# Dynamics Analysis of a Mathematical Model for New Product Innovation Diffusion

Authors: Chunru Li; Zujun Ma
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## Abstract

In this paper, a mathematical model with time-delay-related parameters and media coverage to describe the diffusion process of new products is proposed, in which the time-delay-related parameters denote the stage in which potential customers decide whether to adopt a new product. Then, the stability and the Hopf bifurcation of the proposed model are analyzed in detail. The center manifold theorem and the normal form theory are used to investigate the stability of the bifurcating periodic solution. Moreover, a numerical simulation is conducted to investigate the difference between the model with delay-dependent parameters and that with delay-independent parameters. The results show that there is significant difference between the two models.


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## Body

## 1. Introduction

The diffusion of a product innovation has traditionally been defined as a process of communication among members of a social system overtime in a certain way, consisting of innovation, communication channels, time, and social systems. With the high-pace growth of the launch of new products, decision makers need to pay much attention to how to push these new products into market successfully. Fourt and Woodlock [1] proposed the first purchase model to describe the diffusion process, which is the earliest and quite popular. Bass [2] tried to establish mathematical models to describe the penetration and saturation of the diffusion process of a new product. The Bass model is the main driving force of the diffusion research and is expressed as follows:(1)dNtdt=pm−Nt+qmNm−Nt,where Nt is the total number of adopters of a product for time t, m stands for the total number of potential future customers, and p and q denote coefficients of innovation and imitation, respectively. The first term in the model represents the adoption by innovators, while the latter represents the adoption by imitators.In the past several years, a number of modifications to the Bass model have been proposed and studied, and the Bass model has become more generalized [3–10]. In [6], Kalish investigated a method for maximizing the firm profit cash flow and proposed the optimal advertising strategy, which has issues in innovation. Robinson and Lakhani [7] also considered the impact of the product price in the proposed model. Other variable models were incorporated by Kalish and Lilien [11] in their study of advertising strategies. We refer readers to [12–15] for some other related works.Note that none of the abovementioned models considered the factor of delay. In fact, for a new product, a person usually takes some time to consider accepting or rejecting it. For this reason, Fanelli and Maddalena [16] proposed a model with time-delay parameters to describe the diffusion process of a new product:(2)dAtdt=h+αAt−τC−At−τe−ρτ−δAt−γAtAt−τ,where At is the total population of adopters at time t, C stands for the total number of potential adopters, and h=eηi−c, in which i>0 can be used as a governing incentive, c>0 is the cost of production, and η is a positive constant. α>0 denotes the valid communication between the adopters and potential adopters, δ>0 is used to describe the dismissal rate of the adopters of a product, and γ>0 stands for the rate of valid contact between the adopters at time t and those at time t−τ. kτ=e−ρτ, where ρ>0 denotes the percentage of potential individuals who do not adopt the product after they have evaluated it. The stability of the equilibrium point was also researched by the authors. Then, Ballestra et al. [17] studied stability switches and Hopf bifurcations of the model (2).On the contrary, with the rapid development of communication technology, as the representative of new media, microblogs and WeChat have become the new means of information dissemination. New e-commerce has come under a high degree of concern, with the proliferation of innovative products increasing more and more with new media for marketing communication. The main difference between new media and traditional media is the carrier of communication, which determines the method of information dissemination and whether the user will accept the information in a different state. New media use digital technology, wireless communication network satellite channels, and mobile terminals, to provide users with the dissemination of information and service patterns. Regarding new media as a third channel for the dissemination of product information, the classic model is not considered when assessing what role new media plays in the product information dissemination process and how much of its impact is worth exploring.Joydip Dhar et al. [18] proposed the following model to examine the effect of a media report on the spreading of a product:(3)dNdt=r−d1N−β1+β2Am+ANA+δR,dAdt=β1+β2Am+ANA−d1+νA,dRdt=νA−d1+δR,where Nt, At, and Rt denote the nonadopter class, adopter class, and frustrated class, respectively. β1 is the contact rate before the media alert, and βA=β1+β2A/m+A is the contact rate after the media alert. Their results show that the media has a great effect on the dynamics of the model.Therefore, in this paper, based on the works of [16, 18], a model with time-delay parameters and the media effect is proposed to investigate the impacts of the media on innovation diffusion.The remainder of this paper is presented as follows. In Section2, we have developed and analyzed a model to incorporate the media impact considering two classes of population, namely, potential adopter and adopter. In Section 3, the local stability is investigated, and the existence of Hopf bifurcation is studied. In Section 4, a formula is established and used to determine the direction and stability of bifurcation. Finally, numerical simulations are provided to verify the theoretical predictions in the analysis presented in Sections 3 and 4.


