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Stability of a Continuous Reaction-Diffusion Cournot-Kopel Duopoly Game Model

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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 reaction-diffusion binary system of PDEs, with self and cross diffusion terms. Linear and nonlinear asymptotic L 2-stability, via the Liapunov Direct Methot and a nonautonomous energy functional, are investigated.

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Notes

  1. It has been considered \(\varOmega\subset \mathbb{R}^{3}\) 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.

  3. The value of c 2 can be easily evaluated, by analyzing the time derivative of \(h_{1}(t)= \frac{2\mu_{2}\sqrt{\eta_{2}}}{\eta _{1}}\) and of \(h_{2}(t)= \frac{2\mu_{1}\sqrt{\eta_{1}}}{\eta _{2}}\). Precisely,

    $$\dot{h}_1(t)= \frac{1}{\eta_2^2(t)}\frac{\mu_2\vert 1-2\bar{u}\vert\mu_1 \vert1-2\bar{v}\vert }{(b_2+c_2t)^2(b_1+c_1t)^2} \bigl[c_2c_1(2a_1\bar{\alpha}_2-a_2 \bar{\alpha}_1) t - c_2a_2( \bar{\alpha}_1 b_1+a_1)+2 c_1a_1( \bar{\alpha}_2 b_2+a_2)\bigr] $$

    and

    $$\dot{h}_2(t)= \frac{1}{\eta_1^2(t)}\frac{\mu_2\vert 1-2\bar{u}\vert\mu_1 \vert 1-2\bar{v}\vert}{(b_2+c_2t)^2(b_1+c_1t)^2}\bigl[ c_2c_1(2a_2\bar{\alpha}_1-a_1 \bar{\alpha}_2) t - c_1a_1( \bar{\alpha}_2 b_2+a_2)+2 c_2a_2( \bar{\alpha}_1 b_1+a_1) \bigr]. $$

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Acknowledgements

This paper has been performed under the auspices of the G.N.F.M. of I.N.d.A.M.

I. Torcicollo acknowledges the Project G.N.F.M. Giovani 2013 “Moti fluidi di miscele in strati porosi, immersi in campi termici non isotermi”. S. Rionero acknowledges the Leverhulm Trust “Tipping points: mathematics, metaphors and meanings”. The accuracy of the referees is acknowledged.

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Authors and Affiliations

  1. Dipartimento di Matematica ed Applicazioni “R. Caccioppoli”, Universitá di Napoli Federico II, Complesso Monte S. Angelo, Via Cinzia, Naples, Italy

    Salvatore Rionero

  2. Accademia Nazionale dei Lincei, Via della Lungara, 10, 00165, Rome, Italy

    Salvatore Rionero

  3. Istituto per le Applicazioni del Calcolo “M. Picone”, C.N.R., Via P. Castellino 111, 80131, Naples, Italy

    Isabella Torcicollo

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  1. Salvatore Rionero
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  2. Isabella Torcicollo
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Correspondence to Isabella Torcicollo.

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Rionero, S., Torcicollo, I. Stability of a Continuous Reaction-Diffusion Cournot-Kopel Duopoly Game Model. Acta Appl Math 132, 505–513 (2014). https://doi.org/10.1007/s10440-014-9932-x

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