# Numerical Oscillations Analysis for Nonlinear Delay Differential Equations in Physiological Control Systems

Authors: Qi Wang; Jiechang Wen
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

This paper deals with the oscillations of numerical solutions for the nonlinear delay differential equations in physiological control systems. The exponentialθ-method is applied to p′(t)=β0ωμp(t−τ)/(ωμ+pμ(t−τ))−γp(t) and it is shown that the exponential θ-method has the same order of convergence as that of the classical θ-method. Several conditions under which the numerical solutions oscillate are derived. Moreover, it is proven that every nonoscillatory numerical solution tends to positive equilibrium of the continuous system. Finally, the main results are illustrated with numerical examples.


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## Body

## 1. Introduction

The nonlinear delay differential equation(1.1)p′(t)=β0ωμp(t-τ)ωμ+pμ(t-τ)-γp(t),
where
(1.2)ω>0,β0>γ>0,μ,τ∈R+,
has been proposed by Mackey and Glass [1] as model of hematopoiesis (blood cell production). Here, p(t) denotes the density of mature cells in blood circulation, τ is the time delay between the production of immature cells in the bone marrow and their maturation for release in the circulating blood stream, and the production is a single-humped function of  p(t-τ). Equation (1.1) has been recently studied by many authors. Mackey and Heidn [2] considered the local asymptotic stability of the positive equilibrium by the well-known technique of linearization. Gopalsamy et al. [3] obtained sufficient and also necessary conditions for all positive solutions to oscillate about their positive steady states. They also obtained sufficient conditions for the positive equilibrium to be a global attractor. For more details of (1.1), we refer to Mackey [4, 5], and Su et al. [6].Our aim in this paper is to investigate the oscillations of numerical solutions for (1.1). The oscillatory and asymptotic behavior of solutions of delay differential equations has been the subject of intensive investigations during the past decades. The strong interest in this study is motivated by the fact that it has many useful applications in some mathematical models, such as ecology, biology, and spread of some infectious diseases in humans. The general theory and basic results for this paper have been thoroughly studied in [7, 8]. In recent years, much research has been focused on the oscillations of numerical solutions for delay differential equations [9–12]. Until now, very few results dealing with the corresponding behavior for nonlinear delay differential equations have been presented in the literature except for [13]. In [13], the authors investigate the oscillations of numerical solutions for the nonlinear delay differential equation of population dynamics. Different from [13], in our paper, we will consider another nonlinear delay differential equation (1.1) in physiological control systems and obtain some new results. We not only investigate some sufficient conditions under which the numerical solutions are oscillatory but also consider the asymptotic behavior of nonoscillatory numerical solutions.The structure of this paper is as follows. In Section2, some necessary definitions and results for oscillations of the analytic solutions are given. In Section 3, we obtain the numerical discrete equation by applying the θ-methods to the simplified form which comes from making two transformations on (1.1). Moreover, the oscillations of the numerical solutions are discussed, and conditions under which the numerical solutions oscillate are obtained. In Section 4, we investigate the asymptotic behavior of nonoscillatory solutions. In Section 5, we present numerical examples that illustrate the theoretical results for the numerical methods.


## 2. Preliminaries

Let us state some definitions, lemmas, and theorems that will be used throughout this paper.Definition 2.1.
A functionp of (1.1) is said to oscillate about M* if p-M* has arbitrarily large zeros. Otherwise, p is called non-oscillatory. When M*=0, we say that p oscillates about zero or simply oscillates.Definition 2.2.
A sequence{pn} is said to oscillate about {zn} if {pn-zn} is neither eventually positive nor eventually negative. Otherwise, {pn} is called non-oscillatory. If {zn}={z} is a constant sequence, we simply say that {pn} oscillates about {z}. When {zn}={0}, we say that {pn} oscillates about zero or simply oscillates.Definition 2.3.
We say that (1.1) oscillates if all of its solutions are oscillatory.Theorem 2.4 (see [14]).
Consider the difference equation(2.1)an+1-an+∑j=-klqjan+j=0,
and assume that k,l∈N and qj∈R for j=-k,…,l. Then, the following statements are equivalent:(1)
every solution of (2.1) oscillates;(2)
the characteristic equationλ-1+∑j=-klqjλj=0 has no positive roots.Theorem 2.5 (see [14]).
Consider the difference equation(2.2)an+1-an+san-k+qan=0,
where k>0,p>0, and q>0. Then, the necessary and sufficient conditions for the oscillation of all solutions of (2.2) are  q∈(0,1) and
(2.3)s(k+1)k+1kk>(1-q)k+1.Lemma 2.6.
The inequalityln(1+x)>x/(1+x) holds for x>-1 and x≠0.Lemma 2.7.
The inequalityex<1/(1-x) holds for x<-1 and x≠0.Lemma 2.8 (see [15]).
For allm≥M0, one has(1)
(1+a/(m-θa))m≥ea if and only if 1/2≤θ≤1 for a>0, φ(-1)≤θ≤1 for a<0;(2)
(1+a/(m-θa))m<ea if and only if 0≤θ<1/2 for a<0, 0≤θ≤φ(1) for a>0,where φ(x)=1/x-1/(ex-1) and M0 is a positive constant.


