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Numerical dynamics of a nonstandard finitedifferenceθ method for a red blood cell survival model
Advances in Difference Equations volume 2015, Article number: 86 (2015)
Abstract
In this article, by a nonstandard finitedifferenceθ (NSFDθ) method we study the dynamics of a discrete red blood cell survival model. Firstly, the linear stability of the model is discussed. It is found that the NeimarkSacker bifurcation exists when the delay passes a sequence of critical values. Then the explicit algorithm for determining the direction of the NeimarkSacker bifurcation and the stability of the bifurcating periodic solutions are derived by using the normal form method and center manifold theorem. Our results show the NSFDθ method could inherit the Hopf bifurcation and the asymptotically stability for sufficiently small stepsize \(h=1/m\), where m is a positive integer. In particular, for \(\theta=0,1/2,1\) the results hold for any stepsize \(h=1/m\). Finally, numerical examples are provided to illustrate the theoretical results.
1 Introduction
In order to describe the survival of red blood cells in animal, WazewskaCzyzewska and Lasota [1] proposed the following autonomous functional differential equation:
where \(a\in(0,1) \), \(b,c,\tau\in(0,\infty) \). Here \(x(t) \) denotes the number of red blood cells at time t, a is the probability of death of a red blood cell, b and c are positive constants related to the production of red blood cells per unit time, and τ is the time required to produce a red blood cell. A great deal of research has been devoted to the asymptotic behavior, periodic solutions, global attractivity, and other properties of this model.
The bifurcations of continuoustime model have been discussed in [2–5]. But due to scientific computation and simulation, our interest is focused on the behavior of discrete dynamical system corresponding to (1.1). It is desired that the discretetime model is ‘dynamically consistent’ with the continuoustime model. In [6–12], for many numerical methods, e.g. RungeKutta methods and linear multistep methods, it has been proved that they could retain the local stability and the Hopf bifurcation for some delay differential equations and integrodifferential equations.
In [13, 14], Wulf and Ford showed that, if applying the Euler forward method to solve the delay differential equation, then the discrete scheme is ‘dynamically consistent.’ It means that for all sufficiently small stepsizes the discrete model undergoes a Hopf bifurcation of the same type as the original model. The NSFD scheme [15–18] tries to preserve the significant properties of their continuous analogs and consequently gives reliable numerical results. In [19], by the θmethod Tian investigated the stability properties for the solution of delay differential equations. In [6] by using the Euler method a discrete red blood cell survival model has been discussed. In this paper, we apply a nonstandard finitedifferenceθ (NSFDθ) method to discretize the red blood cell survival model (1.1) with delay. The NSFDθ scheme is a combination of the NSFD scheme and the θ method. So we construct a more general NSFD scheme for model (1.1). We obtain the consistent dynamical results of the corresponding continuoustime model by the NSFDθ method for sufficiently small stepsize. In particular, for \(\theta=0,1/2,1\) the results hold for any stepsize.
The paper is organized as follows. In Section 2, we analyze the distribution of the characteristic equation associated with the discrete red blood cell survival model, and we obtain the existence of the NeimarkSacker bifurcation. In Section 3, the direction and stability of periodic solutions bifurcating from the NeimarkSacker bifurcation of the discrete red blood cell model are determined by using the theories of discrete systems. In the final section, some computer simulations are performed to illustrate the analytical results.
2 Stability analysis
Let \(u(t)=x(\tau t) \). Then (1.1) can be rewritten as
The differential equation
has the general solution \(u(t)=\bar{C}e^{a\tau t}\). We consider a stepsize of the form \(h=1/m\), where \(m\in Z_{+}\). The solution can be written as
This is an exact finitedifference numerical method,
Employ the NSFDθ method to (2.1) and choose the ‘denominator function’ ϕ as
It yields the difference equation
One can see that if \(u_{\ast}\) is a fixed point to (2.3), then \(u_{\ast}\) satisfies
Set \(y_{n}=u_{n}u_{\ast}\). Then \(y_{n}\) satisfies
Introducing a new variable \(Y_{n}=(y_{n},y_{n1},\ldots,y_{nm})^{T}\), we can rewrite (2.5) as
where \(F=(F_{0},F_{1},\ldots,F_{m})^{T}\) and
Clearly the origin is a fixed point of map (2.6), and the linear part of map (2.6) is
where
whose characteristic equation is
Lemma 1
All roots of (2.10) have modulus less than one for sufficiently small \(\tau>0\).