## 2. Mathematical Model

This section describes a delayed mathematical model for new product innovation diffusion. Our goal is to create a realistic model that can provide wide insights into the diffusion of a product innovation.Generally, a social system consists of many people who may not adopt a new product. We divide these people into two classes depending on their different states: one is the potential adopter class, and the other is the adopter class, denoted byPt and At, respectively, at time t. To model the impact of the new product innovation diffusion on a social system, the following assumptions are imposed:(i)
In a social system, not everyone knows about the product; hence, only those who know the product information can become potential adopters. Therefore, we consider that the recruitment rate of the population that will join the nonadopter class is a constantr.(ii)
In a social system, when the potential adopters make contact with the adopters, they usually need some time to consider accepting or rejecting the new product; that is, there exists a delayτ. ρ denotes the percentage of persons who decide not to adopt the technology after they have evaluated it, and e−ρτ denotes the individuals who remain interested but make no decision.(iii)
We assume that the adopters at timet are also affected by the adopters at time t−τ. Let α denote the rate of valid contacts between the adopters at time t and those at time t−τ.(iv)
We assume that the acceptance of new products by people is affected by media reports. Here, we useβ1 and βA=β1+β2A/m+A to represent the contact rate before and after media reports, respectively. The function β2A/m+A is adopted to model the media reports and is used to describe the transmission rate when adopters appear and are reported. Obviously, when A⟶∞, the function β2A/m+A approaches the maximum value β2, and when the reported adopter arrives at m, the function β2A/m+A equals to half the maximum β2.(v)
Both classes have a death rateδ, which is proportional to the existing population. Indeed, if a potential user is not interested in the product, or a person who has adopted the product, after a period of time, he will never use the product.Therefore, the model is governed by the following system of equations:(4)dPdt=r+αAAt−τ−β1+β2Am+APAe−ρτ−δP,dAdt=β1+β2Am+APAe−ρτ−αAAt−τ−δA,where r, α, β1, β2, and δ are all positive constants. Summarising, the meaning of the parameters is shown in Table 1.Table 1
Meaning of the parameters.
ParametersMeaningrRecruitment rate of the population that will join the nonadopter classτPotential adopters need some time to consider accepting or rejecting the new productαRate of valid contacts between the adopters at timet and those at time t−τβ1,β2Contact ratemHalf saturation coefficientδDeath rateρPercentage of persons who decide not to adopt the technology after they have evaluated itIn the following, we study the stability and Hopf bifurcation for system (4) with delay τ as the bifurcation parameter.