## 3. Oscillations of Numerical Solutions

### 3.1. Two Transformations

In order to study (1.1) conveniently, we will impose two transformations on (1.1) in this subsection.Together with (1.1), we will consider the initial condition,
(3.1)p(t)=ψ(t),-τ≤t≤0,
the initial value problem (1.1), and (3.1) has a unique positive solution for all t≥0.We  introduce a similar method in [3]. The change of variables
(3.2)p(t)=ωx(t)
turns (1.1) into the delay differential equation
(3.3)x′(t)=β0x(t-τ)1+xμ(t-τ)-γx(t),
with positive equilibrium M, which is denoted as
(3.4)M=(β0-γγ)1/μ.
The following theorem gives oscillations of the analytic solution of (3.3).Theorem 3.1 (see [3]).
Assume that(3.5)μ>1,β0γ>μμ-1,(3.6)eγτγβ0((μ-1)β0-μγ)τ>1e,
then every positive solution of (3.3) oscillates about its positive equilibrium M.The following corollary is naturally obtained.Corollary 3.2.
Assume that all the conditions in Theorem3.1 hold, then every positive solution of (1.1) oscillates about its positive equilibrium M*=ωM.Next, we introduce an invariant oscillation transformationx(t)=Mey(t), and then (3.3) can be written as
(3.7)y′(t)+β01+Mμ[f1(y(t))f2(y(t-τ))-f1(y(t))-f2(y(t-τ))+2]=0,
where
(3.8)f1(u)=1-e-u,f2(u)=1+(1+Mμ)eu1+Mμeμu.
Then, x(t) oscillates about M if and only if y(t) oscillates about zero.Moreover, since(3.9)β01+Mμ=γ,
then (3.6) and (3.7) become
(3.10)eγτγ(μMμ1+Mμ-1)τ>1e,(3.11)y′(t)=-γf1(y(t))f2(y(t-τ))+γf1(y(t))+γf2(y(t-τ))-2γ,
respectively.For our convenience, denote(3.12)Q=μMμ1+Mμ-1,
then the inequality (3.10) yields
(3.13)eγτQγτ>1e.


### 3.2. The Difference Scheme

Leth=τ/m be a given stepsize with integer m>1. The adaptation of the linear θ-method and the one-leg θ-method to (3.11) leads to the same numerical process of the following type:
(3.14)yn+1=yn-hθγf1(yn+1)f2(yn+1-m)-h(1-θ)γf1(yn)f2(yn-m)+hθγf1(yn+1)+h(1-θ)γf1(yn)+hθγf2(yn+1-m)+h(1-θ)γf2(yn-m)-2hγ,
where 0≤θ≤1,yn+1 and yn+1-m are approximations to y(t) and y(t-τ) of (3.11) at tn+1, respectively.Lettingyn=ln(pn/M*) and using the expressions of f1 and f2, we have
(3.15)pn+1=pnexp(hγωμ(1+Mμ)(θpn+1-mpn+1(ωμ+pn+1-mμ)+(1-θ)pn-mpn(ωμ+pn-mμ))-hγ).Definition 3.3.
We call the iteration formula (3.15) the exponential θ-method for (1.1), where pn+1 and pn+1-m are approximations to p(t) and p(t-τ) of (1.1) at tn+1, respectively.The following theorem, for the proof of which we refer to [16], allows us to obtain the convergence of exponential θ-method.Theorem 3.4.
The exponentialθ-method (3.15) is convergent with order
(3.16)1,whenθ≠12,2,whenθ=12.


