Stability of a Continuous Reaction- Diffusion Cournot-Kopel Duopoly ...

Stability of a Continuous ReactionDiffusion Cournot-Kopel Duopoly Game Model Salvatore Rionero & Isabella Torcicollo

Acta Applicandae Mathematicae An International Research Journal on Applying Mathematics and Mathematical Applications ISSN 0167-8019 Acta Appl Math DOI 10.1007/s10440-014-9932-x

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Author's personal copy Acta Appl Math DOI 10.1007/s10440-014-9932-x

Stability of a Continuous Reaction-Diffusion Cournot-Kopel Duopoly Game Model Salvatore Rionero · Isabella Torcicollo

Received: 17 December 2013 / Accepted: 10 March 2014 © Springer Science+Business Media Dordrecht 2014

Abstract In order to take into account the territory in which the outputs are in the market and the time-depending firms’ strategies, the discrete Cournot duopoly game (with adaptive expectations, modeled by Kopel) is generalized through a non autonomous reactiondiffusion binary system of PDEs, with self and cross diffusion terms. Linear and nonlinear asymptotic L2 -stability, via the Liapunov Direct Methot and a nonautonomous energy functional, are investigated. Keywords Continuous Cournot-Kopel model · Nonlinear duopoly game · Nonlinear stability · Nonautonomous binary dynamical systems of P.D.Es · Self-diffusion · Cross-diffusion · Liapunov Direct Method 1 Introduction Duopoly is the case where the market is controlled by two firms X, Y , producing similar products. As basic continuous model for the evolution of the outputs of the firms, we consider the continuous time-scale ⎧ du ⎪ ⎨ = −α1 u + α1 μ1 v(1 − v) dt (1.1) ⎪ ⎩ dv = −α2 v + α2 μ2 u(1 − u) dt S. Rionero Dipartimento di Matematica ed Applicazioni “R. Caccioppoli”, Universitá di Napoli Federico II, Complesso Monte S. Angelo, Via Cinzia, Naples, Italy e-mail: [email protected] S. Rionero Accademia Nazionale dei Lincei, Via della Lungara, 10, 00165, Rome, Italy e-mail: [email protected]

B

I. Torcicollo ( ) Istituto per le Applicazioni del Calcolo “M. Picone”, C.N.R., Via P. Castellino 111, 80131 Naples, Italy e-mail: [email protected]

Author's personal copy S. Rionero, I. Torcicollo

where: u, v = outputs of the two firms X and Y respectively, μ1 = const. > 0, measure of the intensity of the effect of the action of Y on X, μ2 = const. > 0, measure of the intensity of the effect of the action of X on Y , 0 < αi ≤ 1 (i = 1, 2), adjustment coefficients. Either in the discrete Cournot-Kopel model [1–5], or in the continuous model (1.1), the spatial domain in which the outputs are in the market, is not considered. Motivated by this relevant remark and taking into account that reaction-diffusion systems of PDEs are the best candidates for investigating the diffusion processes in spatial domains {cfr. for instance, [7–32] and references therein}, in [6], it has been introduced the continuous autonomous model  ∂t u = −α1 u + α1 μ1 v(1 − v) + γ11 u + γ12 v (1.2) ∂t v = −α2 v + α2 μ2 u(1 − u) + γ21 u + γ22 v with: Ω ⊂ R3 , bounded domain (territory),1 in which, the outputs are in the market, φ : (x, t) ∈ Ω × R+ → φ(x, t) ∈ R,

φ ∈ {u, v}

∂t = temporal derivative,  = Laplacian operator, γii = const. ≥ 0 (i = 1, 2), self diffusion coefficients and γ12 , γ21 constant cross diffusion coefficients.2 In the present paper, in order to take into account that, according to the market evolution, the strategies of the firms may evolve, we let the adjustment coefficients be time-depending. Precisely, we assume that αi : t ∈ [0, ∞[ → αi (t) ∈ ]0, 1]

(1.3)

with αi ∈ C 1 (R) and follow the stability-instability procedures introduced in [7–9] for the nonautonomous binary systems. Section 2 is devoted to the introduction of – the steady equilibrium solutions (critical points), – the equations governing the perturbations, in Sect. 3, a nonautonomous Liapunov functional is introduced and some its properties are showed. The linear stability is studied in Sect. 4, while the nonlinear stability is performed in Sect. 5.