## 3. Stability and Hopf Bifurcation

In the following, we consider the stability and Hopf bifurcation of the equilibria of system (4). First, we find all possible equilibria of system (4). According to system (4), the equilibria should satisfy(5)r+αA2−β1+β2Am+APAe−ρτ−δP=0,β1+β2Am+APAe−ρτ−αA2−δA=0.Obviously,E0=r/δ,0 is an equilibrium of system (4). For other equilibria, adding the two equations of (5) yields(6)A=r−δPδ.Substituting equation (6) into the first equation of (5), we obtain(7)B1τP3+B2τP2+B3τP+B4τ=0,where(8)B1τ=δ3δ+β1δ2e−ρτ+β2δ3e−ρτ,B2τ=−3δ2αr−δ3αm−2β1δ2re−ρτ−2β2δ2re−ρτ−β1δ3me−ρτ,B3τ=δ3r+3δαr2+β1δe−ρτr2+β2δe−ρτr2+2δ2αmr+β1δ2e−ρτmr,B4τ=−md23r−δ2r2−αmδr2−αr3.Obviously,B1τ>0 and B4τ<0. Therefore, equation (7) has at least a positive solution P∗. If r−δP∗>0, then system (4) has at least a positive equilibrium E∗=P∗,A∗.In the following, we consider the stability of the equilibria of system (4) by analyzing the corresponding characteristic equations. First, we assume the following:(9)H1δ2>β1+β2re−ρτ.Theorem 1.
IfH1 holds, then the equilibrium E0 is globally asymptotically stable.Proof.
ForE0, the characteristic equation becomes(10)λ+δλ−rδβ1e−ρτ+δ=0.
Obviously,λ1=−δ<0 and λ2=r/δβ1e−ρτ−δ. Therefore, if δ2>β1re−ρτ, then λ2<0. Thus, the equilibrium E0 is locally asymptotically stable.
Adding the two equations in model (4), we have(11)dP+Adt=r−δP+A.
Therefore, one obtained thatP+A≤r/δ, which implies that P≤r/δ−A.
On the contrary, from the second equation of system (4), we obtain(12)dAdt=β1+β2Am+APAe−ρτ−αAAt−τ−δA≤β1+β2rδ−AAe−ρτ−αAAt−τ−δA≤Aβ1+β2rδe−ρτ−β1+β2Ae−ρτ−αAt−τ−δ.
BecauseH1 holds, by comparison principle, we have At⟶0t⟶∞. Therefore, for an arbitrary ε>0, there exists t1>τ such that, for any t>t1, At<ε.
From the first equation of system (4), there exists a t2>t1 such that(13)dPdt=r+αAAt−τ−β1+β2Am+APAe−ρτ−δP≤r+ε2−δP.
Again by the comparison principle, we havePt⟶r/δt⟶∞. Based on the above discussions, we find that if H1 holds, then the boundary equilibrium is globally asymptotically stable.Remark 1.
Obviously, ifτ⟶∞, then the term e−ρτ⟶0. Hence, the condition H1 must hold so that the solution of system (4) converges to equilibrium E0. This means that when the period of evaluation of new product adoption becomes very long, the number of adopters decreases since they no longer adopt the product.
Now, we discuss the stability of the positive equilibriumE∗. The linearization of (4) at a constant solution E∗ can be expressed by(14)dPdt=γ1P+−β2P∗e−ρτγ2+γ3A+αA∗At−τ,dPdt=γ4P+β2P∗e−ρτγ2A−αA∗At−τ,where(15)γ1=−r+αA∗2P∗,γ2=Amm+A∗2,γ3=δP∗−rA∗,γ4=αA∗2+d2A∗P∗.
The characteristic equation associated with system (14) is(16)Rλ,τ+Qλ,τe−λτ=0,where(17)Rλ,τ=λ2+aτλ+cτ,Qλ,τ=bτλ+dτ,and aτ, bτ, cτ, and dτ are defined as follows:(18)aτ=−γ1−β2P∗e−ρτγ2,bτ=αA∗,cτ=β2P∗e−ρτγ2γ4+γ1−γ4γ3,dτ=−αA∗γ1+γ4.
Whenτ=0, equation (16) becomes(19)λ2+a0+b0λ+c0+d0=0.
We make the following assumptions:(20)H2β2P∗γ2<αA∗,d2A∗>r,·H3δr+A∗β2P∗e−ρτγ2+αA∗2−δαA∗P∗−rβ2P∗e−ρτγ2<0.Lemma 1.
IfH2 holds, then the positive equilibrium of system (4) is locally asymptotically stable with τ=0.Proof.
Letλ1 and λ2 be two roots of equation (19). If H2 holds, then we have(21)λ1+λ2=−a0+b0=γ1−αA∗+β2P∗γ2<0,λ1λ2=c0+d0=γ4+γ1β2P∗γ2−αA∗−γ4γ3>0.
This means that all the roots of equation (19) have negative real parts. Thus, equilibrium E∗ of system (4) with τ=0 is locally asymptotically stable.Remark 2.
(H2) implies that only whenA∗ is larger than a threshold value, the positive equilibrium of system (4) is locally asymptotic stable with τ=0.
Ifτ>0, then Rλ,τ and Qλ,τ are delay dependent. Based on the above analysis, a necessary condition for the local stability switch of E∗ is that equation (16) has purely imaginary solutions. Assume that iωω>0 is a root of equation (16). Then, ω should satisfy the following equation:(22)−ω2+iaτω+cτ+ibτω+dτcosωτ−isinωτ=0,which implies that(23)ω2−cτ=dτcosωτ+bτωsinωτ,aτω=−bττωcosωτ+dτsinωτ.
It follows that(24)sinωτ=ω2−cτbτω+aτdτωω2b2τ+d2τ,cosωτ=ω2−cτdτ−ω2aτbτω2b2τ+d2τ.
As we know,cos2ωτ+sin2ωτ=1; thus, we obtain(25)ω4+a2τ−2cτ−b2τω2+c2τ−d2τ=0.where aτ, bτ, cτ, and dτ are as defined in (18). Obviously, we have the following Lemma.Lemma 2.
IfH2 and H3 hold, then equation (25) has a unique root.According to equation (25), (26) has a unique root denoted by ω0, where(26)ω02=12b2τ+2cτ−a2τ+Δ1/2,(27)Δ=b2τ+2cτ−a2τ2−4c2τ−d2τ.By equation (23), we have(28)cosω0τ=w02−cτdτ−aτbτw02b2τw02+d2τ.Thus, if we denote(29)τjτ=1ω0arccosw02−cτdτ−aτbτw02b2τw02+d2τ+2jπ,j=0,1,2,…,then ±iω0 is a pair of purely imaginary roots of (16) with τ=τjτ. In addition, the stability switches take place at the zeros of the functions:(30)Sjτ≔τ−τjτ,for somej∈N0.Recently, Beretta and Kuang [19] studied the stability switches of some delay differential systems with delay-dependent parameters and established a geometrical criterion that reveals the existence of purely imaginary roots for a characteristic equation with delay-dependent coefficients. Therefore, according to [19], one has the following results.Lemma 3.
The characteristic equation (16) admits a pair of simple conjugate pure imaginary roots λ=±ω0τj and ω0τj>0 if Sjτ=0 for some j∈N0. If Sjτ=0, this pair of simple conjugate pure imaginary roots crosses the imaginary axis from left to right (as τ increases) if δjτ>0 and from right to left if δjτ<0. The crossing direction of the pair of simple conjugate pure imaginary roots through the imaginary axis is determined by(31)δτj≔signdReλdτ|λ=iω0τj=−signdSjτdτ|τ=τj.
Based on the above discussions, we have the following results.Theorem 2.
IfH2 and H3 hold, assume further that Sjτ=0 and δτj>0. Then, when τ=τj, the Hopf bifurcation occurs. That is, system (4) has a branch of periodic solutions bifurcating from E∗ near τ=τj.Remark 3.
If the functionSjτ=0 has two or more roots, then a stability switch may occur in system (4).