### 3.3. Oscillation Analysis

It is not difficult to know thatpn oscillates about M* if and only if yn is oscillatory. In order to study oscillations of (3.15), we only need to consider the oscillations of (3.14). The following conditions which are taken from [3] will be used in the next analysis:
(3.17)uf1(u)>0,foru≠0,limu→0f1(u)u=1,f2(u)>0,for   everyu,limu→0f2(u)=2,f1(u)≤u,foru>0,f1(0)=0,f2(u)≤2,foru≥0,μ>2,Mμ>1,f2(0)=2.
The linearized form of (3.14) is given by
(3.18)yn+1=yn-hθγyn+1-h(1-θ)γyn+hθγ(1-μMμ1+Mμ)yn+1-m+h(1-θ)γ(1-μMμ1+Mμ)yn-m.
Then by (3.12), (3.18) gives
(3.19)yn+1=1-h(1-θ)γ1+hθγyn-hθγQ1+hθγyn+1-m-h(1-θ)γQ1+hθγyn-m.
It follows from [14] that (3.14) oscillates if (3.19) oscillates under the condition (3.17).Definition 3.5.
Equation (3.15)   is said to be oscillatory if all of its solutions are oscillatory.Definition 3.6.
We say that the exponentialθ-method preserves the oscillations of (1.1) if (1.1) oscillates, then there is a h->0 or h-=∞, such that (3.15) oscillates for h<h-. Similarly, we say that the exponential θ-method preserves the nonoscillations of (1.1) if (1.1) non-oscillates, then there is a h->0 or h¯=∞, such that (3.15) nonoscillates for h<h-.In the following, we will study whether the exponentialθ-method inherits the oscillations of (1.1). Equivalently, when Corollary 3.2 holds, we will investigate the conditions under which (3.15) is oscillatory.Lemma 3.7.
The characteristic equation of (3.18) is given by
(3.20)ξ=R(-hγ(1+Qξ-m)).Proof.
Lettingyn=ξny0 in (3.18), we have
(3.21)ξn+1y0=ξny0-hθγξn+1y0-h(1-θ)γξny0+hθγ(1-μMμ1+Mμ)ξn+1-my0+h(1-θ)γ(1-μMμ1+Mμ)ξn-my0,
that is,
(3.22)ξ=1-hθγξ(1-(1-μMμ1+Mμ)ξ-m)-h(1-θ)γ(1-(1-μMμ1+Mμ)ξ-m),
which is equivalent to
(3.23)ξ=1-h(1-θ)γ(1-(1-(μMμ)/(1+Mμ))ξ-m)1+hθγ(1-(1-(μMμ)/(1+Mμ))ξ-m)=1-hγ(1-(1-(μMμ)/(1+Mμ))ξ-m)1+hθγ(1-(1-(μMμ)/(1+Mμ))ξ-m).
In view of [17], we know that the stability function of the θ-method is
(3.24)R(x)=1+(1-θ)x1-θx=1+x1-θx.
By noticing (3.12), thus the characteristic equation of (3.18) is given by (3.20). The proof is completed.Lemma 3.8.
If condition (3.13) holds, then the characteristic equation (3.20) has no positive roots for 0≤θ≤1/2.Proof.
Letf(ξ)=ξ-R(-hγ(1+Qξ-m)). By Lemma 2.8, we know that
(3.25)R(-hγ(1+Qξ-m))≤exp(-hγ(1+Qξ-m))
holds for ξ>0 and 0≤θ≤1/2. In the following, we will prove that g(ξ)=ξ-exp(-hγ(1+Qξ-m))>0 for ξ>0. Suppose the opposite, that is, there exists a ξ0>0 such that W(ξ0)≤0, then we get ξ0≤exp(-hγ(1+Qξ0-m)), and
(3.26)ξ0m≤exp(-γτ-γτQξ0-m).
Multiplying both sides of the inequality (3.26) by eγτQγτeξ0-m, we have
(3.27)eγτQγτe≤Qγτξ0-mexp(1-Qγτξ0-m)
Thus, we have the following two cases.
Case  1. If 1-Qγτξ0-m=0, then eγτQγτe≤1, which contradicts the condition (3.13).
Case  2. If 1-Qγτξ0-m≠0, then in view of Lemma 2.7, we obtain
(3.28)exp(1-Qγτξ0-m)<11-(1-Qγτξ0-m)=1Qγτξ0-m,
that is,
(3.29)Qγτξ0-mexp(1-Qγτξ0-m)<1,
so eγτQγτe<1, which is also a contradiction to (3.13).
In summary, we have, forξ>0,
(3.30)f(ξ)=ξ-R(-hγ(1+Qξ-m))≥ξ-exp(-hγ(1+Qξ-m))=g(ξ)>0,
which implies that the characteristic equation (3.20) has no positive roots. The proof is complete.Without loss of generality, in the case of1/2<θ≤1, we assume that m>1.Lemma 3.9.
If condition (3.13) holds and 1/2<θ≤1, then the characteristic equation (3.20) has no positive roots for h<h^, where
(3.31)h^={∞,forQγτ≥1,τ(1+γτ+lnQγτ)1+γτ(1-lnQγτ),forQγτ<1.Proof.
SinceR(-hγ(1+Qξ-m)) is an increasing function of θ when ξ>0, then, for ξ>0 and 1/2<θ≤1,
(3.32)R(-hγ(1+Qξ-m))=1-h(1-θ)γ(1+Qξ-m)1+hθγ(1+Qξ-m)≤11+hγ(1+Qξ-m).
In the following, we will prove that the inequality
(3.33)ξ-11+hγ(1+Qξ-m)>0
holds under certain conditions.
The left side of inequality (3.33) can be rewritten as
(3.34)ξ-11+hγ(1+Qξ-m)=(1+hγ)ξ1-m1+hγ(1+Qξ-m)Γ(ξ),
where
(3.35)Γ(ξ)=ξm-11+hγξm-1+hγQ1+hγ,
so we only need to prove that Γ(ξ)>0 for ξ>0. It is easy to know that Γ(ξ)=0 is the characteristic equation of the following difference equation(3.36)yn+1-yn+hγQ1+hγyn+1-m+hγ1+hγyn=0.
By Theorems 2.4 and 2.5, we know that Γ(ξ) has no positive roots if and only if
(3.37)hγQ1+hγmm(m-1)m-1>(1-hγ1+hγ)m,
which is equivalent to
(3.38)lnQγτ+(m-1)ln(1+1+γτm-1)>0.
We examine two cases depending on the position of Qγτ: either Qγτ≥1 or Qγτ<1.
Case  1. If Qγτ≥1, by m>1, (3.38) holds true.
Case  2. If Qγτ<1 and
(3.39)h<τ(1+γτ+lnQγτ)1+γτ(1-lnQγτ),
then by Lemma 2.6, we have
(3.40)lnQγτ+(m-1)ln(1+1+γτm-1)>lnQγτ+(m-1)(1+γτ)/(m-1)1+(1+γτ)/(m-1)=lnQγτ+(m-1)(1+γτ)m+γτ>0.
Therefore, the inequality (3.33) holds for h<h^, where h^ is defined in (3.31). So, we get that the following inequality
(3.41)f(ξ)=ξ-R(-hγ(1+Qξ-m))≥ξ-11+hγ(1+Qξ-m)>0
holds for h<h^ and ξ>0, which implies that the characteristic equation (3.20) has no positive roots. This completes the proof.Remark 3.10.
By inequality (3.38) and condition Qγτ<1, we have
(3.42)τ(1+γτ+lnQγτ)1+γτ(1-lnQγτ)>0,
thus h^ is meaningful.In view of (3.17), Lemmas 3.8 and 3.9, and Theorem 2.4, we present the first main theorem of this paper.Theorem 3.11.
If condition (3.13) holds, then (3.15) is oscillatory for
(3.43)h<h~={∞,when0≤θ≤12,h^,when12<θ≤1,
where h^ is defined in Lemma 3.9.


