12M \le 0\).
This means that \(f'(y) \ge 0\). Hence \(f(y)\) is monotonically increasing on \([0, + \infty )\). Based on the above analysis, we know that the positive root is unique.
If (ii) holds, it means that \(\Delta =  12M > 0\), then \(f'(y) = 0\) has two roots:
We obtain \(r_{1} < 0\), \(r_{2} > 0\) and \(r_{1} < r_{2}\), then \(f(y)\) is monotonically increasing on \(( \infty,r_{1})\) and \([r_{2}, + \infty )\), or decreasing on \([r_{1},r_{2})\), this together with \(f(r_{1}) < 0\) imply that \(f(y)\) has a unique positive root.
Similarly, if (iii) holds\(,f(y)\) has a unique positive root.
It means that under one of the assumptions (i)(iii), there is a unique positive equilibrium point \(( x^{ *},y^{ *} ) \) in the system (2.3). This completes the proof. □
Let \(u(n) = \tilde{x}(n)  x^{ *}\), \(v(n) = \tilde{y}(n)  y^{ *}\). Substituting them into equation (2.3), we have
Introducing a variable \(X(n) = ( u(n),v(n),v(n  1), \ldots,v(n  d) ) ^{T}\), equation (2.5) can be transformed into the following form:
where \(F = (F_{0},F_{1}, \ldots,F_{d})^{T}\). We have
Then we can get the linear part of equation (2.6) as follows:
where
The characteristic equation of equation (2.7) is given by
where
According to the knowledge of the dynamics, we know that the equilibrium stability of equation (2.6) is determined by the roots distribution of equation (2.9). We will utilize a lemma of Zhang et al. [18] to analyze the roots distribution of equation (2.9).
Lemma 2.2
There exists a \(\bar{\tau } > 0\) such that, for \(0 < \tau < \bar{\tau }\), all roots of equation (2.9) have modulus less than one.
Proof
When \(\tau = 0\), we can obtain \(a_{1} =  2\), \(a_{2} = 1\), \(a_{3} = 0\).
Then equation (2.9) becomes
The equation has dfold roots \(\lambda = 0\), and twofold roots \(\lambda =1\).
Consider the root \(\lambda (\tau )\) such that \(\vert \lambda (0) \vert =1\). This root depends continuously on τ and is a differentiable function of τ.
Differentiating both sides of equation (2.9) with respect to τ yields
where \(\lambda ' = \frac{d\lambda }{d\tau }\), \(a_{1}^{\prime } = \frac{da_{1}}{d\tau } = h(b_{1} +n_{3})\), \(a_{2}^{\prime } = \frac{da_{2}}{d\tau } = 2\tau h^{2}b_{1}n_{3}  h(b_{1} + n_{3})\), \(a_{3}^{\prime } = \frac{da_{3}}{d\tau } = 4\tau h^{2}b_{2}\).
Differentiating both sides of equation (2.10) with respect to τ yields
From equation (2.11), we obtain
Substituting \(\lambda = 1\), \(\tau = 0\) into equation (2.12), we have
Solving equation (2.13), we have
By calculation, we have
Consequently, \(\vert \lambda \vert < 1\) holds for all sufficiently small \(\tau > 0\). Thus, all roots of equation (2.9) lie in \(\vert z \vert < 1\) for sufficiently small positive τ, and existence of the maximal τ̄ follows. This completes the proof. □
Denote by \(e^{i\omega ^{ *}}\) a root of equation (2.9) at \(\tau = \tau ^{ *}\), then we have
in which \(a_{1}^{ *},a_{2}^{ *},a_{3}^{ *}\) can be obtained by substituting \(\tau ^{ *}\) into \(a_{1},a_{2},a_{3}\). Separating the real and imaginary parts, we have
So
where \(\Delta _{1} = a_{1}^{*2}(a_{2}^{*}  1)^{2}  4a_{2}^{*}(1 2a_{2}^{*} + a_{2}^{*2}  a_{3}^{*2})\),
Lemma 2.3
Assume the step size h is sufficiently small. If \(\Delta _{1} \ge 0\), and \(0 < 16a_{2}^{*}(  a_{1}^{*}a_{2}^{*}  a_{1}^{*} \pm \sqrt{\Delta _{1}} ) < 1\), then equation (2.9) has no root with modulus one for all \(\tau > 0\).
Proof
Assume that the step size h is sufficiently small.
For sufficiently small \(h > 0\), if \(\Delta _{1} \ge 0\), and \(0 < 16a_{2}^{*}(  a_{1}^{*}a_{2}^{*}  a_{1}^{*} \pm \sqrt{\Delta _{1}} ) < 1\), then we obtain \(\cos \omega ^{*} > 1\). This is a contradiction. So we complete proof. □
From Lemma 2.3, we know that if \(16a_{2}^{*}(  a_{1}^{*}a_{2}^{*}  a_{1}^{*} \pm \sqrt{\Delta _{1}} ) > 1\), then the roots \(e^{ \pm i\omega ^{ *}}\) of (2.9) with modulus one satisfy
Here \(\Delta _{1} = a_{1}^{*2}(a_{2}^{*}  1)^{2}  4a_{2}^{*}(1  2a_{2}^{*} + a_{2}^{*2}  a_{3}^{*2})\),
It is easy to see that there is a sequence of time delays \(\tau _{0} < \tau _{1} < \tau _{2} < \cdots < \tau _{n}\) satisfies equation (2.14).