## 4. Stability and Direction of the Hopf Bifurcation

In this section, we investigate the direction and stability of period solutions bifurcating from the positive equilibriumE∗ by applying the center manifold theorem and normal form theory developed in [20].Denoteτj by τ∗ and introduce the new parameter μ=τ−τ∗. Then, normalize the delay τ by the time-scaling t⟶t/τ. Thus, system (4) can be rewritten as(32)dUtdt=Lτ∗Ut+FUt,μ,where(33)Lμφ=μγ1φ10+γ3−β2P∗e−ρτ∗γ2φ20+αφ2−1γ4φ10+β2P∗e−ρτ∗γ2φ20−αφ2−1,Fφ,μ=Lμφ+fφ,μ,fφ,μ=τ∗+μ−e−ρτ∗β1+β2γ5φ10φ20−β2P∗e−ρτ∗γ6φ220+αφ20φ2−1e−ρτ∗β1+β2γ5φ10φ20+β2P∗e−ρτ∗γ6φ220−αφ20φ2−1+h.o.t.γ5=2Am+A−A2m+A,γ6=2m2m+A3,for φ=φ1,φ2T∈C.Then, the linearized equation of (32) at the origin 0,0 is(34)dUtdt=Lτ∗Ut.LetC≔C−1,0,ℝ2. Consider the following FDE on C:(35)z˙=Lτ∗zt,where Lτ∗ is a continuous linear function mapping C−1,0,ℝ2 to ℝ2. By the Riesz representation theorem, there is a 2×2 matrix function ηθ,τ−1≤θ≤0 whose elements are of bounded variation such that(36)Lτ∗φ=∫−10dηθ,τ∗φθ,forφ∈C.Actually, we can choose(37)ηθ,τ∗=τ∗γ1γ3−β2P∗e−ρτ∗γ2γ4β2P∗e−ρτ∗γ2δθ−τ∗0α0−αδθ+1,where δ is the Dirac delta function.LetAτ∗ represent the infinitesimal generator of the semigroup induced by the solutions of (35) and A∗ be the formal adjoint of Aτ∗ under the bilinear pairing(38)ψ,ϕ=ψ0,ϕ0−∫−10∫ξ=0θψξ−θdηθϕξdξ=ψ0,ϕ0+τ∗∫−10ψθ+10α0−αϕθdθfor ϕ∈C,ψ∈C∗=C0,1,R2. Then, Aτ∗ and A∗ are a pair of adjoint operators. It is easily obtained that Aτ∗ has a pair of simple purely imaginary eigenvalues ±iω0τ∗. Aτ∗ and A∗ are a pair of adjoint operators, so they are also eigenvalues of A∗.LetP and P∗ be the center spaces. Then, P∗ is the adjoint space of P and dimP=dimP∗=2.Lemma 4.
Let(39)σ=iω0−γ1γ3−β2P∗γ2e−ρτ+αe−ω0τ,σ∗=iω−γ1ω4.Then, for−1≤θ≤0,(40)r1θ=1,σTeiω0τ∗θ,r2θ=r¯1θ,are a basis of P associated with Λ0, and for 0≤s≤1,(41)q1s=e−iω0τ∗s1,σ∗,q2s=q¯1s,are a basis of Q associated with Λ0.LetΦ=Φ1,Φ2 and Ψ∗=Ψ1∗,Ψ2∗T with(42)Φ1θ=r1θ+r2θ2=Reeiω0τ∗θReσeiω0τ∗θ=cosω0τ∗θL1cosω0τ∗θ+L2ωsinω0τ∗θh12+α2,where(43)h1=γ3−β2P∗γ2e−ρτ+αcosωτ,L1=−h1γ1−αωsinωτ,L2=−h1ω+αγ1sinωτ,Φ2θ=r1θ−r2θ2i=Imeiω0τ∗θImσ∗eiω0τ∗θ=sinω0τ∗θL3cosω0τ∗θ+L4sinω0τ∗θh12+α2,where(44)L3=h1ω−αγ1sinωτ,L4=−h1γ1−ωsinωτ,for θ∈−1,0 and(45)Ψ1∗s=q1s+q2s2=Ree−iω0τ∗sReσ∗e−iω0τ∗s=cosω0τ∗s−ωsinωτ∗s−γ1cosωτ∗sγ4,Ψ2∗s=q1s−q2s2i=Ime−iω0τ∗sImσ∗e−iω0τ∗s=−sinω0τ∗sγ1sinω0τ∗s+ω0cosω0τ∗sγ4,for s∈0,1. According to (38), we get Ψ1∗,Φ1 and Ψ1∗,Φ2. It is noted that(46)q1,r1=Ψ1∗,Φ1−Ψ2∗,Φ2+iΨ1∗,Φ2+Ψ2∗,Φ1,q1,r1=1+σσ∗+στ∗α1−σ∗e−iω0τ∗≔W∗.Consequently, we obtain(47)Ψ1∗,Φ1−Ψ2∗,Φ2=ReW∗,Ψ1∗,Φ2+Ψ2∗,Φ1=ImW∗.LetΨ∗,Φ=Ψj∗,Φkj,k=1,2. Clearly,(48)Ψ=Ψ1,Ψ2T=Ψ∗,Φ−1Ψ∗is a new basis ψ of Q.Furthermore, we definef0=ξ01,ξ02, where(49)ξ01=10,ξ02=01.Letc⋅f0 be defined by(50)c⋅f0=c1ξ01+c2ξ02,for c=c1,c2T,cj∈Rj=1,2.Then, the center space of equation (34) is given by PCNC, where(51)PCNφ=ΦΨ,φ,f0⋅f0,φ∈c,and C=PCNC⊕PSC, where PSC represents the complementary subspace of PCNC.AssumeAτ∗ is defined as follows:(52)Aτ∗φθ=φ.