## 3.1. Two Transformations

In order to study (1.1) conveniently, we will impose two transformations on (1.1) in this subsection.Together with (1.1), we will consider the initial condition,
(3.1)p(t)=ψ(t),-τ≤t≤0,
the initial value problem (1.1), and (3.1) has a unique positive solution for all t≥0.We  introduce a similar method in [3]. The change of variables
(3.2)p(t)=ωx(t)
turns (1.1) into the delay differential equation
(3.3)x′(t)=β0x(t-τ)1+xμ(t-τ)-γx(t),
with positive equilibrium M, which is denoted as
(3.4)M=(β0-γγ)1/μ.
The following theorem gives oscillations of the analytic solution of (3.3).Theorem 3.1 (see [3]).
Assume that(3.5)μ>1,β0γ>μμ-1,(3.6)eγτγβ0((μ-1)β0-μγ)τ>1e,
then every positive solution of (3.3) oscillates about its positive equilibrium M.The following corollary is naturally obtained.Corollary 3.2.
Assume that all the conditions in Theorem3.1 hold, then every positive solution of (1.1) oscillates about its positive equilibrium M*=ωM.Next, we introduce an invariant oscillation transformationx(t)=Mey(t), and then (3.3) can be written as
(3.7)y′(t)+β01+Mμ[f1(y(t))f2(y(t-τ))-f1(y(t))-f2(y(t-τ))+2]=0,
where
(3.8)f1(u)=1-e-u,f2(u)=1+(1+Mμ)eu1+Mμeμu.
Then, x(t) oscillates about M if and only if y(t) oscillates about zero.Moreover, since(3.9)β01+Mμ=γ,
then (3.6) and (3.7) become
(3.10)eγτγ(μMμ1+Mμ-1)τ>1e,(3.11)y′(t)=-γf1(y(t))f2(y(t-τ))+γf1(y(t))+γf2(y(t-τ))-2γ,
respectively.For our convenience, denote(3.12)Q=μMμ1+Mμ-1,
then the inequality (3.10) yields
(3.13)eγτQγτ>1e.


## 3.2. The Difference Scheme

Leth=τ/m be a given stepsize with integer m>1. The adaptation of the linear θ-method and the one-leg θ-method to (3.11) leads to the same numerical process of the following type:
(3.14)yn+1=yn-hθγf1(yn+1)f2(yn+1-m)-h(1-θ)γf1(yn)f2(yn-m)+hθγf1(yn+1)+h(1-θ)γf1(yn)+hθγf2(yn+1-m)+h(1-θ)γf2(yn-m)-2hγ,
where 0≤θ≤1,yn+1 and yn+1-m are approximations to y(t) and y(t-τ) of (3.11) at tn+1, respectively.Lettingyn=ln(pn/M*) and using the expressions of f1 and f2, we have
(3.15)pn+1=pnexp(hγωμ(1+Mμ)(θpn+1-mpn+1(ωμ+pn+1-mμ)+(1-θ)pn-mpn(ωμ+pn-mμ))-hγ).Definition 3.3.
We call the iteration formula (3.15) the exponential θ-method for (1.1), where pn+1 and pn+1-m are approximations to p(t) and p(t-τ) of (1.1) at tn+1, respectively.The following theorem, for the proof of which we refer to [16], allows us to obtain the convergence of exponential θ-method.Theorem 3.4.
The exponentialθ-method (3.15) is convergent with order
(3.16)1,whenθ≠12,2,whenθ=12.