Proof
For \(\tau=0\), (2.10) becomes
The equation has an mfold root \(\lambda=0\) and a simple root \(\lambda = 1\).
Consider the root \(\lambda(\tau) \) such that \(\lambda(0)= 1\). This root is a \(C^{1}\) function of τ. For (2.10), we have
Consequently, all roots of (2.10) lie in \(\lambda<1\) for sufficiently small \(\tau>0\). □
Lemma 2
For any stepsize h, if \(cu_{\ast}<1\), then (2.10) has no root with modulus one for all \(\tau>0\).
Proof
A NeimarkSacker bifurcation occurs when two roots of the characteristic equation (2.10) cross the unit circle. We have to find values of τ such that there exist roots on the unit circle. The roots on the unit circle are given by \(e^{\mathrm{i}w},w\in(\mathrm{\pi },\pi]\). Since we are dealing with a real polynomial, complex roots occur in complex conjugate pairs and we have only to look for \(w\in(0,\pi]\). For \(w\in(0,\pi]\), \(e^{\mathrm{i}w}\) is a root of (2.10) if and only if
Therefore
If \(cu_{\ast}<1\), then \(\cos w>1\), which yields a contradiction. So (2.10) has no root with modulus one for all \(\tau>0\). □
If \(cu_{\ast}>1\), for any stepsize h, \(\cos\omega<1\) and \(\tau>0\) is positive real, from (2.13) we know that
where \([\cdot]\) denotes the greatest integer function. It is clear that there exists a sequence of the time delay parameters \(\tau_{k}\) satisfying (2.12) according to \(\omega=\omega_{k}\).
Let \(\lambda_{i}(\tau)=r_{i}(\tau)e^{\mathrm{i}w_{i}(\tau)}\) be a root of (2.10) near \(\tau=\tau_{i}\) satisfying \(r_{i}(\tau_{i})=1\) and \(w_{i}(\tau_{i})=w_{i}\). We have the following result.
Lemma 3
For sufficiently small stepsize h, if \(cu_{\ast}>1\), then
Particularly, if \(\theta=0,1/2,1\), the above results hold for any stepsize h.
Proof
From (2.10), we obtain
where
Substituting (2.13) into (2.15), we obtain for sufficiently small stepsize h
Especially, when \(\theta=0\)
when \(\theta=1/2\)
and when \(\theta=1\)
Therefore, if \(\theta=0,1/2,1\), the results hold for any stepsize h. This completes the proof. □
Theorem 1
For system (2.6), the following statements are true:

(i)
If \(cu_{\ast}<1\), then \(u=u_{\ast}\) is asymptotically stable for any \(\tau>0\).

(ii)
If \(cu_{\ast}>1\), then \(u=u_{\ast}\) is asymptotically stable for \(\tau\in(0,\tau_{0}) \), and unstable for \(\tau>\tau_{0}\). Equation (2.6) undergoes a NeimarkSacker bifurcation at \(u_{\ast }\) when \(\tau=\tau_{i}\), for \(i=0,1,2,\ldots,[\frac{m1}{2}]\).
Proof
(i) If \(cu_{\ast}<1\), from Lemmas 1 and 2, we know that (2.10) has no root with modulus one for all \(\tau>0\). Applying Corollary 2.4 in [20], all roots of (2.10) have modulus less than one for all \(\tau>0\). The conclusion follows.
(ii) If \(cu_{\ast}>1\), applying Lemma 3, we know that all roots of (2.10) have modulus less than one when \(\tau\in(0,\tau_{0}) \), and (2.10) has at least a couple of roots with modulus greater than one when \(\tau>\tau_{0}\). The conclusion follows. □
Remark 1
According to the conclusions of Theorem 1, we have results that are consistent with those for the corresponding continuoustime model (see [5]).
3 Direction and stability of the NeimarkSacker bifurcation in discrete model
In the previous section, we obtained conditions for NeimarkSacker bifurcation to occur when \(\tau=\tau_{i}\), for \(i=0,1,2,\ldots,[\frac{m1}{2}]\). In this section we study the direction of the NeimarkSacker bifurcation and the stability of the bifurcating periodic solutions when \(\tau=\tau_{0}\), using techniques from normal form and center manifold theory [21–23].