θ+X0θLτ∗φθ−φ.0,φ∈BC,where X0:−1,0⟶BX,X is given by(53)X0θ=0,−1≤θ<0,I,θ=0.Therefore, the solutions of (32) and (34) induce the infinitesimal generator Aτ∗, and it can be described by the following equation:(54)Ut.=Aτ∗Ut+X0FUt,μ.ByC=PCNC⊕PSC and (51), the solution of (32) can be described as(55)Ut=Φx1t,x2tT⋅f0+hx1,x2,μ,where(56)x1t,x2tT=Ψ,Ut,f0,hx1,x2,μ∈Psc,and h0,0,0=Dh0,0,0=0. Particularly, on the center manifold, the solution of (32) is as follows:(57)Ut∗=hx1,x2,0+Φx1t,x2tT⋅f0.Letz=x1−ix2. By r1=Φ1+iΦ2, then (57) is rewritten as(58)Ut∗=Wz,z¯+12r1z+r1¯z¯⋅f0,where Wz,z¯=hz+z¯/2,−z−z¯/2i,0. In addition, z satisfies the following equation:(59)z˙=iω0τ∗z+gz,z¯,(60)gz,z¯=Ψ10−iΨ20FUt∗,0,f0.Denote(61)Wz,z¯=W20z22+W11zz¯+W02z¯22+⋯,(62)gz,z¯=g20z22+g11zz¯+g02z¯22+⋯.By (58), we get(63)FUt∗,0,f0=τ∗z24−e−ρτ∗β1+β2γ5−β2P∗e−ρτ∗γ6σ2+ασ2e−iωτ∗e−ρτ∗β1+β2γ5+β2P∗e−ρτ∗γ6σ2−ασ2e−iωτ∗+τ∗zz¯4−e−ρτ∗β1+β2γ5σ+σ¯−2β2P∗e−ρτ∗γ6σσ¯+ασσ¯e−iωτ∗+eiωτ∗e−ρτ∗β1+β2γ5σ+σ¯+2β2P∗e−ρτ∗γ6σσ¯−ασσ¯e−iωτ∗+eiωτ∗+τ∗z¯24−e−ρτ∗β1+β2γ5σ¯−β2P∗e−ρτ∗γ6σ¯2+ασ¯2eiωτ∗e−ρτ∗β1+β2γ5σ¯+β2P∗e−ρτ∗γ6σ¯2−ασ¯2eiωτ∗+τ∗4−e−ρτ∗β1+β2γ52w1120+w2020+2σw1110+σ¯w2010−β2P∗e−ρτ∗γ62σ¯w2020+4σw1120+α2σw112−1+σ¯w202−1+2σw1120e−iωτ∗+σ¯eiωτ∗w2010,1e−ρτ∗β1+β2γ52w1120+w2020+2σw1110+σ¯w2010+β2P∗e−ρτ∗γ62σ¯w2020+4σw1120−α2σw112−1+σ¯w202−1+2σw1120e−iωτ∗+σ¯eiωτ∗w2010,1z2z¯+⋯<Wijnθ,1>=1π∫0πWijnθxdx.Letψ1,ψ2,ψ3=Ψ10−iΨ20. Thus, by (60)–(62), we can obtain the following quantities:(64)g20=τ∗2−e−ρτ∗β1+β2γ5−β2P∗e−ρτ∗γ6σ2+ασ2e−iωτ∗ψ1+e−ρτ∗β1+β2γ5+β2P∗e−ρτ∗γ6σ2−ασ2e−iωτ∗ψ2,g11=τ∗4−e−ρτ∗β1+β2γ5σ+σ¯−2β2P∗e−ρτ∗γ6σσ¯+ασσ¯e−iωτ∗+eiωτ∗ψ1+e−ρτ∗β1+β2γ5σ+σ¯+2β2P∗e−ρτ∗γ6σσ¯−ασσ¯e−iωτ∗+eiωτ∗e−ρτ∗β1+β2γ5σ+σ¯+2β2P∗e−ρτ∗γ6σσ¯−ασσ¯e−iωτ∗+eiωτ∗ψ2,g02=τ∗2−e−ρτ∗β1+β2γ5σ¯−β2P∗e−ρτ∗γ6σ¯2+ασ¯2eiωτ∗ψ1¯+e−ρτ∗β1+β2γ5σ¯+β2P∗e−ρτ∗γ6σ¯2−ασ¯2eiωτ∗ψ2¯,g21=−e−ρτ∗β1+β2γ52w1120+w2020+2σw1110+σ¯w2010−β2P∗e−ρτ∗γ62σ¯w2020+4σw1120+α2σw112−1+σ¯w202−1+2σw1120e−iωτ∗+σ¯eiωτ∗w2010,1ψ1+e−ρτ∗β1+β2γ52w1120+w2020+2σw1110+σ¯w2010+β2P∗e−ρτ∗γ62σ¯w2020+4σw1120−α2σw112−1+σ¯w202−1+2σw1120e−iωτ∗+σ¯eiωτ∗w2010,1ψ2.Notice thatg21 concludes W20θ,W11θ for θ∈−1,0. From (61), we can easily obtain that(65)W˙z,z¯=W20zz˙+W11z˙z¯+zz˙¯+W02z¯z˙¯+⋯,(66)Aτ∗W=Aτ∗W20z22+Aτ∗W11zz¯+Aτ∗W02z¯22+⋯.Furthermore, by [20], Wzt,z¯t satisfies(67)W˙=Aτ∗W+Hz,z¯,where(68)Hz,z¯=H20z22+H11zz¯+H02z¯22+⋯,with Hij∈PSC,i+j=2. Thus, from (58) and (65)–(67), we