## 3.3. Oscillation Analysis

It is not difficult to know thatpn oscillates about M* if and only if yn is oscillatory. In order to study oscillations of (3.15), we only need to consider the oscillations of (3.14). The following conditions which are taken from [3] will be used in the next analysis:
(3.17)uf1(u)>0,foru≠0,limu→0f1(u)u=1,f2(u)>0,for   everyu,limu→0f2(u)=2,f1(u)≤u,foru>0,f1(0)=0,f2(u)≤2,foru≥0,μ>2,Mμ>1,f2(0)=2.
The linearized form of (3.14) is given by
(3.18)yn+1=yn-hθγyn+1-h(1-θ)γyn+hθγ(1-μMμ1+Mμ)yn+1-m+h(1-θ)γ(1-μMμ1+Mμ)yn-m.
Then by (3.12), (3.18) gives
(3.19)yn+1=1-h(1-θ)γ1+hθγyn-hθγQ1+hθγyn+1-m-h(1-θ)γQ1+hθγyn-m.
It follows from [14] that (3.14) oscillates if (3.19) oscillates under the condition (3.17).Definition 3.5.
Equation (3.15)   is said to be oscillatory if all of its solutions are oscillatory.Definition 3.6.
We say that the exponentialθ-method preserves the oscillations of (1.1) if (1.1) oscillates, then there is a h->0 or h-=∞, such that (3.15) oscillates for h<h-. Similarly, we say that the exponential θ-method preserves the nonoscillations of (1.1) if (1.1) non-oscillates, then there is a h->0 or h¯=∞, such that (3.15) nonoscillates for h<h-.In the following, we will study whether the exponentialθ-method inherits the oscillations of (1.1). Equivalently, when Corollary 3.2 holds, we will investigate the conditions under which (3.15) is oscillatory.Lemma 3.7.
The characteristic equation of (3.18) is given by
(3.20)ξ=R(-hγ(1+Qξ-m)).Proof.
Lettingyn=ξny0 in (3.18), we have
(3.21)ξn+1y0=ξny0-hθγξn+1y0-h(1-θ)γξny0+hθγ(1-μMμ1+Mμ)ξn+1-my0+h(1-θ)γ(1-μMμ1+Mμ)ξn-my0,
that is,
(3.22)ξ=1-hθγξ(1-(1-μMμ1+Mμ)ξ-m)-h(1-θ)γ(1-(1-μMμ1+Mμ)ξ-m),
which is equivalent to
(3.23)ξ=1-h(1-θ)γ(1-(1-(μMμ)/(1+Mμ))ξ-m)1+hθγ(1-(1-(μMμ)/(1+Mμ))ξ-m)=1-hγ(1-(1-(μMμ)/(1+Mμ))ξ-m)1+hθγ(1-(1-(μMμ)/(1+Mμ))ξ-m).
In view of [17], we know that the stability function of the θ-method is
(3.24)R(x)=1+(1-θ)x1-θx=1+x1-θx.
By noticing (3.12), thus the characteristic equation of (3.18) is given by (3.20). The proof is completed.Lemma 3.8.
If condition (3.13) holds, then the characteristic equation (3.20) has no positive roots for 0≤θ≤1/2.Proof.
Letf(ξ)=ξ-R(-hγ(1+Qξ-m)). By Lemma 2.8, we know that
(3.25)R(-hγ(1+Qξ-m))≤exp(-hγ(1+Qξ-m))
holds for ξ>0 and 0≤θ≤1/2. In the following, we will prove that g(ξ)=ξ-exp(-hγ(1+Qξ-m))>0 for ξ>0. Suppose the opposite, that is, there exists a ξ0>0 such that W(ξ0)≤0, then we get ξ0≤exp(-hγ(1+Qξ0-m)), and
(3.26)ξ0m≤exp(-γτ-γτQξ0-m).
Multiplying both sides of the inequality (3.26) by eγτQγτeξ0-m, we have
