Innovation diffusion model in patch environment

Applied Mathematics and Computation 134 (2003) 51–67 www.elsevier.com/locate/amc

Innovation diffusion model in patch environment Wang Wendi

a,b,*,1

, P. Fergola c, C. Tenneriello

c

a

b

Department of Mathematics, Xi
an Jiaotong University, Xi
an 710049, China Department of Mathematics, Southwest Normal University, Chongqing 400715, China c Dipartimento di Matematica e Applicazioni, ‘‘R. Caccioppoli’’, Universit a di Napoli ‘‘Federico II’’, Italy

Abstract A mathematical model is proposed to describe the dynamics of users of one product in two different patches. Advertisement force, contact rate between users and non-users, population dispersal rate and returning rate from user class to non-user class are chosen as key parameters. For the product with a long life-span, it is assumed that the returning rate is proportional to the number of users. It is shown that this model has a unique positive equilibrium which is globally stable. For the product with a short life-span, time delays are introduced to represent the duration of the product in two different markets and the stability of a positive equilibrium is analyzed in one reasonable case. Periodic advertisements are also incorporated and the existence and uniqueness of positive periodic solutions are investigated. Ó 2002 Elsevier Science Inc. All rights reserved. Keywords: Innovation; Patch; Stability; Periodic; Delay

1. Introduction A problem of interest in market behavior is that of forecasting how a social system reacts to the offer of two different products. This problem can be

*

Corresponding author. E-mail address: [email protected] (W. Wendi). 1 The author was supported by Doctorate Foundation of Xi
an Jiaotong University.

0096-3003/02/$ - see front matter Ó 2002 Elsevier Science Inc. All rights reserved. PII: S 0 0 9 6 - 3 0 0 3 ( 0 1 ) 0 0 2 6 8 - 5

52

W. Wendi et al. / Appl. Math. Comput. 134 (2003) 51–67

formulated as an innovation diffusion problem according to the modeling ideas developed in many recent papers as, for instance, in [1–7]. By supposing that the channels of communication are represented by external sources (mass media like radio, TV, newspaper, poster, internet, etc.) and interpersonal contacts (word-of-mouth), we assume that the overall population at time t can be divided into two different classes, non-users of innovation and users of innovation. Furthermore, in order to take into account that individuals can give up the use of an innovation, we allow the possibility for users to leave their class and pass to non-user class after a given time period which we can interpret as the safe period of innovation. Therefore, we introduce delay terms in the model representing this process. Since it is important to consider the sales of products in different regions, we also include space structure into our model by incorporating patch environment. As a beginning, we will only consider two patches in this paper. When two patches are isolated, we suppose that the dynamics of both classes are governed by dN1 ¼ c1 N1  k1 A1 N1  d1 N1 þ m1 A1 þ b1 ; dt dA1 ¼ c1 N1 þ k1 A1 N1  d1 A1  m1 A1 ; dt

ð1:1Þ

where N1 is the number of non-users in the first patch, A1 is the number of users in the first patch, b1 is the birth rate of the population in the first patch, d1 is the death rate of the population in the first patch, c1 represents the intensity of advertisement for promoting product-users in the first patch, and k1 is the valid contact rate of users of the product with non-users in the first patch. Similarly, we suppose that the dynamics of non-user-class and user-class in the second patch are governed by dN2 ¼ c2 N2  k2 A2 N2  d2 N2 þ m2 A2 þ b2 ; dt dA2 ¼ c2 N2 þ k2 A2 N2  d2 A2  m2 A2 ; dt

ð1:2Þ

where N2 is the number of non-users in the second patch, A2 is the number of users in the second patch, b2 is the birth rate of the population in the second patch, d2 is the death rate of the population in the second patch, c2 represents the intensity of advertisement for promoting product-users in the second patch, and k2 is the valid contact rate of users of the product with non-users in the second patch. We now suppose that, when the two patches are connected, h1 is the probability that an individual migrates from the first patch to the second patch, h2 the probability that an individual migrates from the second patch to the first patch. Further, we assume that k2 is the fraction of new immigrants from the

W. Wendi et al. / Appl. Math. Comput. 134 (2003) 51–67

53

second patch who remain in user-class and that k1 is the fraction of new immigrants from the first patch who remain in user-class. Under these assumptions, our model is: dN1 ¼ c1 N1  k1 A1 N1  d1 N1 þ m1 A1 þ b1 þ h2 ðN2 þ ð1  k2 ÞA2 Þ  h1 N1 ; dt dA1 ¼ c1 N1 þ k1 A1 N1  d1 A1  m1 A1  h1 A1 þ k2 h2 A2 ; dt dN2 ¼ c2 N2  k2 A2 N2  d2 N2 þ m2 A2 þ b2 þ h1 ðN1 þ ð1  k1 ÞA1 Þ  h2 N2 ; dt dA2 ¼ c2 N2 þ k2 A2 N2  d2 A2  m2 A2  h2 N2 þ k1 h1 A1 : dt ð1:3Þ We will suppose that all the parameters are positive constants except that ki are non-negative constants. If we set Ti ¼ Ni þ Ai , (1.3) implies that T10 ¼ b1  ðd1 þ h1 ÞT1 þ h2 T2 ; T20 ¼ b2  ðd2 þ h2 ÞT2 þ h1 T1 :