2 Preliminaries Denoted by (u, ¯ v) ¯ the generic equilibrium point of (1.1), system (1.1) admits the zero solution (u¯ = 0, v¯ = 0) and the constant steady states,   Yi Yi 2 Y2 2 − + , − 2i , (2.1) (u, ¯ v) ¯ = 3 μ1 μ2 9 μ1 μ1 μ2 1 It has been considered Ω ⊂ R3 also for taking into account territories containing mountains. 2 The cross-diffusion is introduced in order to take into account the influence of an output on the diffusion of

the other output, also via the diffusion terms.

Author's personal copy Stability of a Continuous Reaction-Diffusion Cournot-Kopel Duopoly

with (i = 1, 2, 3), and

where

⎧ Y1 = Y+ + Y− ⎪ ⎪ ⎪ ⎪ ⎨ Y+ − Y− √ Y + + Y− +i Y2 = − 3 2 2 ⎪ ⎪ ⎪ Y+ − Y− √ Y + Y− ⎪ ⎩ Y3 = − + −i 3 2 2

(2.2)

⎧



⎪ 2 3 ⎪ 3 3 p q q q q 2 p3 ⎪ ⎨Y+ = − + + , Y− = − − + , 2 4 27 2 4 27 ⎪ 3 3 3 2 2 2 ⎪ μμ 2μ μ ⎪ ⎩p = − μ1 μ2 + μ2 μ2 , q = − 1 2 + 1 2 − μ21 μ2 . 1 3 27 3

(2.3)

¯ = q + p ≤ 0 implies the existence of three non zero real steady soWe remark that,  4 27 2 3 ¯ = q + p > 0 implies the existence of only one non zero real solution. lutions, while  4 27 Denoting by U and V the perturbations 2

3

U = u − u, ¯

V = v − v, ¯

U and V are governed by  ¯ + γ11 U + γ12 V − α1 (t)μ1 V 2 ∂t U = −α1 (t)U + α1 (t)μ1 (1 − 2v)V ∂t V = α2 (t)μ2 (1 − 2u)U ¯ − α2 (t)V + γ21 U + γ22 V − α2 (t)μ2 U 2 and hence setting

one obtains



⎧ a12 (t) = α1 (t)μ1 (1 − 2v) ¯ ⎪ ⎨a11 (t) = −α1 (t), ¯ a22 (t) = −α2 (t) a21 (t) = α2 (t)μ2 (1 − 2u), ⎪ ⎩ F1 = −α1 (t)μ1 V 2 , F2 = −α2 (t)μ2 U 2 ,

∂t U = a11 (t)U + a12 (t)V + γ11 U + γ12 V + F1 (U, V ) ∂t V = a21 (t)U + a22 (t)V + γ21 U + γ22 V + F2 (U, V ).

To (2.7) we append the initial data  U (x, 0) = U0 (x) = u0 (x) − u¯ V (x, 0) = V0 (x) = v0 (x) − v¯

(2.4)

(2.5)

(2.6)

(2.7)

x∈Ω

(2.8)

(1 − β) ∇φ · n = 0 on ∂Ω × R+ β

(2.9)

and the Robin boundary conditions φ+

with φ ∈ {U, V } and 0 < β < 1. We denote by: – – – – –

· L2 (Ω)-norm;

· ∂Ω L2 (∂Ω)-norm; ·, · scalar product in L2 (Ω); ·, · ∂Ω scalar product in L2 (∂Ω); H 1 (Ω, β) functional space such that φ ∈ H 1 (Ω, β) → {φ 2 + (∇φ)2 ∈ L(Ω), βφ + (1 − β)∇φ · n = 0 on ∂Ω};

Author's personal copy S. Rionero, I. Torcicollo

– α¯ = α(Ω, ¯ β) > 0 positive constant, appearing in the following inequality

∇φ 2 +

β 2

φ 2 ∂Ω ≥ α φ

¯ , 1−β

(2.10)

is the lowest eigenvalue α of the spectral problem φ + αφ = 0

(2.11)

1

in H (Ω, β), {cfr. [34]}. Remark 1 We remark that, (2.1) are steady states either of (1.1) or of (1.2) or of (2.7).