(3.27)eγτQγτe≤Qγτξ0-mexp(1-Qγτξ0-m)
Thus, we have the following two cases.
Case  1. If 1-Qγτξ0-m=0, then eγτQγτe≤1, which contradicts the condition (3.13).
Case  2. If 1-Qγτξ0-m≠0, then in view of Lemma 2.7, we obtain
(3.28)exp(1-Qγτξ0-m)<11-(1-Qγτξ0-m)=1Qγτξ0-m,
that is,
(3.29)Qγτξ0-mexp(1-Qγτξ0-m)<1,
so eγτQγτe<1, which is also a contradiction to (3.13).
In summary, we have, forξ>0,
(3.30)f(ξ)=ξ-R(-hγ(1+Qξ-m))≥ξ-exp(-hγ(1+Qξ-m))=g(ξ)>0,
which implies that the characteristic equation (3.20) has no positive roots. The proof is complete.Without loss of generality, in the case of1/2<θ≤1, we assume that m>1.Lemma 3.9.
If condition (3.13) holds and 1/2<θ≤1, then the characteristic equation (3.20) has no positive roots for h<h^, where
(3.31)h^={∞,forQγτ≥1,τ(1+γτ+lnQγτ)1+γτ(1-lnQγτ),forQγτ<1.Proof.
SinceR(-hγ(1+Qξ-m)) is an increasing function of θ when ξ>0, then, for ξ>0 and 1/2<θ≤1,
(3.32)R(-hγ(1+Qξ-m))=1-h(1-θ)γ(1+Qξ-m)1+hθγ(1+Qξ-m)≤11+hγ(1+Qξ-m).
In the following, we will prove that the inequality
(3.33)ξ-11+hγ(1+Qξ-m)>0
holds under certain conditions.
The left side of inequality (3.33) can be rewritten as
(3.34)ξ-11+hγ(1+Qξ-m)=(1+hγ)ξ1-m1+hγ(1+Qξ-m)Γ(ξ),
where
(3.35)Γ(ξ)=ξm-11+hγξm-1+hγQ1+hγ,
so we only need to prove that Γ(ξ)>0 for ξ>0. It is easy to know that Γ(ξ)=0 is the characteristic equation of the following difference equation(3.36)yn+1-yn+hγQ1+hγyn+1-m+hγ1+hγyn=0.
By Theorems 2.4 and 2.5, we know that Γ(ξ) has no positive roots if and only if
(3.37)hγQ1+hγmm(m-1)m-1>(1-hγ1+hγ)m,
which is equivalent to
(3.38)lnQγτ+(m-1)ln(1+1+γτm-1)>0.
We examine two cases depending on the position of Qγτ: either Qγτ≥1 or Qγτ<1.
Case  1. If Qγτ≥1, by m>1, (3.38) holds true.
Case  2. If Qγτ<1 and
(3.39)h<τ(1+γτ+lnQγτ)1+γτ(1-lnQγτ),
then by Lemma 2.6, we have
(3.40)lnQγτ+(m-1)ln(1+1+γτm-1)>lnQγτ+(m-1)(1+γτ)/(m-1)1+(1+γτ)/(m-1)=lnQγτ+(m-1)(1+γτ)m+γτ>0.
Therefore, the inequality (3.33) holds for h<h^, where h^ is defined in (3.31). So, we get that the following inequality
(3.41)f(ξ)=ξ-R(-hγ(1+Qξ-m))≥ξ-11+hγ(1+Qξ-m)>0
holds for h<h^ and ξ>0, which implies that the characteristic equation (3.20) has no positive roots. This completes the proof.Remark 3.10.
By inequality (3.38) and condition Qγτ<1, we have
(3.42)τ(1+γτ+lnQγτ)1+γτ(1-lnQγτ)>0,
thus h^ is meaningful.In view of (3.17), Lemmas 3.8 and 3.9, and Theorem 2.4, we present the first main theorem of this paper.Theorem 3.11.
If condition (3.13) holds, then (3.15) is oscillatory for
(3.43)h<h~={∞,when0≤θ≤12,h^,when12<θ≤1,
where h^ is defined in Lemma 3.9.