ð1:4Þ

Let us define C1 ¼

d2 b 1 þ h2 b 1 þ h2 b 2 ; d1 d2 þ d1 h2 þ h1 d2

C2 ¼

h1 b 1 þ d1 b 2 þ h1 b 2 : d1 d2 þ d1 h2 þ h1 d2

It is easy to see that any positive solution ðT1 ðtÞ; T2 ðtÞÞ of (1.4) satisfies Ti ðtÞ ! Ci

as t ! 1:

Since we are interested in the asymptotic behavior of (1.3), we may regard N1 þ A1 as C1 and N2 þ A2 as C2 . Thus, we have dA1 ¼ ðc1 þ k1 A1 ÞðC1  A1 Þ  ðd1 þ m1 þ h1 ÞA1 þ k2 h2 A2 ; dt dA2 ¼ ðc2 þ k2 A2 ÞðC2  A2 Þ  ðd2 þ m2 þ h2 ÞA2 þ k1 h1 A1 : dt For simplicity, we denote ðdi þ mi þ hi Þ by ai and denote ki hi by di , i ¼ 1; 2. Hence, we obtain dA1 ¼ ðc1 þ k1 A1 ÞðC1  A1 Þ  a1 A1 þ d2 A2 ; dt dA2 ¼ ðc2 þ k2 A2 ÞðC2  A2 Þ  a2 A2 þ d1 A1 : dt

ð1:5Þ

54

W. Wendi et al. / Appl. Math. Comput. 134 (2003) 51–67

The organization of this paper is as follows. In Section 2, we show that (1.5) has a positive equilibrium which is globally stable. In Section 3, we introduce time delays into the model which represent the duration of the product in two patches and study its stability. In Section 4, we will suppose that the advertisements are periodic in time and will establish sufficient conditions that ensure the existence and uniqueness of positive periodic solutions.

2. Stability of the model Let us begin from d1 ¼ d2 ¼ 0. In this case, (1.5) has a unique positive equilibrium ðA10 ; A20 Þ, where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2a þ k C þ 4a21 þ k21 C12 1 1 1 1 ; A10 ¼ 2 k1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2a þ k C þ 4a22 þ k22 C22 2 2 2 1 A20 ¼ ; 2 k2 and (1.5) can be rewritten as   dA1 a 1 A1 ¼ ðc1 þ k1 A1 Þ C1  A1  dt c 1 þ k 1 A1   a1 c1 ðA1  A10 Þ ¼ ðc1 þ k1 A1 Þ  ðA1  A10 Þ  ; ðc1 þ k1 A1 Þðc1 þ k1 A10 Þ   dA2 a 2 A2 ¼ ðc2 þ k2 A2 Þ C2  A2  dt c 2 þ k 2 A2   a2 c2 ðA2  A20 Þ ¼ ðc2 þ k2 A2 Þ  ðA2  A20 Þ  : ðc2 þ k2 A2 Þðc2 þ k2 A20 Þ

ð2:1Þ

Since each equation of this system is independent, it is sufficient to consider the stability of each equation separately. Set Z A1 A1 ðsÞ  A10 ds: V1 ¼ c A10 1 þ k1 A1 ðsÞ Calculating the derivative of V1 along the positive solutions of (2.1), we obtain   a1 c 1 V_1 6  1 þ ðA1  A10 Þ2 : ðc1 þ k1 A1 Þðc1 þ k1 A10 Þ It follows that A10 is globally stable. Similarly, we see that A20 is globally stable. Thus, ðA10 ; A20 Þ is globally stable.

W. Wendi et al. / Appl. Math. Comput. 134 (2003) 51–67

Suppose d1 ¼ 0 and d2 > 0. Then (1.5) is reduced to   dA1 a 1 A1 ¼ ðc1 þ k1 A1 Þ C1  A1  ; dt c 1 þ k 1 A1   dA2 a 2 A2 ¼ ðc2 þ k2 A2 Þ C2  A2  þ d2 A1 : dt c 2 þ k 2 A2

55

ð2:2Þ

Since A10 is globally stable for the first equation of the system, the limiting equation of the system is   dA2 a 2 A2 ¼ ðc2 þ k2 A2 Þ C2  A2  ð2:3Þ þ d2 A10 : dt c 2 þ k 2 A2 Suppose A21 is the positive solution of the following equation:   a2 A2 ðc2 þ k2 A2 Þ C2  A2  þ d2 A10 ¼ 0; c 2 þ k 2 A2 where A2 is the variable of the equation. Since it can be written as   dA2 ða2 c2 þ d2 A10 k2 ÞðA2  A21 Þ ¼ ðc2 þ k2 A2 Þ  ðA2  A21 Þ  : ðc2 þ k2 A2 Þðc2 þ k2 A21 Þ dt Define V2 ¼