Lemma 2.4
The inequality \(d_{h} = \frac{d\vert \lambda \vert ^{2}}{d\tau }_{\lambda = \lambda ^{ *},\tau = \tau ^{ *}} > 0\) holds for sufficiently small h.
Proof
For sufficiently small h, we have
Here
Note that
This completes the proof. □
From Lemmas 2.12.4, we obtain the following theorem.
Theorem 2.1
When the parameters of system (2.3) are defined, we have:

(i)
When \(\tau \in [0,\tau _{0})\), the equilibrium \((x^{*},y^{*}) \) of system (2.3) is asymptotically stable.
When \(\tau > \tau _{0}\), the equilibrium \(( x^{ *},y^{ *} ) \) of system (2.3) is unstable.

(ii)
When \(\tau = \tau _{0}\), for the equilibrium \(( x^{ *},y^{ *} ) \) of system (2.3) there will exist a NeimarkSacker bifurcation. This is to say, for system (2.3) there exists a cluster of periodic solutions bifurcating near the equilibrium at \(\tau = \tau _{0}\).
Remark
According to the above discussions and applying the NeimarkSacker bifurcation theory presented in Kuznetsov [19], we obtain Theorem 2.1. We can see that the stability of the equilibrium \(( x^{ *},y^{ *} ) \) varies as the parameter τ varies. It is shown that the equilibrium is asymptotically stable for \(\tau \in [0,\tau _{0})\), and unstable for \(\tau > \tau _{0}\). We can observe that the NeimarkSacker bifurcation occurs when the time delay crosses the critical value\(\tau _{0}\). According to the results of Theorem 2.1, we can see that the conclusions of the discrete system (2.3) are consistent with those of the continuous models (see [8, 10, 11]).
Direction and stability of the NeimarkSacker bifurcation
In this section, we will use bifurcation theory [19–21] to discuss the direction and stability of the NeimarkSacker bifurcation of system (2.3) for considering the delay time as a bifurcation parameter.
Let \(\tau ^{ *}\) is the critical value \(\tau _{j}\) (\(j = 0,1,2, \ldots\)) of the origin, at which system (2.6) undergoes a NeimarkSacker bifurcation.
If it is sufficiently smooth, equation (2.6) can be transformed into the form
where
and
Lemma 3.1
If \(q = ( q_{1},q_{2},q_{3},q_{4}, \ldots,q_{d + 2} ) ^{T}\) is the eigenvector of A and its eigenvalue is \(e^{i\omega ^{*}}\); \(p = ( p_{1},p_{2}, \ldots,p_{d + 2} ) ^{T}\) is the eigenvector of \(A^{T}\), and its eigenvalue is \(e^{  i\omega ^{ *}}\), then there exist
Proof
Let \(q(\tau ^{ *} ) \in C^{d + 2}\) be a complex eigenvector of \(A(\tau ^{ *} )\) corresponding to \(e^{i\omega ^{ *}}\), \(p(\tau ^{ *} ) \in C^{d + 2}\) be a complex eigenvector of transposed matrix \(A^{T}(\tau ^{ *} )\) corresponding to \(e^{  i\omega ^{ *}}\), then we have
It is noted that \(q(\tau ^{ *} )\), \(p(\tau ^{ *} )\) should satisfy \(\langle p,q \rangle = 1\), where \(\langle p,q \rangle = \sum_{j = 0}^{d} \bar{p}_{j} q_{j}\).
Let \(q = ( q_{1},q_{2}, \ldots,q_{d + 2} ) ^{T}\) and \(p = ( p_{1},p_{2}, \ldots,p_{d + 2} ) ^{T}\).
Let \(q_{1} = 1\), then we can get the eigenvector q.
It can be obtained following by calculating:
Then p can be obtained in the same way:
Then the proof of Lemma 3.1 is complete. □
Let a real two dimensional eigenspace of \(e^{ \pm \omega ^{ *} j}\) be \(T^{c}\), and \(T^{s}\) be a d dimensional eigenspace which is other than \(T^{c}\).
Take any \(x \in R^{d + 2}\), whose decomposition is
Regarding the variable z as a coordinate on \(T^{C}\),
So, we have a format of the map F at \(\tau = \tau ^{*}\) in the above coordinate as follows:
Then we can obtain the following formulas by applying Taylor expansions:
Here \(g_{ij} \in C\), \(G_{10},G_{01} \in C^{d + 2}\), and
So, we can obtain the format of the center manifold
where \(\langle q^{ *},w_{ij} \rangle = 0\).