Denote
Since \(G(0,0,0,0)=0\) and \(G_{y_{n+1}}(0,0,0,0)=1>0\), by the implicit function theory, there exists a function \(y_{n+1}=\tilde{y}(y_{n},y_{nm+1},y_{nm}) \) satisfying \(G(\tilde {y}(y_{n},y_{nm+1},y_{nm}),y_{n},y_{nm+1},y_{nm})=0\) and
Therefore
In the sequence, we can calculate all the secondorder and thirdorder partial derivatives of \(\tilde{y}\) with respect to \(y_{n}\), \(y_{nm+1}\) and \(y_{nm}\) at \((0,0,0) \), respectively. So, we can rewrite map (2.6) as
where
Here \(B(\phi, \psi) \) and \(C(\phi, \psi, \eta) \) are symmetric multilinear vector functions of \(\phi, \psi, \eta\in\mathbb{C}^{m+1}\). In coordinates we have
in which \(g(\xi)=\tilde{y}(\xi_{l}, \xi_{m}, \xi_{m+1}) \) for \(\xi\in \mathbb{C}^{m+1}\).
Let \(q=q(\tau_{0})\in \mathbb{C}^{m+1}\) be an eigenvector of A corresponding to \(e^{\mathrm{i}w_{0}}\), then
We also introduce an adjoint eigenvector \(q^{\ast}=q^{\ast}(\tau)\in {\mathbb{C}^{m+1}}\) having the properties
and satisfying the normalization \(\langle q^{\ast},q\rangle=1\), where \(\langle q^{\ast},q\rangle=\sum_{i=0}^{m}\overline{q}_{i}^{\ast}q_{i}\).
Lemma 4
Define a vector valued function \(q: \mathbb {C}\longrightarrow\mathbb{C}^{m+1}\) by
If ξ is an eigenvalue of A, then \(Ap(\xi)=\xi p(\xi) \).
In view of Lemma 4, we have
Lemma 5
Suppose \(q^{\ast}=(q_{0}^{\ast},q_{1}^{\ast},\ldots ,q_{m}^{\ast})^{T}\) is the eigenvector of \(A^{T}\) corresponding to the eigenvalue \(e^{\mathrm{i}w_{0}}\), and \(\langle q^{\ast},q\rangle =1\). Then
where \(a_{m}=e^{a\tau h}\), \(a_{0}=cu_{\ast}(1e^{a\tau h})(1\theta) \), and \(a_{1}=cu_{\ast}(1e^{a\tau h})\theta\) are the coefficients of λ in the characteristic equation (2.10), and
Proof
Assume \(q^{\ast}\) satisfies \(A^{T}q^{\ast}=\overline{z}q^{\ast}\) with \(\overline{z}=e^{\mathrm {i}w_{0}}\), then we have
Let \(q_{m1}^{\ast}=e^{\mathrm{i}w_{0}}\), by the normalization \(\langle q^{\ast},q\rangle =1\) and direct computation, the lemma follows. □
Let \(T_{\mathrm{center}}\) denote a real eigenspace corresponding to \(e^{\pm \mathrm{i}w_{0}}\), which is twodimensional and is spanned by \(\{ \operatorname{Re}(q),\operatorname{Im}(\bar{q})\}\), and \(T_{\mathrm{stable}}\) be a real eigenspace corresponding to all eigenvalues of \(A^{T}\), other than \(e^{\pm\mathrm{i}w_{0}}\), which is \((m1) \)dimensional [23].
All vectors \(x\in\mathbb{R}^{m+1}\) can be decomposed as
where \(v\in C\), \(vq+\bar{v} \bar{q}\in T_{\mathrm{center}}\), and \(y\in T_{\mathrm{stable}}\). The complex variable v can be viewed as a new coordinate on \(T_{\mathrm{center}}\), so we have
Let \(a(\lambda) \) be characteristic polynomial of A and \(\lambda_{0}=e^{\mathrm{i}w_{0}}\), following the algorithms in [23] and using a computation process similar to [13, 14], we have
where
So, we can compute an expression for the critical coefficient \(C_{1}(\tau_{0}) \):
By (3.1), (3.2), and Lemma 5, we get
Thus applying the NeimarkSacker bifurcation theorem [24], the stability of the closed invariant curve can be summarized as follows.
Theorem 2
If \(cu_{\ast}>1\), then \(u=u_{\ast}\) is asymptotically stable for any \(\tau\in[0,\tau_{0}) \) and unstable for \(\tau>\tau_{0}\). An attracting (repelling) invariant closed curve exists for \(\tau>\tau_{0}\) if \(\Re[e^{\mathrm{i}w_{0}}C_{1}(\tau_{0})]<0\) (>0).