## 4. Asymptotic Behavior of Nonoscillatory Solutions

In this section, we will study the asymptotic behavior of non-oscillatory solutions of (3.15). We first recall the following result about asymptotic behavior of (3.3).Lemma 4.1 (see [3]).
Assume that(4.1)τ>0,μ>1,β0>γ,
then every solution x(t) of the initial value problem is
(4.2)x′(t)=β0x(t-τ)1+xμ(t-τ)-γx(t),x(t)=φ,-τ≤t≤0,
which is non-oscillatory about  M satisfies
(4.3)limt→∞x(t)=M.From (3.3) and (3.7), we know that the non-oscillatory solution of (3.7) satisfies limt→∞y(t)=0 if Lemma 4.1 is satisfied. Furthermore, limt→∞p(t)=M* is also obtained. In the following, we will prove that the numerical solution of (1.1) can inherit this property.Lemma 4.2.
Letyn be a non-oscillatory solution of (3.14), then limn→∞yn=0.Proof.
Without loss of generality, we assume thatyn>0 for sufficiently large n. Then by condition (3.17), we have
(4.4)f1(yi)>0,f2(yi)-1>0,f2(yi)-2<0,
for sufficiently large i. Moreover, it is can be seen from (3.14) that
(4.5)yn+1-yn=-hθγf1(yn+1)[f2(yn+1-m)-1]-h(1-θ)γf1(yn)[f2(yn-m)-1]+hθγ[f2(yn+1-m)-2]+h(1-θ)γ[f2(yn-m)-2],
which gives
(4.6)yn+1-yn-hθγ[f2(yn+1-m)-2]-h(1-θ)γ[f2(yn-m)-2]<0,
thus we have
(4.7)yn+1-yn<hθγ[f2(yn+1-m)-2]+h(1-θ)γ[f2(yn-m)-2]<0,
then the sequence {yn} is decreasing, and, therefore,
(4.8)limn→∞yn=η∈[0,∞).
Next, we prove that η=0. If η>0, then there existN∈N and ε>0 such that 0<η-ε<yn<η+ε for n-m>N. Thus, yn-m>η-ε and yn-m+1>η-ε. So, inequality (4.6) yields
(4.9)yn+1-yn-hθγ[f2(η-ε)-2]-h(1-θ)γ[f2(η-ε)-2]<0,
which implies that yn+1-yn<C<0, where
(4.10)C=hγ[f2(η-ε)-2]=hγ(eη-ε-1)+Mμeη-ε(1-e(μ-1)(η-ε))1+Mμeμ(η-ε).
Thus, yn→-∞ as n→∞, which is a contradiction to (4.8). This completes the proof.As a consequence, the second main theorem of this paper is as follows.Theorem 4.3.
Lettingpn be a positive solution of (3.15), which does not oscillate about M*, then limn→∞pn=M*, where M* is the positive equilibrium of the continuous system (1.1).