Z

A2

A21

A2 ðsÞ  A21 ds: c2 þ k2 A2 ðsÞ

Then  V_2 ¼  1 þ

 ða2 c2 þ d2 A10 k2 Þ 2 ðA2  A21 Þ : ðc2 þ k2 A2 Þðc2 þ k2 A21 Þ

It follows from [11] that the positive equilibrium ðA10 ; A21 Þ of (2.2) is globally stable. Suppose d1 > 0 and d2 > 0. We will show that (1.5) has a unique positive equilibrium. By setting the right-hand side of (1.5) to 0, we obtain A2 ¼ ½ðc1 þ k1 A1 ÞðC1  A1 Þ  a1 A1 =d2 ; A1 ¼ ½ðc2 þ k2 A2 ÞðC2  A2 Þ  a2 A2 =d1 : These two curves are parabolic. Set f1 ¼ ½ðc1 þ k1 A1 ÞðC1  A1 Þ  a1 A1 =d2 ; f2 ¼ ½ðc2 þ k2 A2 ÞðC2  A2 Þ  a2 A2 =d1 : We have f10 ðA1 Þ ¼

k1 C1 þ 2k1 A1 þ c1 þ a1 ; d2

ð2:4Þ

56

W. Wendi et al. / Appl. Math. Comput. 134 (2003) 51–67

f20 ðA2 Þ ¼

k2 C2 þ 2k2 A2 þ c2 þ a2 ; d1

f100 ðA1 Þ ¼ 2

k1 ; d2

f200 ðA2 Þ ¼ 2

k2 : d1

By the second equation of (2.4), we obtain  1 k2 C 2  a2  c 2 A2 ¼ 2k2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  þ k22 C22  2k2 C2 a2 þ 2k2 C2 c2 þ a22 þ 2a2 c2 þ c22 þ 4k2 A1 d1 : Denote the right-hand side of this equation by g2 ðA1 Þ. It is easy to see that 3 g2 ðA1 Þ > 0, g20 ðA1 Þ > 0 for A1 P 0. Then, since g200 ðA1 Þ ¼ f200 ðA2 Þ=ðf20 ðA2 ÞÞ < 0, g2 ðA1 Þ is concave. Notice that f1 ð0Þ < 0, g2 ð0Þ > 0, limA1 !þ1 ½f1 ðA1 Þ  g2 ðA1 Þ ¼ þ1 and that f1 is convex. The parabolic curves have one and only one intersection point in the interior of first quadrant. Consequently, (1.5) has one and only one positive equilibrium ðA1 ; A2 Þ. Let us check the vector field of (1.5) at ðA; AÞ. When A is positive and sufficiently small, ðc1 þ k1 AÞðC1  AÞ  a1 A þ d2 A ¼ ðc1 þ k1 AÞ½C1  A þ ðd2  a1 ÞA=ðc1 þ k1 AÞ > 0; ðc2 þ k2 AÞðC2  AÞ  a2 A þ d1 A ¼ ðc2 þ k2 AÞ½C2  A þ ðd1  a2 ÞA=ðc2 þ k2 AÞ > 0: When A is sufficiently large, ðc1 þ k1 AÞðC1  AÞ  a1 A þ d2 A ¼ ðc1 þ k1 AÞ½C1  A þ ðd2  a1 ÞA=ðc1 þ k1 AÞ < 0; ðc2 þ k2 AÞðC2  AÞ  a2 A þ d1 A ¼ ðc2 þ k2 AÞ½C2  A þ ðd1  a2 ÞA=ðc2 þ k2 AÞ < 0: Notice that (1.5) is cooperative. It follows that (1.5) is permanent. Since the positive equilibrium of (1.5) is unique, it follows from [12] that it is globally attractive. At this time, we can state the main result of this section. Theorem 2.1. System (1.5) has one positive equilibrium which is globally attractive.

W. Wendi et al. / Appl. Math. Comput. 134 (2003) 51–67

57

3. The model with time delays The purpose of this section is to modify (1.3) to incorporate duration of the product. The assumption that returning rate is proportional to the number of users seems reasonable when the length of life-span of the product is large. Otherwise, the effect of the length will be significant. Because of this, we introduce time delays due to the life-span of the product into (1.3) and obtain: dN1 ¼ c1 N1 ðtÞ  k1 A1 ðtÞN1 ðtÞ  d1 N1 ðtÞ þ ed1 s1 ðc1 N1 ðt  s1 Þ dt þ k1 A1 ðt  s1 ÞN1 ðt  s1 ÞÞ þ b1 þ h2 ðN2 þ ð1  k2 ÞA2 Þ  h1 N1 ; dA1 ¼ c1 N1 ðtÞ þ k1 A1 ðtÞN1 ðtÞ  d1 A1 ðtÞ  ed1 s1 ðc1 N1 ðt  s1 Þ dt þ k1 A1 ðt  s1 ÞN1 ðt  s1 ÞÞ  h1 A1 ðtÞ þ k2 h2 A2 ðtÞ; dN2 ¼ c2 N2 ðtÞ  k2 A2 ðtÞN2 ðtÞ  d2 N2 ðtÞ þ ed2 s2 ðc2 N2 ðt  s2 Þ dt þ k2 A2 ðt  s2 ÞN2 ðt  s2 ÞÞ þ b2 þ h1 ðN1 þ ð1  k1 ÞA1 Þ  h2 N2 ; dA2 ¼ c2 N2 ðtÞ þ k2 A2 ðtÞN2 ðtÞ  d2 A2 ðtÞ  ed2 s2 ðc2 N2 ðt  s2 Þ dt þ k2 A2 ðt  s2 ÞN2 ðt  s2 ÞÞ  h2 N2 ðtÞ þ k1 h1 A1 : ð3:1Þ Here s1 represents the life-span of the product in the first patch and s2 represents the life-span of the product in the second patch. By the same arguments as those in Section 1, we only need to consider the asymptotic behavior of the following system: dA1 ¼ ðc1 þ k1 A1 ðtÞÞðC1  A1 ðtÞÞ  ðd1 þ h1 ÞA1 ðtÞ dt  ed1 s1 ðc1 þ k1 A1 ðt  s1 ÞÞðC1  A1 ðt  s1 ÞÞ þ k2 h2 A2 ðtÞ; dA2 ¼ ðc2 þ k2 A2 ðtÞÞðC2  A2 ðtÞÞ  ðd2 þ h2 ÞA2 ðtÞ dt