3 Nonautonomous L2 -Energy Properties The main theorem of the Direct Method for nonautonomous systems [33] guarantees that, if there exists a function E which is positive definite, and its temporal derivative E˙ along the solutions is non positive, then the null solution is stable; while, if E admits an upper bound which is infinitely small at the origin (i.e. E ≤ m1 ( U 2 + V 2 ), m1 = const > 0) and its temporal derivative along the solutions is negative definite {i.e. E˙ < 0 for U 2 +

V 2 = 0}, then the null solution is asymptotically stable. The stability analysis, performed in the following, is based on the functional

1 η1 (t) U 2 + η2 (t) V 2 , (3.1) 2 with ηi (i = 1, 2) positive derivable functions in R+ and bounded there together with the derivative η˙ i to be chosen suitably later. Following the procedure used by Rionero in [7–9], we set, for any function f : R+ → R E=

f ∗ = sup f.

f∗ = inf f, R+

(3.2)

R+

Lemma 1 If (ηi )∗ > 0, i = 1, 2, then, E is definite positive. Proof In view of (3.1)–(3.2) it follows that, E>



  1 min (η1 )∗ , (η2 )∗ U 2 + V 2 . 2

Lemma 2 Let

 η1 γ11 η2 γ22 −

η1 γ12 + η2 γ21 2



2 > 0.

(3.3)

Then the time derivative of E along the solutions of (2.7)–(2.9) verifies the inequality 1 E˙ ≤ (η˙ 1 + 2a11 η1 − 2λα) U ¯

2 + (η˙ 2 + 2a22 η2 − 2λα) V ¯

2 + 2(η1 a12 + η2 a21 )U, V 2     − 2α2 η2 μ2 U 2 , V − 2α1 η1 μ1 U, V 2 , (3.4) with λ given by (η1 γ11 + η2 γ22 ) − λ< 2





η1 γ11 − η2 γ22 2



2 +

η1 γ12 + η2 γ21 2

2 .

(3.5)

Author's personal copy Stability of a Continuous Reaction-Diffusion Cournot-Kopel Duopoly

Proof The time derivative of E along the solutions of (2.7)–(2.9) is given by 1 E˙ = (η˙ 1 + 2a11 η1 ) U 2 + (η˙ 2 + 2a22 η2 ) V 2 + 2(η1 a12 + η2 a21 )U, V 2 + 2η1 γ11 U, U + 2η1 γ12 U, V + 2η2 γ21 V , U + 2η2 γ22 V , V     − 2α2 η2 μ2 U 2 , V − 2α1 η1 μ1 U, V 2 . (3.6) By using the divergence theorem and (2.9), (2.10) leads to 2η1 γ11 U, U + 2η2 γ22 V , V + 2η2 γ21 V , U + 2η1 γ12 U, V     β β = 2η1 γ11 −

U 2∂Ω − ∇U 2 + 2η2 γ22 −

V 2∂Ω − ∇V 2 1−β 1−β β U, V ∂Ω − 2(η1 γ12 + η2 γ21 )∇U, ∇V . − 2(η1 γ12 + η2 γ21 ) (3.7) 1−β On taking into account that

  −η1 γ11 W 2 − (η1 γ12 + η2 γ21 )W Z − η2 γ22 Z 2 ≤ −λ W 2 + Z 2

(3.8)

is satisfied, ∀W, Z, for ⎧ ⎪ ⎨λ < min(η1 γ11 , η2 γ22 )