## 5. Numerical Examples

In order to verify our results, three numerical examples are examined in this section.Example 5.1.
Consider the following equation:(5.1)p′(t)=3×2.65p(t-2.5)2.65+p5(t-2.5)-1.5p(t),
subject to the initial condition
(5.2)p(t)=1.7,fort≤0.
In (5.1), it can be seen that condition (3.6) holds true and Qγτ≈5.6250>1. That is, the analytic solutions of (5.1) are oscillatory. In Figures 1–3, we draw the figures of the analytic solutions and the numerical solutions of (5.1), respectively. Set m=25, θ=0.25 in Figure 2 and m=10, θ=0.6 in Figure 3, respectively. From the two figures, we can see that the numerical solutions of (5.1) oscillate about M*≈2.6, which are in agreement with Theorem 3.11.Figure 1
The analytic solution of (5.1).Figure 2
The numerical solution of (5.1) with m=25 and θ=0.25.Figure 3
The numerical solution of (5.1) with m=10 and θ=0.6.Example 5.2.
Let us consider the equation(5.3)p′(t)=1.77×1.983p(t-2)1.983+p3(t-2)-p(t),
with the initial value p(t)=1.9 for t≤0. In (5.3), it is easy to see that condition (3.6) is fulfilled and Qγτ≈0.6102<1. That is, the analytic solutions of (5.3) are oscillatory. In Figures 4–6, we draw the figures of the analytic solutions and the numerical solutions of (5.3), respectively. Set m=20, θ=0.4 in Figure 5 and m=40, θ=0.7 in Figure 6, respectively. We can see from the three figures that the numerical solutions of (5.3) oscillate about M*≈1.8149, which are consistent with Theorem 3.11. On the other hand, by direct calculation, we get h^≈1.2568, so the stepsize h^ is not optimal.Figure 4
The analytic solution of (5.3).Figure 5
The numerical solution of (5.3) with m=20 and θ=0.4.Figure 6
The numerical solution of (5.3) with m=40 and θ=0.7.Example 5.3.
Consider the following equation:(5.4)p′(t)=1.5×3.72p(t-1)3.72+p2(t-1)-0.5p(t),
subject to the initial condition
(5.5)p(t)=9,fort≤0.
For (5.4), it is easy to see that Qγτeγτ+1≈0.7469<1, so the condition (3.6) is not satisfied. That is, the analytic solutions of (5.4) are non-oscillatory. In Figures 7–9, we draw the figures of the analytic solutions and the numerical solutions of (5.4), respectively. In Figure 7, we can see that p(t)→M*≈5.2326 as t→∞. From Figures 8 and 9, we can also see that the numerical solutions of (5.4) satisfy pn→M*≈5.2326 as n→∞. That is, the numerical method inherits the asymptotic behavior of non-oscillatory solutions of (5.4), which coincides with Theorem 4.3.Figure 7
The analytic solution of (5.4).Figure 8
The numerical solution of (5.4) with m=15 and θ=0.3.Figure 9
The numerical solution of (5.4) with m=20 and θ=0.8.Furthermore, according to Definition3.6, we can see from these figures that the exponential θ-method preserves the oscillations of (5.1) and (5.3) and the non-oscillations of (5.4), respectively.



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*Source: 348384-2012-12-25.xml*