ð3:2Þ

 ed2 s2 ðc2 þ k2 A2 ðt  s2 ÞÞðC2  A2 ðt  s2 ÞÞ þ k1 h1 A1 ðtÞ: First, we analyze the local stability of (3.2). In order to make mathematics tractable, we consider the case where k1 ¼ k2 ¼ 0, which means that all the new immigrants are the non-users of the product. In this case, (3.2) is reduced to

58

W. Wendi et al. / Appl. Math. Comput. 134 (2003) 51–67

dA1 ¼ ðc1 þ k1 A1 ðtÞÞðC1  A1 ðtÞÞ  p1 A1 ðtÞ dt  g1 ðc1 þ k1 A1 ðt  s1 ÞÞðC1  A1 ðt  s1 ÞÞ; dA2 ¼ ðc2 þ k2 A2 ðtÞÞðC2  A2 ðtÞÞ  p2 A2 ðtÞ dt  g2 ðc2 þ k2 A2 ðt  s2 ÞÞðC2  A2 ðt  s2 ÞÞ;

ð3:3Þ

where pi ¼ di þ hi and gi ¼ edi si . It is easy to see that (3.3) has one and only one positive equilibrium ðA1 ; A2 Þ, where   1  A1 ¼  ð1  g1 Þc1 þ ð1  g1 Þk1 C1  p1 þ ð1  g1 Þ2 c21 2ð1  g1 Þk1 þ 2ð1  g1 Þ2 k1 c1 C1 þ 2ð1  g1 Þc1 p1 þ ð1  g1 Þ2 k21 C12 1=2  2  2ð1  g1 Þk1 C1 p1 þ p1 ; A2 ¼

  1  ð1  g2 Þc2 þ ð1  g2 Þk2 C2  p2 þ ð1  g2 Þ2 c22 2ð1  g2 Þk2 2

2

þ 2ð1  g2 Þ k2 c2 C2 þ 2ð1  g2 Þc2 p2 þ ð1  g2 Þ k22 C22 1=2   2ð1  g2 Þk2 C2 p2 þ p22 : Linearizing (3.3) at ðA1 ; A2 Þ, we obtain dB1 ¼ q1 B1 ðtÞ  g1 ðq1 þ p1 ÞB1 ðt  s1 Þ; dt dB2 ¼ q2 B2 ðtÞ  g2 ðq2 þ p2 ÞB2 ðt  s2 Þ; dt

ð3:4Þ

where qi ¼ ki Ci  ci  pi  2ki Ai : Note that the two equations of (3.4) are independent. It is sufficient to concentrate on one equation. Let us consider the first equation. The characteristic equation of it is k ¼ q1  g1 ðq1 þ p1 Þ eks : The location of the roots of this equation is indicated by the Hayes theorem (see [9, p. 444]). For convenience, we state this theorem here. Theorem 3.1 (Hayers). If A; B 2 R; then all roots of A ez þ B  z ez ¼ 0

W. Wendi et al. / Appl. Math. Comput. 134 (2003) 51–67

59

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi have negative real parts if and only if A < 1 and A < B < a2 þ A2 , where a is the root of a ¼ A tan a, such that 0 < a < p. (If A ¼ 0, then a ¼ p=2). By this theorem, we obtain: Lemma 3.2. Assume: (i) s1 ðk1 C1  c1  p1  2k1 A1 Þ < 1. (ii) ð1  g1 Þðk1 C1  c1  2k1 A1 Þ < p1 . qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 (iii) s1 g1 ðk1 C1  c1  2k1 A1 Þ < a21 þ ½s1 ðk1 C1  c1  p1  2k1 A1 Þ , where a1 is  the root of a1 ¼ s1 ðk1 C1  c1  p1  2k1 A1 Þ tan a1 , such that 0 < a1 < p. (If k1 C1  c1  p1  2k1 A1 ¼ 0, then a1 ¼ p=2.) Then A1 is asymptotically stable. Similarly, we have: Lemma 3.3. Assume: (i) s2 ðk2 C2  c2  p2  2k2 A2 Þ < 1. (ii) ð1  g2 Þðk2 C2  c2  2k2 A2 Þ < p2 . qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (iii) s2 g2 ðk2 C2  c2  2k2 A2 Þ < a22 þ ½s2 ðk2 C2  c2  p2  2k2 A2 Þ 2 , where a2 is the root of a2 ¼ s2 ðk2 C2  c2  p2  2k2 A2 Þ tan a2 , such that 0 < a2 < p. (If k2 C2  c2  p2  2k2 A2 ¼ 0, then a2 ¼ p=2.) Then A2 is asymptotically stable. By Lemmas 3.2 and 3.3, we have: Theorem 3.4. Assume: (i) si ðki Ci  ci  pi  2ki Ai Þ < 1; i ¼ 1; 2. (ii) ð1  gi Þðki Ci  ci  2ki Ai Þ < pi ; i ¼ 1; 2. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (iii) si gi ðki Ci  ci  2ki Ai Þ < a2i þ ½si ðki Ci  ci  pi  2ki Ai Þ 2 , where a is the root of ai ¼ si ðki Ci  ci  pi  2ki Ai Þ tan ai , such that 0 < ai < p. (If ki Ci  ci  pi  2ki Ai ¼ 0, then ai ¼ p=2.) Then the equilibrium ðA1 ; A2 Þ of (3.3) is asymptotically stable. Example. Suppose k2 ¼ 1, C2 ¼ 10, c2 ¼ 1, p2 ¼ 6 and s2 ¼ 1. By numerical calculations, we obtain A2 ¼ 4:356378697, a2 ¼ 2:434890483 and g2 ¼ 0:1353352832. Then we have k2 C2  c2  2k2 A2  p2  1=s2 ¼ 3:851792217; ð1  g2 Þðk2 C  c2  2kA2 Þ  p2 ¼ 4:009953137;