can obtain that(69)2iω0τ∗−Aτ∗W20=H20,−Aτ∗W11=H11.It is noted thatAτ∗ has two pure imaginary roots ±iω0τ∗. Thus, (67) has a unique solution Wiji+j=2 in PSC given by(70)W20=2iω0τ∗−Aτ∗−1H20,W11=−Aτ∗−1H11.From (68), we obtain that(71)Hz,z¯=−ΦθΨ0<FUt∗,0,f0>⋅f0=−14g20r1θ+g¯02r2θz2⋅f0−12g11r1θ+g¯11r2θzz¯⋅f0+⋯,for −1≤θ<0.Therefore, for−1≤θ<0,(72)H20θ=−12g20r1θ+g¯02r2θ⋅f0.H11θ=−12g11r1θ+g¯11r2θ⋅f0.Hz,z¯0=FUt∗,0−ΦΨ,FUt∗,0,f0⋅f0,H200=τ∗2−e−ρτ∗β1+β2γ5−β2P∗e−ρτ∗γ6σ2+ασ2e−iωτ∗e−ρτ∗β1+β2γ5+β2P∗e−ρτ∗γ6σ2−ασ2e−iωτ∗−12g20r10+g¯02r20⋅f0,H110=τ∗4−e−ρτ∗β1+β2γ5σ+σ¯−2β2P∗e−ρτ∗γ6σσ¯+ασσ¯e−iωτ∗+eiωτ∗e−ρτ∗β1+β2γ5σ+σ¯+2β2P∗e−ρτ∗γ6σσ¯−ασσ¯e−iωτ∗+eiωτ∗−12g11r10+g¯11r20⋅f0.According to the definition ofAτ∗, and from (70), we get that(73)W˙20θ=2iω0τ∗W20θ+12g20r1θ+g¯02r2θ⋅f0,−1≤θ<0.It is noted thatr1θ=r10eiω0τ∗, −1≤θ≤0. Consequently,(74)W20θ=i2g20ω0τ∗r1θ+g¯023ω0τ∗r2θ⋅f0+Ee2iω0τ∗θ,(75)E=W200−i2g20ω0τ∗r10+g¯023ω0τ∗r20⋅f0.Combining (70) and (75), we get(76)2iω0τ∗ig202ω0τ∗r10⋅f0+ig¯026ω0τ∗r20⋅f0+E−Lτ∗ig202ω0τ∗r1θ⋅f0+ig¯026ω0τ∗r2θ⋅f0+Ee2iω0τ∗θ=τ∗2−e−ρτ∗β1+β2γ5−β2P∗e−ρτ∗γ6σ2+ασ2e−iωτ∗e−ρτ∗β1+β2γ5+β2P∗e−ρτ∗γ6σ2−ασ2e−iωτ∗−12g20r10+g¯02r20⋅f0.Notice that(77)Lτ∗r1θ⋅f0=iω0τ∗r10⋅f0,Lτ∗r2θ⋅f0=−iω0τ∗r20⋅f0.Thus, we get(78)2iω0τ∗E−τ∗DΔE−Lτ∗Ee2iω0τ∗θ=τ∗2−e−ρτ∗β1+β2γ5−β2P∗e−ρτ∗γ6σ2+ασ2e−iωτ∗e−ρτ∗β1+β2γ5+β2P∗e−ρτ∗γ6σ2−ασ2e−iωτ∗.From the above equation, it is easily obtained that(79)E=122iω0−γ1−γ3+β2P∗e−ρτγ2−αA∗e−2iω0τ∗−γ42iω0−β2P∗e−ρτγ2+αA∗e−2iω0τ∗−1×−e−ρτ∗β1+β2γ5−β2P∗e−ρτ∗γ6σ2+ασ2e−iωτ∗e−ρτ∗β1+β2γ5+β2P∗e−ρτ∗γ6σ2−ασ2e−iωτ∗.Similarly, we obtain(80)W˙11θ=12g11r1θ+g¯11r2θ⋅f0,−1≤θ<0,W11θ=i2ω0τ∗−g11r1θ+g¯11r2θ⋅f0+F.In the similarly way, we get(81)F=14−γ1−γ3+β2P∗e−ρτγ2−γ4−β2P∗e−ρτγ2−1×−e−ρτ∗β1+β2γ5σ+σ¯−2β2P∗e−ρτ∗γ6σσ¯+ασσ¯e−iωτ∗+eiωτ∗e−ρτ∗β1+β2γ5σ+σ¯+2β2P∗e−ρτ∗γ6σσ¯−ασσ¯e−iωτ∗+eiωτ∗.As a result,g21 can be obtained explicitly, and we have the following result.Theorem 3.
System (4) has the following Poincare´ normal form(82)ξ˙=iω0τ∗ξ+c10ξξ2+oξ5,where(83)c10=i2ω0τ∗g20g11−2g112−g0223+g212.Hence, we can compute the following results:(84)σ2=−Rec10Reλ′τ∗,β2=2Rec10,T2=−Imc10+σ2Imλ′τ∗ω0τ∗,which determine the properties of bifurcating periodic solutions at the critical values τ∗; i.e., σ2 determines the directions of the Hopf bifurcation: if σ2>0σ2<0, then the Hopf bifurcation is supercritical (subcritical) and the bifurcating periodic solutions exist for τ>τ∗; β2 determines the stability of the bifurcating periodic solutions: the bifurcating periodic solutions on the center manifold are stable (unstable) if β2<0β2>0; and T2 determines the period of the bifurcating periodic solutions: the periodic increase (decrease) occurs if T2>0T2<0.


