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.
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:where is the total number of adopters of a product for time , stands for the total number of potential future customers, and and 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:where is the total population of adopters at time , stands for the total number of potential adopters, and , in which can be used as a governing incentive, is the cost of production, and is a positive constant. denotes the valid communication between the adopters and potential adopters, is used to describe the dismissal rate of the adopters of a product, and stands for the rate of valid contact between the adopters at time and those at time . , where 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:where , , and denote the nonadopter class, adopter class, and frustrated class, respectively. is the contact rate before the media alert, and 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 Section 2, 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 by and , respectively, at time . 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 constant .(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 delay . denotes the percentage of persons who decide not to adopt the technology after they have evaluated it, and denotes the individuals who remain interested but make no decision.(iii)We assume that the adopters at time are also affected by the adopters at time . Let denote the rate of valid contacts between the adopters at time and those at time .(iv)We assume that the acceptance of new products by people is affected by media reports. Here, we use and to represent the contact rate before and after media reports, respectively. The function is adopted to model the media reports and is used to describe the transmission rate when adopters appear and are reported. Obviously, when , the function approaches the maximum value , and when the reported adopter arrives at , the function equals to half the maximum .(v)Both classes have a death rate , 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:where , , , , and are all positive constants. Summarising, the meaning of the parameters is shown in Table 1.
In the following, we study the stability and Hopf bifurcation for system (4) with delay 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
Obviously, is an equilibrium of system (4). For other equilibria, adding the two equations of (5) yields
Substituting equation (6) into the first equation of (5), we obtainwhere
Obviously, and . Therefore, equation (7) has at least a positive solution . If , then system (4) has at least a positive equilibrium .
In the following, we consider the stability of the equilibria of system (4) by analyzing the corresponding characteristic equations. First, we assume the following:
Theorem 1. If holds, then the equilibrium is globally asymptotically stable.
Proof. For , the characteristic equation becomesObviously, and . Therefore, if , then . Thus, the equilibrium is locally asymptotically stable.
Adding the two equations in model (4), we haveTherefore, one obtained that , which implies that .
On the contrary, from the second equation of system (4), we obtainBecause holds, by comparison principle, we have . Therefore, for an arbitrary , there exists such that, for any , .
From the first equation of system (4), there exists a such thatAgain by the comparison principle, we have . Based on the above discussions, we find that if holds, then the boundary equilibrium is globally asymptotically stable.
Remark 1. Obviously, if , then the term . Hence, the condition must hold so that the solution of system (4) converges to equilibrium . 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 equilibrium . The linearization of (4) at a constant solution can be expressed bywhereThe characteristic equation associated with system (14) iswhereand , , , and are defined as follows:When , equation (16) becomesWe make the following assumptions:
Lemma 1. If holds, then the positive equilibrium of system (4) is locally asymptotically stable with .
Proof. Let and be two roots of equation (19). If holds, then we haveThis means that all the roots of equation (19) have negative real parts. Thus, equilibrium of system (4) with is locally asymptotically stable.
Remark 2. (H2) implies that only when is larger than a threshold value, the positive equilibrium of system (4) is locally asymptotic stable with .
If , then and are delay dependent. Based on the above analysis, a necessary condition for the local stability switch of is that equation (16) has purely imaginary solutions. Assume that is a root of equation (16). Then, should satisfy the following equation:which implies thatIt follows thatAs we know, ; thus, we obtainwhere , , , and are as defined in (18). Obviously, we have the following Lemma.
Lemma 2. If and hold, then equation (25) has a unique root.
According to equation (25), (26) has a unique root denoted by , where
By equation (23), we have
Thus, if we denotethen is a pair of purely imaginary roots of (16) with . In addition, the stability switches take place at the zeros of the functions:
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 and if for some . If , this pair of simple conjugate pure imaginary roots crosses the imaginary axis from left to right (as increases) if and from right to left if . The crossing direction of the pair of simple conjugate pure imaginary roots through the imaginary axis is determined byBased on the above discussions, we have the following results.
Theorem 2. If and hold, assume further that and . Then, when , the Hopf bifurcation occurs. That is, system (4) has a branch of periodic solutions bifurcating from near .
Remark 3. If the function 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 equilibrium by applying the center manifold theorem and normal form theory developed in [20].
Denote by and introduce the new parameter . Then, normalize the delay by the time-scaling . Thus, system (4) can be rewritten aswherefor .