60

W. Wendi et al. / Appl. Math. Comput. 134 (2003) 51–67

 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2 g2 k2 C2  c2  2k2 A2  a22 þ ðs2 ðk2 C2  c2  2k2 A2  p2 ÞÞ2 ¼ 2:591693812: Thus, A2 is asymptotically stable. If k1 ¼ 3, c1 ¼ 0:5, C1 ¼ 5, p1 ¼ 4, d1 ¼ 1 and h1 ¼ 3, then  ð1  g1 Þ k1 C  c1  2kA1  p1 ¼ ð1  expðs1 ÞÞ  1 0  1=2  3  21 þ 29 expðs1 Þ  561  1458 expðs1 Þ þ 961 expðs1 Þ2 C B  4C B A: @14:5  6 þ 6 expðs1 Þ

By means of maple, we see that it is negative. Thus, we only need to consider conditions (i) and (iii) of Lemma 3.2. Note that k1 C1  c1  2k1 A1  p1  1=s1 ¼ 10:5 

3½21 þ 29 expðs1 Þ  ð561  1458 expðs1 Þ þ 961 expð2s1 ÞÞ1=2  1=s1

: 6 þ 6 expðs1 Þ

By numerical calculations, we see that it is negative if 0 < s1 < 0:1005 or 0:388 < s1 and is positive if 0:1005 < s1 < 0:388. Furthermore, condition (iii) is valid if 0:4427159803 < s or 0 < s1 < 0:08347740397. Thus, the positive equilibrium is asymptotically stable if s1 > 0:4427159803 or 0 < s1 < 0:08347740397 and is unstable if 0:08347740397 < s1 < 0:4427159803. This example shows that stability switch can occur in model (3.3). Let us turn to the global stability of the positive equilibrium. Since the first and second equations of (3.3) have the same form and are independent, we will concentrate on the first one. Let us rewrite the first equation as  dA1 ¼ ðc1 þ k1 A1 ðtÞÞ C1  A1 ðtÞ dt  p1 A1 ðtÞ þ g1 ðc1 þ k1 A1 ðt  s1 ÞÞðC1  A1 ðt  s1 ÞÞ  c1 þ k1 A1 ðtÞ    c1 A1 ðtÞ  A1 ¼ ðc1 þ k1 A1 ðtÞÞ  A1 ðtÞ  A1  p1 ðc1 þ k1 A1 ðtÞÞðc1 þ k1 A1 Þ h    g1  c1 A1 ðt  s1 Þ  A1 þ k1 C1 A1 ðt  s1 Þ  A1 2    k1 A1 ðt  s1 Þ  A1  2k1 A1 ðt  s1 Þ  A1 A1  i.       C1 k1 A1 ðtÞ  A1 þ k1 A1 A1 ðtÞ  A1 ðc1 þ k1 A1 ðtÞÞ : ð3:5Þ

W. Wendi et al. / Appl. Math. Comput. 134 (2003) 51–67

61

Let us define Liapunov function V1 by Z A1 A1 ðsÞ  A1 ds: V1 ¼ A1 c1 þ k1 A1 s Calculating the derivative of V1 along the positive solutions of (3.5), we have  2  c1 A1 ðtÞ  A1  2 _ V1 ¼  A1 ðtÞ  A1  p1 ðc1 þ k1 A1 ðtÞÞðc1 þ k1 A1 Þ h     g1 A1 ðtÞ  A1  c1 A1 ðt  s1 Þ  A1 þ k1 C1 A1 ðt  s1 Þ  A1    2  k1 A1 ðt  s1 Þ  A1  2k1 A1 ðt  s1 Þ  A1 A1  C1 k1 A1 ðtÞ  A1  i. þ k1 A1 A1 ðtÞ  A1 ð3:6Þ ðc1 þ k1 A1 ðtÞÞ: By (3.3), it is easy to see that 0 < A1 ðtÞ < C1 if t is large. It follows that for all large t, we have 2  V_1 6  A1 ðtÞ  A1 

2 g1 C1 k1  A1 ðtÞ  A1 c1 ðc þ k1 C1 Þ h 1     g1 A1 ðtÞ  A1  c1 A1 ðt  s1 Þ  A1 þ k1 C1 A1 ðt  s1 Þ  A1   2 i.  k1 A1 ðt  s1 Þ  A1  2k1 A1 ðt  s1 Þ  A1 A1 ðc1 þ k1 A1 ðtÞÞ: 

p 1 c1

2

A1 ðtÞ  A1

2

þ

ð3:7Þ By Cauchy inequality, we have ! 2 g1 C1 k1  1þ  A1 ðtÞ  A1 2 c1 ðc1 þ k1 C1 Þ h  2 g1 c1 þ k1 C1 þ k1 þ 2k1 A1 A1 ðtÞ  A1 þ 2ðc1 þ k1 A1 ðtÞÞ   2 4 i  þ c1 þ k1 C1 þ 2k1 A1 A1 ðt  s1 Þ  A1 þ k1 A1 ðt  s1 Þ  A1 :