## 5. Numerical Simulation

In this section, we present numerical simulations of some examples to illustrate our theoretical results.


### 5.1. Hopf Bifurcations Induced by Delay

Consider system (4) with the following parameters: r=0.8, α=0.3, β1=0.3, β2=0.2, δ=0.04, m=10, and ρ=0.4. We plot the graph of P∗ and A∗ and that of Sjτ versus τ in the interval 0,10 in Figures 1 and 2, respectively.Figure 1
Graph ofP∗ and A∗ versus τ for system (4).Figure 2
Graph ofSjτ versus τ for system (4).From Figure1, we find that, with increasing delay τ, P∗ gradually increases and A∗ gradually decrease, and when τ≈11.5, the positive equilibrium disappears. In fact, the positive equilibrium and the boundary equilibrium E0 are inconsistent at this time. This outcome occurs because when the evaluation period of a new product becomes very long, the individuals do not become adopters since they do not adopt the product.Apparently, no zero point exits forS1τ, and for S0τ, there are two zeros: the first at τ1=1.0879 and the other at τ2=6.6616. These indicate that, at the point τ∈0,τ1, the positive equilibrium of system (4) is stable (see Figure 3), and when τ∈τ1,τ2, the positive equilibrium of system (4) is unstable (see Figure 4). As τ increases through the critical value τ2=6.6616, the stability switch occurs again, and the positive equilibrium of system (4) becomes stable, as shown in Figure 5.Figure 3
The equilibriumE∗ of system (4) is stable with τ=0.9.Figure 4
The equilibriumE∗ of system (4) is unstable with τ=6.Figure 5
The equilibriumE∗ of system (4) is stable with τ=8.4.Whenτ passes through the zero point τ1=1.0879, the positive equilibrium becomes unstable, and a Hopf bifurcation occurs, i.e., a family of periodic solutions bifurcate from the positive equilibrium, as shown in Figure 4. We can compute μ2=−Rec10/Reλ′τ∗=22243672.8653>0 and β2=2Rec10=−109.2516<0. Therefore, the orbitally asymptotically bifurcated periodic solutions of the system are stable (4) when τ1=1.1929 in the whole phase space, and the Hopf bifurcation is supercritical for μ2>0. As τ increases further and reaches the critical value τ2=6.6616, it can be computed that β2=0.3038. Hence, the limit cycle becomes unstable based on the discussion in Section 3, and the positive equilibrium recovers it stability, as shown in Figure 5. In summary, with the increase in τ, the positive equilibrium of the system will finally reach stability by two Hopf bifurcations. However, if the delay τ increases further, then the positive equilibrium will disappear, and all solutions of system (4) will reach equilibrium E0, as shown in Figure 6.Figure 6
All solutions of system (4) converge to the equilibrium E0 with τ=13.