Putting equation (3.2) into equation (3.1), then we have
We can obtain the Taylor coefficients as follows:
Define
It easy to obtain \(c_{1} ( \tau ^{*} ) \) by substituting \(z = e^{  i\omega^{*}}\) into equation (3.3).
Theorem 3.1
When \(l > 0\) (<0), where \(l =  \operatorname{Re} [ e^{i\omega ^{ *}} c_{1} ( \tau ^{ *} ) ] / d_{h}\), we know that if the curve exists for \(\tau > \tau ^{ *}\), the bifurcation of equation (2.4) is supercritical (subcritical); when \(\operatorname{Re} [e^{  i\omega ^{ *}} c_{1}(\tau ^{ *} )] < 0\), or \(\operatorname{Re} [e^{  i\omega ^{ *}} c_{1}(\tau ^{ *} )] > 0\) the bifurcation is orbitally stable or unstable.
Numerical simulation
To illustrate the analytical results obtained, we consider the following particular case of equation (2.5). Let \(n_{1} = 0.7\), \(n_{2} = 0.1\), \(n_{3} = 0.1\), \(n_{4} = 1\), \(n_{5} = 1\), \(n_{6} = 0.2\), \(h = 0.2\), \(d = 5\). Then \(\tau _{0} \approx 1.4667\) is the critical value such that NeimarkSacker bifurcation occurs. Figures 15 are the waveform plots and phase diagrams of equation (2.5).
In Figure 1, we show that the waveform plot and phase diagram for equation (2.5) when \(\tau = 1 < \tau _{0}\) and for the equilibrium \(( x^{ *},y^{*} ) \) are asymptotically stable.
In Figure 2, we show that the waveform plot and phase diagram for equation (2.5) when \(\tau = 1.467 > \tau _{0}\) and near \(\tau _{0}\) (\(x^{ *},y^{*}\)) are unstable. From Theorems 2.1 and 3.1, there is a NeimarkSacker bifurcation.
In Figure 3, we show that the waveform plot and phase diagram for equation (2.5) when \(\tau = 1.5 > \tau _{0}\) and \(( x^{ *},y^{ *} ) \) are unstable. There are multiple NeimarkSacker bifurcations.
In Figure 4, we show that the waveform plot and phase diagram for equation (2.5) when \(\tau = 2 > \tau _{0}\) and \(( x^{ *},y^{ *} ) \) are unstable. There is a NeimarkSacker bifurcation.
In Figure 5, we show that for the waveform plot and phase diagram for equation (2.5) when \(\tau = 4.1 > \tau _{0}\), the system (2.5) appears to show chaotic behavior. Figures 5(c) and (d) present the bifurcation diagram and the Lyapunov spectrum for system (2.5), for \(\tau\in [1,4.4]\). Figure 5 presents the complex behavior of system (2.5) in a neighborhood of the origin, for \(\tau \in [1,4.4]\), revealing the routes to chaos starting from the NeimarkSacker and other bifurcations.
From Figures 15, we can observe that a NeimarkSacker bifurcation occurs when the time delay crosses the critical value\(\tau _{0}\), It is shown that the positive equilibrium is asymptotically stable for \(\tau \in [0,\tau _{0})\), and unstable for \(\tau > \tau _{0}\) which is entirely consistent with the results in Theorem 2.1. It can be found that there are different kinds of NeimarkSacker bifurcations for different time delays and when the system undergoes a variety of bifurcations, for the system can be found the chaos phenomenon. Chaos has important significance to the life system. It shows that the parameter time delay has important influence on the dynamic behaviors of the system, which may easily generate complex oscillations and chaos.
Conclusions
A discretetime genetic network with delay was considered in this paper. Some sufficient conditions of local stability of equilibrium points were given. By choosing the time delay as a bifurcation parameter, we show that the NeimarkSacker bifurcation would occur when the bifurcation parameter crosses some critical values. We obtained a formula for determining the direction and stability of a NeimarkSacker bifurcation.
The theoretical studies on discretetime genetic network model may not only contribute to the understanding of dynamic relation among different elements.
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Acknowledgements
The authors are very grateful to the editor and anonymous referees for their valuable comments and helpful suggestions, which have led to a great improvement of the original manuscript. This work was partially supported by National Natural Science Foundation (NNSF) of China under Grant 61472374, 61503053, 61672112, 11401110, 61603358, the fund of research center for advanced control of complex systems & intelligent geosciences instrument, China University of Geosciences (Wuhan) (AU2016CJ021).
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FL and XY wrote the first draft of a paper and did revising and editing; all authors were in charge of the choice of the topic, the method, revising, and editing. All authors read and approved the final manuscript.
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Liu, F., Yin, X., Sun, F. et al. Bifurcation analysis and chaotic behavior of a discretetime delayed genetic oscillator model. Adv Differ Equ 2017, 3 (2017). https://doi.org/10.1186/s1366201610536
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Keywords
 stability
 bifurcation
 Euler discretized
 genetic oscillator model