V_1 6 

p1 c1

ð3:8Þ Notice that 0 < A1 ðtÞ < C1 for large t and 0 < A1 < C1 . We have ! 2 p1 c1 g1 C1 k1  _V1 6  1 þ  A1 ðtÞ  A1 2 c1 ðc1 þ k1 C1 Þ  2 g1 h þ ðc1 þ k1 C1 þ k1 þ 2k1 C1 Þ A1 ðtÞ  A1 2c1   2 i þ c1 þ k1 C1 þ 2k1 C1 þ k1 C12 A1 ðt  s1 Þ  A1 :

ð3:9Þ

62

W. Wendi et al. / Appl. Math. Comput. 134 (2003) 51–67

Let us define Liapunov functional V2 by Z t  2 g  A1 ðsÞ  A1 ds: V2 ¼ 1 c1 þ k1 C1 þ 2k1 C1 þ k1 C12 2c1 ts1 It follow from (3.9) that for all large t, we have, " p1 c1 g C 1 k1 g1 0 ðV1 þ V2 Þ 6  1 þ  1  ðc þ k1 C1 þ k1 2 c1 2c1 1 ðc1 þ k1 C1 Þ #  2 g1  2 þ 2k1 C1 Þ c1 þ k1 C1 þ 2k1 C1 þ k1 C1 A1 ðtÞ  A1 : 2c1 ð3:10Þ By [8] we see that A1 is asymptotically stable if p 1 c1 g C1 k1 g1  1  ðc þ k1 C1 þ k1 þ 2k1 C1 Þ 2 c1 2c1 1 ðc1 þ k1 C1 Þ g   1 c1 þ k1 C1 þ 2k1 C1 þ k1 C12 > 0: 2c1

b,1þ

ð3:11Þ

If V ¼ V1 þ V2 , we have Z t  2 A1 ðsÞ  A1 ds 6 V ð0Þ < 1: V ðtÞ þ b 0

It is easy to see that A1 ðtÞ is uniformly continuous on ½0; 1Þ. It follows from Barbalat lemma [10] that limt!1 ðA1 ðtÞ  A1 Þ2 ¼ 0. Consequently, A1 is globally stable. We are now in a stage to state one main result of this section. Theorem 3.5. Assume gi C i gi  ðc þ ki Ci þ ki þ 2ki Ci Þ ci 2ci i ðci þ ki Ci Þ g   i ci þ ki Ci þ 2ki Ci þ ki Ci2 > 0; i ¼ 1; 2: 2ci

1þ

p i ci

2



ð3:12Þ

Then the positive equilibrium ðA1 ; A2 Þ of (3.3) is globally stable.

4. Periodic model In this section, we suppose that the advertisement intensity is periodic in time. This seems reasonable in reality. By introducing periodic advertisements into (3.3), we obtain

W. Wendi et al. / Appl. Math. Comput. 134 (2003) 51–67

dA1 ¼ ðc1 ðtÞ þ k1 A1 ðtÞÞðC1  A1 ðtÞÞ  p1 A1 ðtÞ  g1 ðc1 ðt  s1 Þ dt þ k1 A1 ðt  s1 ÞÞðC1  A1 ðt  s1 ÞÞ; dA2 ¼ ðc2 ðtÞ þ k2 A2 ðtÞÞðC2  A2 ðtÞÞ  p2 A2 ðtÞ dt  g2 ðc2 þ k2 A2 ðt  s2 ÞÞðC2  A2 ðt  s2 ÞÞ;

63

ð4:1Þ

where ci ðtÞ, i ¼ 1; 2, are non-negative, continuous and periodic functions with common period T and all the other parameters have the same meaning as the last two sections. Theorem 4.1. Suppose ci ðtÞ > gi ðci ðt  s1 Þ þ ki Ci Þ;

i ¼ 1; 2; t 2 ½0; T :

ð4:2Þ

Then there exists at least a positive periodic solution in (4.1). Proof. Since the two equations of this model are independent and have the same structure, we will concentrate on the first one. By (4.2), we can choose  > 0 small enough such that c1 ðtÞ >

p1  g C1 ðc1 ðt  s1 Þ þ k1 C1 Þ  1 ; C1   C1  

t 2 ½0; T :

Define M ¼ maxfjðc1 ðtÞ þ k1 AÞðC1  AÞ  p1 A  ðc1 ðt  s1 Þ þ k1 BÞðC1  BÞj : 0 6 A 6 C1 ; 0 6 B 6 C1 ; t 2 ½0; T g: Then set D ¼ f/ :  6 /ðhÞ 6 C1 ; h 2 ½s1 ; 0 ; j/ðh1 Þ  /ðh2 Þj 6 Mjh1  h2 j; h1 ; h2 2 ½s; 0 g: Let Aðt; 0; /Þ be the solution of the first equation of (4.1) through ð0; /Þ. Suppose that P is the Poincar
e map of the first equation of (4.1). We show that D is positively invariant. Let us consider a positive solution Aðt; 0; /Þ of the first equation of (4.1) with / 2 D. If t is the first time such that A1 ðt; 0; /Þ ¼ C1 , we have dA1 ðt; 0; /Þ ¼ p1 C1  g1 ðc1 ðt  s1 Þ þ k1 A1 ðt  s1 ; 0; /ÞÞ dt  ðC1  A1 ðt  s1 ; 0; /ÞÞ < 0:

ð4:3Þ

64

W. Wendi et al. / Appl. Math. Comput. 134 (2003) 51–67

If t is the first time such that A1 ðt; 0; /Þ ¼ , we have  dA1 ðt; 0; /Þ p1  ¼ ðC1  Þ c1 ðtÞ þ k1   dt C1    ðc1 ðt  s1 Þ þ k1 A1 ðt  s1 ÞÞðC1  A1 ðt  s1 ÞÞ  g1 C1     p1  ðc1 ðt  s1 Þ þ k1 C1 ÞC1  g1 > 0: > ðC1  Þ c1 ðtÞ þ k1   C1   C1   ð4:4Þ It follows from (4.3) and (4.4) that D is positively invariant. By Schauder fixed point theorem, we conclude that the Poincar
e map has a fixed point in D, which shows the first equation of (4.1) has a positive periodic solution. The existence of positive periodic solution in the second equation of (4.2) follows in the same way. The proof is complete.  Let us consider the global stability of positive periodic solution of (4.1). Suppose that A1 ðtÞ is a positive periodic solution of the first equation of (4.1). Set k ¼ maxt2½0;T c1 ðtÞ. Theorem 4.2. Let (4.2) be satisfied. Then A1 ðtÞ is globally stable if "   # 2 c1 ðt  s1 ÞC1 þ k1 C12 8k1 C12 þ 2c1 C1 p1  k1  g1 > 0; a,  g1  Þ2  Þ2 C1 ðC1  A ðC1  A  > 0 satisfies where A  Þ  p1 A  ÞðC1  A  ¼ 0: ðk þ k1 A Proof. The first equation of (4.1) can be rewritten as  dA1 ðtÞ p1 A1 ðtÞ ¼ ðC1  A1 ðtÞÞ c1 ðtÞ þ k1 A1 ðtÞ  dt C1  A1 ðtÞ  ðc ðt  s1 Þ þ k1 A1 ðt  s1 ÞÞðC1  A1 ðt  s1 ÞÞ  g1 1 : C1  A1 ðtÞ

ð4:5Þ

Obviously,  dA1 ðtÞ  p1 A1 ðtÞ ¼ C1  A1 ðtÞ c1 ðtÞ þ k1 A1 ðtÞ  dt C1  A1 ðtÞ     c1 ðt  s1 Þ þ k1 A1 ðt  s1 Þ C1  A1 ðt  s1 Þ  g1 : C1  A1 ðtÞ

ð4:6Þ

Let A1 ðtÞ be a positive solution of the first equation of (4.1). Then define    V1 ðtÞ ¼  lnðC1  A1 ðtÞÞ  ln C1  A1 ðtÞ :

W. Wendi et al. / Appl. Math. Comput. 134 (2003) 51–67

65

It is not hard to obtain     p1 C1  A1 ðtÞ  A ðtÞ  þ k1  A1 ðtÞ  A ðtÞ  1 1  ðC1  A1 ðtÞÞðC1  A1 ðtÞÞ   ðc ðt  s1 Þ þ k1 A1 ðt  s1 ÞÞðC1  A1 ðt  s1 ÞÞ þ g1  1 C1  A1 ðtÞ     c1 ðt  s1 Þ þ k1 A1 ðt  s1 Þ C1  A1 ðt  s1 Þ   : C1  A1 ðtÞ

Dþ V1 6 

Note that  ðc1 ðt  s1 Þ þ k1 A1 ðt  s1 ÞÞðC1  A1 ðt  s1 ÞÞ C1  A1 ðtÞ    c1 ðt  s1 Þ þ k1 A1 ðt  s1 Þ C1  A1 ðt  s1 Þ  C1  A1 ðtÞ h   ¼  k1 C1 A1 ðtÞ A1 ðt  s1 Þ  A1 ðt  s1 Þ  2k1 C1 A1 ðt  s1 Þ A1 ðt  s1 Þ  A1 ðt  s1 Þ þ 2k1 A1 ðtÞA1 ðt  s1 ÞðA1 ðt  s1 Þ  A1 ðt  s1 ÞÞ  2 þ k1 A1 ðt  s1 ÞC1 A1 ðtÞ  A1 ðtÞ  k1 A1 ðt  s1 Þ A1 ðtÞ  A1 ðtÞ    c1 ðt  s1 ÞC1 A1 ðt  s1 Þ  A1 ðt  s1 Þ þ c1 ðt  s1 ÞA1 ðtÞ A1 ðt  s1 Þ    A1 ðt  s1 Þ þ k1 C12 A1 ðt  s1 Þ  A1 ðt  s1 Þ  k1 C1 A1 ðt  s1 Þ 2  2  A1 ðt  s1 Þ þ k1 A1 ðtÞ A1 ðt  s1 Þ  A1 ðt  s1 Þ   i þ c1 ðt  s1 ÞC1 A1 ðtÞ  A1 ðtÞ  c1 ðt  s1 ÞA1 ðt  s1 Þ A1 ðtÞ  A1 ðtÞ .  ½ðC1  A1 ðtÞÞ C1  A1 ðtÞ : Since 0 < A1 ðtÞ < C1 , 0 < A1 ðtÞ < C1 and c1 ðtÞ 6 k for all large t, we have   ðc1 ðt  s1 Þ þ k1 A1 ðt  s1 ÞÞðC1  A1 ðt  s1 ÞÞ   C1  A1 ðtÞ    c1 ðt  s1 Þ þ k1 A1 ðt  s1 Þ C1  A1 ðt  s1 Þ    C1  A1 ðtÞ       2   2 6 2 kC1 þ k1 C1 A1 ðtÞ  A1 ðtÞ þ 8k1 C1 þ 2kC1 A1 ðt  s1 Þ     A1 ðt  s1 Þ ðC1  A1 ðtÞÞ C1  A1 ðtÞ : In order to estimate eventual lower bound of the denominator, let us consider the auxiliary equation dA1 ¼ ðk þ k1 A1 ÞðC1  A1 Þ  p1 A1 : dt