  η1 γ12 + η2 γ21 2 2 ⎪ − (η γ + η γ )λ + η γ η γ − ≥ 0, λ ⎩ 1 11 2 22 1 11 2 22 2

which, on the other hand, is verified when, (3.3) holds, for λ < λ− with     (η1 γ11 + η2 γ22 ) η1 γ12 + η2 γ21 2 η1 γ11 − η2 γ22 2 − + > 0, λ− = 2 2 2

(3.9)

(3.10)

then, (3.6), collecting (3.7) and (3.8), becomes  1 E˙ ≤ (η˙ 1 + 2a11 η1 ) U 2 + (η˙ 2 + 2a22 η2 ) V 2 + 2(η1 a12 + η2 a21 )U, V 2    β  2 2 2 2 − 2λ ∇U + ∇V +

U ∂Ω + V ∂Ω (1 − β)      − 2α2 η2 μ2 U 2 , V − 2α1 η1 μ1 U, V 2 . (3.11) By using (2.10), (3.11) gives 1 E˙ ≤ (η˙ 1 + 2a11 η1 ) U 2 + (η˙ 2 + 2a22 η2 ) V 2 + 2(η1 a12 + η2 a21 )U, V 2       − 2λα¯ U 2 + V 2 − 2α2 η2 μ2 U 2 , V − 2α1 η1 μ1 U, V 2 (3.12) and hence (3.4) is obtained.



4 Linear Stability Disregarding the nonlinear terms, (3.12) reduces to 1 ¯

2 + (η˙ 2 + 2a22 η2 − 2λα) V ¯

2 E˙ ≤ (η˙ 1 + 2a11 η1 − 2λα) U 2

+ 2(η1 a12 + η2 a21 )U, V .

(4.1)

Author's personal copy S. Rionero, I. Torcicollo

From now on, for the sake of simplicity and concreteness, we assume that ai αi (t) = α¯ i + , 0 < αi ≤ 1 bi + ci t with ai , bi , ci (i = 1, 2) positive constants such that 0 < α¯ i ≤ 1 − theorem holds.

ai bi

(4.2)

. Then the following

Theorem 1 Under the assumption (3.3), either (a12 a21 )∗ < 0,

∀t ∈ R+ ,

(4.3)

or (a12 a21 )∗ > 0,

a11 a22 − a12 a21 > 0

∀t ∈ R+ ,

(4.4)

imply the linear stability. Proof Let us assume that (4.3) holds. We give the proof in the case {(a12 )∗ > 0, (a21 )∗ < 0, ∀t ∈ R+ }; the same steps can be followed in the case {(a12 )∗ < 0, (a21 )∗ > 0, ∀t ∈ R+ }. For η1 = −a21 , and η2 = a12 , it follows that η1 a12 + η2 a21 = −a21 a12 + a12 a21 = 0 then, (4.1) reduces to

1 ¯

2 + (η˙ 2 + 2a22 η2 − 2λα) V ¯

2 , (4.5) E˙ ≤ (η˙ 1 + 2a11 η1 − 2λα) U 2 where, ∀t ≥ 0 ⎧ η˙ 1 + 2a11 η1 − 2λα¯ ⎪ ⎪     ⎪ ⎪ a1 a2 c2 a2 ⎪ ⎪ ⎪ =− + 2 α ¯ + + ¯ − 2λα¯ < 0, α ¯ μ2 |1 − 2u| 1 2 ⎨ (b2 + c2 t)2 b1 + c1 t b2 + c2 t ⎪η˙ 2 + 2a22 η2 − 2λα¯ ⎪ ⎪     ⎪ ⎪ ⎪ a2 a1 c1 a1 ⎪ ⎩ =− + 2 α ¯ + + ¯ − 2λα¯ < 0. α ¯ μ1 |1 − 2v| 2 1 (b1 + c1 t)2 b2 + c2 t b1 + c1 t Under the assumption (4.4), on choosing η1 = |a21 |, and η2 = |a12 |, 2(η1 a12 + η2 a21 )U, V   = 2 |a21 |a12 + |a12 |a21 U, V       ≤ 2 |a21 ||a12 | + |a12 ||a21 | |U |, |V | = 4 |a12 ||a21 | |U |, |V |       √ √ = 4 |a12 ||a21 | |a12 ||a21 | |U |, |V | = 4 η1 η2 a12 a21 |U |, |V |     √ √ < 4 η1 η2 a11 a22 |U |, |V | ≤ 2 η1 |a11 | U 2 + η2 |a22 | V 2 ,

