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 [<xref ref-type="bibr" rid="B3">3</xref>]).
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 Theorem 3.1 hold, then every positive solution of (1.1) oscillates about its positive equilibrium M*=ωM.

Next, we introduce an invariant oscillation transformation x(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
Let h=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.

Letting yn=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 exponential θ-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 that pn 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, for u≠0, limu→0f1(u)u=1,f2(u)>0, for every u, limu→0f2(u)=2,f1(u)≤u, for u>0, f1(0)=0,f2(u)≤2, for u≥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 exponential θ-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 exponential θ-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.
Letting yn=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.
Let f(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, for ξ>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 of 1/2<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^={∞,for Qγτ≥1,τ(1+γτ+lnQγτ)1+γτ(1-lnQγτ),for Qγτ<1.

Proof.
Since R(-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~={∞,when 0≤θ≤12,h^,when 12<θ≤1,
where h^ is defined in Lemma 3.9.