### 5.2. Effect of Parameterρ

We now investigate the effect of the parameterρ. Therefore, let the parameters be the same as in Section 5.1, except ρ=0, which means that the system parameters are constants independent of time delay. The graph of Sjτ versus τ is plotted in Figure 7. It is shown that the positive equilibrium is stable with τ∈0,τ1.Figure 7
Graph ofSjτ versus τ for the neuron model (4) with delay-independent parameters.However, with each crossing of every critical delayτjj=1,2,…,, the number of eigenvalues with positive real parts can be increased by 2, and the equilibrium becomes unstable. This is quite different from the analysis in Figure 2. In this analysis, with the increase in τ, a limited number of stable switches can occur, and the system can be stabilized with a moderately large delay.Figure8 shows the graph of Sjτ versus τ for different ρ. Clearly, the larger the value of ρ is, the wider the interval of the time delay τ is under system stability.Figure 8
The effect of parameterρ on the stability of system.
(a)(b)(c)(d)


## 5.1. Hopf Bifurcations Induced by Delay

Consider system (4) with the following parameters: r=0.8, α=0.3, β1=0.3, β2=0.2, δ=0.04, m=10, and ρ=0.4. We plot the graph of P∗ and A∗ and that of Sjτ versus τ in the interval 0,10 in Figures 1 and 2, respectively.Figure 1
Graph ofP∗ and A∗ versus τ for system (4).Figure 2
Graph ofSjτ versus τ for system (4).From Figure1, we find that, with increasing delay τ, P∗ gradually increases and A∗ gradually decrease, and when τ≈11.5, the positive equilibrium disappears. In fact, the positive equilibrium and the boundary equilibrium E0 are inconsistent at this time. This outcome occurs because when the evaluation period of a new product becomes very long, the individuals do not become adopters since they do not adopt the product.Apparently, no zero point exits forS1τ, and for S0τ, there are two zeros: the first at τ1=1.0879 and the other at τ2=6.6616. These indicate that, at the point τ∈0,τ1, the positive equilibrium of system (4) is stable (see Figure 3), and when τ∈τ1,τ2, the positive equilibrium of system (4) is unstable (see Figure 4). As τ increases through the critical value τ2=6.6616, the stability switch occurs again, and the positive equilibrium of system (4) becomes stable, as shown in Figure 5.Figure 3
The equilibriumE∗ of system (4) is stable with τ=0.9.Figure 4
The equilibriumE∗ of system (4) is unstable with τ=6.Figure 5
The equilibriumE∗ of system (4) is stable with τ=8.4.Whenτ passes through the zero point τ1=1.0879, the positive equilibrium becomes unstable, and a Hopf bifurcation occurs, i.e., a family of periodic solutions bifurcate from the positive equilibrium, as shown in Figure 4. We can compute μ2=−Rec10/Reλ′τ∗=22243672.8653>0 and β2=2Rec10=−109.2516<0. Therefore, the orbitally asymptotically bifurcated periodic solutions of the system are stable (4) when τ1=1.1929 in the whole phase space, and the Hopf bifurcation is supercritical for μ2>0. As τ increases further and reaches the critical value τ2=6.6616, it can be computed that β2=0.3038. Hence, the limit cycle becomes unstable based on the discussion in Section 3, and the positive equilibrium recovers it stability, as shown in Figure 5. In summary, with the increase in τ, the positive equilibrium of the system will finally reach stability by two Hopf bifurcations. However, if the delay τ increases further, then the positive equilibrium will disappear, and all solutions of system (4) will reach equilibrium E0, as shown in Figure 6.Figure 6
All solutions of system (4) converge to the equilibrium E0 with τ=13.


## 5.2. Effect of Parameterρ

We now investigate the effect of the parameterρ. Therefore, let the parameters be the same as in Section 5.1, except ρ=0, which means that the system parameters are constants independent of time delay. The graph of Sjτ versus τ is plotted in Figure 7. It is shown that the positive equilibrium is stable with τ∈0,τ1.Figure 7
Graph ofSjτ versus τ for the neuron model (4) with delay-independent parameters.However, with each crossing of every critical delayτjj=1,2,…,, the number of eigenvalues with positive real parts can be increased by 2, and the equilibrium becomes unstable. This is quite different from the analysis in Figure 2. In this analysis, with the increase in τ, a limited number of stable switches can occur, and the system can be stabilized with a moderately large delay.Figure8 shows the graph of Sjτ versus τ for different ρ. Clearly, the larger the value of ρ is, the wider the interval of the time delay τ is under system stability.Figure 8
The effect of parameterρ on the stability of system.
(a)(b)(c)(d)


## 6. Conclusions

In this paper, the stability switches and Hopf bifurcation of a delayed nonlinear mathematical model are analyzed in detail. The model is designed with a stage structure used to simulate the stages of the process of adopting a new product. Moreover, the normal and the center manifold theory are used to investigate the stability and the direction of the bifurcating periodic solutions. The results of the numerical simulations show that, in the range of zero to infinity for the delay, there may exist a number of stability switches for the model with delay-dependent parameters. Moreover, the system could also be stabilized by a moderately large delay. This characteristic of the models is quite different from that of systems without delay-dependent parameters.



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