ð4:7Þ

66

W. Wendi et al. / Appl. Math. Comput. 134 (2003) 51–67

 is a positive equilibrium of this equation and is globally It is easy to see that A  þ  and A ðtÞ 6 stable. Then by comparison principle, we see that A1 ðtÞ 6 A 1  A þ  for all large t, where  is a very small positive constant. For simplicity in  and A ðtÞ 6 A  for all large t. This will not innotation, we suppose A1 ðtÞ 6 A 1 fluence our final results. As a consequence,   ðc1 ðt  s1 Þ þ k1 A1 ðt  s1 ÞÞðC1  A1 ðt  s1 ÞÞ   C1  A1 ðtÞ    c1 ðt  s1 Þ þ k1 A1 ðt  s1 Þ C1  A1 ðt  s1 Þ    C1  A1 ðtÞ       6 2 kC1 þ k1 C12 A1 ðtÞ  A1 ðtÞ þ 8k1 C12 þ 2kC1 A1 ðt  s1 Þ   Þ2 :  A ðt  s1 Þ =ðC1  A 1

Hence, we have "

 #  2 kC1 þ k1 C12  p1 D V1 6   k1  g1 A1 ðtÞ  A1 ðtÞ 2 Þ C1 ðC1  A   8k1 C12 þ 2kC1  þ g1 A1 ðt  s1 Þ  A1 ðt  s1 Þ: 2 Þ ðC1  A þ

Set 

8k1 C12 þ 2kC1 V2 ¼ g1  Þ2 ðC1  A

Z

t

  A1 ðsÞ  A ðsÞ ds: 1

ts

Then we have "

 2 kC1 þ k1 C12 p1 D ðV1 þ V2 Þ 6   k1  g1  Þ2 C1 ðC1  A #   8k1 C12 þ 2kC1   g1 A1 ðtÞ  A1 ðtÞ: 2 Þ ðC1  A þ

ð4:8Þ

It follows from [8] that A1 ðtÞ is asymptotically stable. If V ¼ V1 þ V2 , an integration of (4.8) leads to Z t   A1 ðsÞ  A ðsÞ ds 6 V ð0Þ < 1: V ðtÞ þ a 1 0

It is easy to see that A1 ðtÞ and A1 ðtÞ are uniformly continuous on ½0; 1Þ. It follows from Barbalat lemma [10] that limt!1 jA1 ðtÞ  A1 ðtÞj ¼ 0. Hence, the positive periodic solution is globally stable. The proof is complete. 

W. Wendi et al. / Appl. Math. Comput. 134 (2003) 51–67

67

References [1] V. Mahajan, E. Muller, R.A. Kerin, Introduction strategy for a new products with positive and negative word-of-mouth, Management Sci. 30 (1984) 1389–1404. [2] V. Mahajan, R. Peterson, Models for Innovation Diffusion, Sage, Beverly Hills, CA, 1985. [3] V. Mahajan, Y. Wind, Innovation Diffusion Models of New Product Acceptance, Bellinger, Cambridge, 1986. [4] R.W. Mizerski, An attribution explanation of the disproportionate influence of unfavorable information, J. Consumer. Res. 9 (1982) 301–310. [5] P. Fergola, C. Tenneriello, Z. Ma, F. Petrillo, Delayed innovation diffusion processes with positive and negative word-of-mouth, Int. J. Diff. Equa. Appl. 1 (2000) 131–147. [6] P. Fergola, C. Tenneriello, Z. Ma, F. Petrillo, An innovation diffusion model with time delay: positive and negative word of mouth, to appear. [7] P. Fergola, F. Petrillo, Some new results on the stability of a delayed innovation diffusion model, to appear. [8] J.K. Hale, S.M.V. Lunel, Introduction to Functional Differential Equations, Springer, New York, 1993. [9] R. Bellman, K.L. Cooke, Differential–Difference Equations, Academic Press, New York, 1963. [10] I. Barbalat, Systems d’equations differentielles d’oscillations non-lineaires, Rev. Roumaine. Math. Pures Appl. 4 (1959) 267–270. [11] J.P. Lasalle, The stability of dynamical systems, Hamilton Press, Berlin, 1976. [12] H.L. Smith, Monotone Dynamical Systems, Mathematical Surverys and Monographs, vol. 41, American Mathematical Society, Providence, RI, 1995.