4 Numerical simulations
The purpose of this section is to test the results in Sections 2 and 3 by examples. Let \(a=0.5\), \(b=2\), \(c=3.695\), then \(u_{\ast}=0.5413\). \(cu_{\ast}\approx2>1\) is satisfied. We compute the bifurcation points and direction of bifurcation at the fixed point of the numerical solutions of (1.1) with the NSFDθ method (2.6) for some stepsize. The results refer to Figures 14.
Firstly, let us see \(\theta=0\). We obtain \(h=1/2\), \(\tau_{0}=1.8229\); \(h=1/10\), \(\tau_{0}=2.2950\); \(h=1/40\), \(\tau_{0}=2.3879\). The numerical solution refers to Figure 1.
Secondly, let \(\theta=1/4\). We obtain \(h=1/2\), \(\tau_{0}=2.2519\); \(h=1/10\), \(\tau_{0}=2.3653\); \(h=1/40\), \(\tau_{0}=2.4038\). The numerical solution is shown in Figure 2.
Thirdly, we test \(\theta=1/2\). We obtain \(h=1/2\), \(\tau_{0}=2.7722\); \(h=1/10\), \(\tau_{0}=2.4302\); \(h=1/40\), \(\tau_{0}=2.4191\). The numerical solution refers to Figure 3.
Finally, let \(\theta=1\). We have \(h=1/2\), \(\tau_{0}=2.7725\); \(h=1/10\), \(\tau _{0}=2.5346\); \(h=1/40\), \(\tau_{0}=2.4483\). The numerical solution is in Figure 4.
At the same time, by the Euler method we compute \(h=1/2\), \(\tau _{0}=1.4641\); \(h=1/10\), \(\tau_{0}=2.1682\); \(h=1/40\), \(\tau_{0}=2.3526\). The numerical solution is in Figure 5.
We see that there exists a sequence of \(\tau_{i}\), and with the increasing of m, \(\tau_{0}\) asymptotically converges to \(\tau_{0}=2.4184\), which is the true value.
When \(\theta=0,1/4,1/2,1\), we obtain Figures 14. From the above analysis and Figures 14, one comes to a better conclusion for \(\theta=1/2\) than the other values by means of describing approximately the dynamics of the original system with the same stepsize. Through Figure 5 we could argue that the NSFDθ method is better than the Euler method by means of describing approximately the dynamics of the original system with the same stepsize.
5 Conclusions
From the above analysis we could draw the biological conclusions. We can find that the delay does not influence the system’s stability when the coefficients of the system satisfy the condition \(cu_{\ast}<1\). But when the coefficients of system satisfy the condition \(cu_{\ast}>1\), for a small delay the positive fixed point of system is stable, the number of red blood cells reaches an equilibrium. With the increasing of the delay (the critical point \(\tau_{0}\)), the positive fixed point loses its stability and a family of periodic solutions occurs, the number of red blood cells oscillates around the unstable equilibrium. In real life, we will try to control τ so that it does not exceed the critical point \(\tau_{0}\). Therefore it can produce a stable system. These results are very useful to the biologists. The existence of a NeimarkSacker bifurcation shows the periodic oscillatory behavior of the discrete red blood cell survival system.
It is noted that system (1.1) is a delay differential equation, making it an impossible task to obtain its analytical solutions to study its qualitative properties; therefore it is necessary to solve numerical solutions or approximate solutions of system (1.1) according to different discrete difference schemes. It is common to change a continuous dynamical system into a discretetime dynamical system. However, the derived difference equation will be acceptable only if it preserves the dynamical feature of the continuoustime models. In this paper, through the NSFDθ method, we obtain general results. Our results show the NSFDθ method could inherit the Hopf bifurcation and the asymptotically stability for sufficiently small stepsize. In particular, for \(\theta=0,1/2,1\) the results hold for any stepsize.
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Acknowledgements
The author is grateful to the referees for their helpful comments and constructive suggestions. This work was supported by the National Natural Science Foundation of China (Grant No. 11401586) and by the Fundamental Research Funds for the Central Universities of China (14CX02159A) and by the NNSF Shandong Province (Nos. ZR2014AL008, ZR2014AQ004, ZR2014AQ014).
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Wang, Y. Numerical dynamics of a nonstandard finitedifferenceθ method for a red blood cell survival model. Adv Differ Equ 2015, 86 (2015). https://doi.org/10.1186/s1366201504328
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DOI: https://doi.org/10.1186/s1366201504328
Keywords
 nonstandard finitedifferenceθ (NSFDθ) method
 delay
 NeimarkSacker bifurcation
 Hopf bifurcation