(4.6)

then, (4.1) reduces to

1 ¯

2 + (η˙ 2 − 2λα) V ¯

2 E˙ ≤ (η˙ 1 − 2λα) U 2 with

(4.7)

⎧ ⎪ ⎨η˙ 1 − 2λα¯ = −

c2 a2 μ2 |1 − 2u| ¯ − 2λα¯ < 0, (b2 + c2 t)2 c a 1 1 ⎪ ⎩η˙ 2 − 2λα¯ = − μ1 |1 − 2v| ¯ − 2λα¯ < 0. (a1 + c1 t)2

(4.8)

Author's personal copy Stability of a Continuous Reaction-Diffusion Cournot-Kopel Duopoly Table 1 Values of the parameters and corresponding steady states guaranteeing the condition (4.10)

Table 2 Values of the parameters and corresponding steady states guaranteeing the condition (4.11)

μ1

μ2

u¯

v¯ 3.275

1

20

0.478

1/3

3

0.109

0.962

1

3

0.247

0.953

1

25

0.039

6.07

1/3

30/11

0.077

0.882

μ1

μ2

u¯

v¯

μ1 μ2 (1 − 2v)(1 ¯ − 2u) ¯

5

1/3

0.128

0.305

0.76

2

7/8

0.181

0.441

0.132

2

7/9

0.140

0.416

0.188

2

7/10

0.097

0.394

0.239

5

2/5

0.199

0.322

0.429

In both of cases, E˙ ≤ 0 and recalling that E is positive definite, there exists a constant c1 > 0 such that E˙ ≤ −c1 E

⇔

E ≤ E(0)e−c1 t .

(4.9) 

Remark 2 In view of (2.6), we remark that: (4.3) is equivalent to ¯ − 2u) ¯ 0, α1 α2 μ1 μ2 (1 − 2v)(1

μ1 μ2 (1 − 2v)(1 ¯ − 2u) ¯ 12 ). Precisely, it shows, for some values of the parameters μ1 and μ2 , the corresponding positive steady states. The Table 2 is concerned with the stability condition (4.11) for the steady solution such that (u¯ < 12 ; v¯ < 12 ).

5 Nonlinear Stability Theorem 2 Let the assumption of Theorem 1, hold. Then the local nonlinear stability in the L2 -norm, holds. Proof In order to obtain the nonlinear stability of null solution, let us consider the nonlinear terms in (3.12) and let us set     (5.1) Φ = −2α2 (t)μ2 η2 (t) U 2 , V − 2α1 (t)μ1 η1 (t) U, V 2 ,

Author's personal copy S. Rionero, I. Torcicollo

which can be written as follows √ √  2α1 (t)μ1 η1 (t)   2α2 (t)μ2 η2 (t)  2 Φ= η2 (t)V , −η1 (t)U + η1 (t)U, −η2 (t)V 2 . η1 (t) η2 (t) From Φ ≤ c2 where c2 = max(maxt

√

η1 |U | +

√

2μ2 η2 η1

√

√

η2 |V |, η1 |U |2 + η2 |V |2



(5.2)

√

, maxt

η1 |U | +

√

2μ1 η1 3 ), η2

η2 |V | ≤

since

√  1 2 η1 |U |2 + η2 |V |2 2

from (5.2) it follows that √  3 Φ ≤ c2 2 η1 U 2 + η2 V 2 2 . In view of (4.9), it follows that, (3.12) leads to √  √  E˙ ≤ −c1 E + c2 2E 1+ε ≤ c2 2E ε − c1 E. For E ε (0)