Then, the linearized equation of (32) at the origin is
Let . Consider the following FDE on :where is a continuous linear function mapping to . By the Riesz representation theorem, there is a matrix function whose elements are of bounded variation such that
Actually, we can choosewhere is the Dirac delta function.
Let represent the infinitesimal generator of the semigroup induced by the solutions of (35) and be the formal adjoint of under the bilinear pairingfor . Then, and are a pair of adjoint operators. It is easily obtained that has a pair of simple purely imaginary eigenvalues . and are a pair of adjoint operators, so they are also eigenvalues of .
Let and be the center spaces. Then, is the adjoint space of and .
Lemma 4. Let
Then, for ,are a basis of associated with , and for ,are a basis of associated with .
Let and withwherewherefor andfor . According to (38), we get and . It is noted that
Consequently, we obtain
Let . Clearly,is a new basis of .
Furthermore, we define , where
Let be defined byfor .
Then, the center space of equation (34) is given by , whereand , where represents the complementary subspace of .
Assume is defined as follows:where is given by
Therefore, the solutions of (32) and (34) induce the infinitesimal generator , and it can be described by the following equation:
By and (51), the solution of (32) can be described aswhereand . Particularly, on the center manifold, the solution of (32) is as follows:
Let . By , then (57) is rewritten aswhere . In addition, satisfies the following equation:
Denote
By (58), we get
Let . Thus, by (60)–(62), we can obtain the following quantities:
Notice that concludes for . From (61), we can easily obtain that
Furthermore, by [20], satisfieswherewith . Thus, from (58) and (65)–(67), we can obtain that
It is noted that has two pure imaginary roots . Thus, (67) has a unique solution in given by
From (68), we obtain thatfor .
Therefore, for ,
According to the definition of , and from (70), we get that
It is noted that , . Consequently,
Combining (70) and (75), we get
Notice that
Thus, we get
From the above equation, it is easily obtained that
Similarly, we obtain
In the similarly way, we get
As a result, can be obtained explicitly, and we have the following result.
Theorem 3. System (4) has the following normal formwhere
Hence, we can compute the following results:which determine the properties of bifurcating periodic solutions at the critical values ; i.e., determines the directions of the Hopf bifurcation: if , then the Hopf bifurcation is supercritical (subcritical) and the bifurcating periodic solutions exist for ; determines the stability of the bifurcating periodic solutions: the bifurcating periodic solutions on the center manifold are stable (unstable) if ; and determines the period of the bifurcating periodic solutions: the periodic increase (decrease) occurs if .
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: , , , , , , and . We plot the graph of and and that of versus in the interval in Figures 1 and 2, respectively.
From Figure 1, we find that, with increasing delay , gradually increases and gradually decrease, and when , the positive equilibrium disappears. In fact, the positive equilibrium and the boundary equilibrium 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 for , and for , there are two zeros: the first at and the other at . These indicate that, at the point , the positive equilibrium of system (4) is stable (see Figure 3), and when , the positive equilibrium of system (4) is unstable (see Figure 4). As increases through the critical value , the stability switch occurs again, and the positive equilibrium of system (4) becomes stable, as shown in Figure 5.
When passes through the zero point , 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 and . Therefore, the orbitally asymptotically bifurcated periodic solutions of the system are stable (4) when in the whole phase space, and the Hopf bifurcation is supercritical for . As increases further and reaches the critical value , it can be computed that . 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 , the positive equilibrium of the system will finally reach stability by two Hopf bifurcations. However, if the delay increases further, then the positive equilibrium will disappear, and all solutions of system (4) will reach equilibrium , as shown in Figure 6.
5.2. Effect of Parameter
We now investigate the effect of the parameter . Therefore, let the parameters be the same as in Section 5.1, except , which means that the system parameters are constants independent of time delay. The graph of versus is plotted in Figure 7. It is shown that the positive equilibrium is stable with .
However, with each crossing of every critical delay , the number of eigenvalues with positive real parts can be increased by , and the equilibrium becomes unstable. This is quite different from the analysis in Figure 2. In this analysis, with the increase in , a limited number of stable switches can occur, and the system can be stabilized with a moderately large delay.
Figure 8 shows the graph of versus for different . Clearly, the larger the value of is, the wider the interval of the time delay is under system stability.
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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.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The work was supported by the National Science Foundation of China under Grant no. 71672154.