Bifurcation Analysis of a Duopoly Game with R&D Spillover, Price Competition and Time Delays

: The aim of this study is to analyse a discrete-time two-stage game with R&D competition by considering a continuous-time set-up with ﬁxed delays. The model is represented in the form of delay differential equations. The stability of all the equilibrium points is studied. It is found that the model exhibits very complex dynamical behaviours, and its Nash equilibrium is destabilised via Hopf bifurcations.


Introduction
Research and development (R&D) is one of the main strengths of firms growth. Firms need to pursue R&D as an effective way to reduce production costs and improve quality of products, so as to increase the competitiveness of firms in the market [1]. R&D behaviour is eventually followed by R&D spillover. R&D spillovers are likely because of the exchange of information on R&D between firms and the distribution of human resources. Over the last few years, the topic of competitiveness and collaboration throughout R&D spending has drawn growing interest from entrepreneurs and economists. The AJ model proposed by d'Aspremont and Jacquemin [2] and the KMZ model proposed by Kamien et al. [3] are two representative models for simulating the spillover effect of R&D. Such two models are two-stage game models and, respectively, addressed the spillover of R&D and the spillover of R&D production. Nowadays, the two-stage game has attracted the attention of many academics. Bischi and Lamantia [4,5] suggested a two-stage system to represent firms R&D networks in the marketplace. Matsumura et al. [6] proposed a two-stage Cournot model where companies select R&D spending at the first step and choose production amounts at the second stage. Shibata [7] analysed spillovers of R&D spending across different market structures. In particular, he expanded the work of Matsummura et al. [6] to integrate R&D investment spillovers. The implementation of chaos theory in structural dynamic economics developed by Day [8] presented a theoretical basis for the analysis of a complex model. The synthesis of dynamic theory and oligopoly theory has become a primary tool for economists and mathematicians to research economic phenomena. In recent years, it has attracted the attention of a growing number of researchers to investigate the evolution of the economic system and describe the complex economic phenomenon using chaos theory.
In this paper, we reconsider the discrete duopoly game model of R&D competition between two high-tech enterprises as introduced by Zhou and Wang [29], where the combination of game theory and nonlinear dynamics theory is applied to a monopoly market with R&D spillover. Their model happens to be described by where There are two firms, labelled by m (m = 1, 2), in a market, which conduct R&D and produce complementary goods. Here, x m is the R&D effort of firm m. The parameters a > 0, b > 0 and c > 0 represent the market size, the price sensitivity of consumers and the unit cost of produced goods without R&D efforts, respectively. β ∈ (0, 1) is related to the R&D spillover, whereas γ > 0 is the cost parameter of firm's technological innovation, which indicates the efficiency of using or producing the unique technology or knowledge resources for an enterprise. The smaller the parameter, γ, the stronger the innovation ability of firm m. Finally, α m > 0 is the speed of adjustment for firm m. A symmetry of parameters α 1 and α 2 exists in this system. Assuming continuous time scales and replacing x m (t + 1) − x m (t) (m = 1, 2) in (1) withẋ m (t) = dx m (t)/dt, system (1) can be transformed into a continuous-time model, which may be further extended to a dynamic environment characterised by differential equations with two fixed delays. Within this framework, we show how the introduction of delays may cause chaotic dynamics that cannot be observed when time delays are absent, therefore providing a starting point for building on more sophisticated models with R&D. The structure of this article is organised as follows. In Section 2, the continuous two-stage Cournot model with R&D spillover is established. In Section 3, the corresponding model with time delays is considered. The stability of its equilibrium points is discussed in case of one or two delays, and the occurrence of Hopf bifurcations is shown. Section 4 outlines the conclusions.

Existence of Equilibria and Local Bifurcations with Homogeneous Time Delays
By setting τ 1 = τ 2 = τ ≥ 0, system (5) becomeṡ To examine the stability of E * , we consider the characteristic equation of the linearisation of (6) at which writes as If all the roots of (7) have negative real parts, then the equilibrium E * of (6) is locally asymptotically stable, and it is unstable if (7) has at least one root with positive real part. In case τ = 0, assume that E * is stable. Let τ > 0. For computational purpose, we multiply both sides of (7) by e λτ and get We use this equation to yield purely imaginary roots iω to the characteristic Equation (7).
As each crossing of the real part of characteristic roots at τ * is from left to right as τ increases, based on the above analysis, we have the following result.

Existence of Equilibria and Local Bifurcations with Heterogeneous Time Delays
The aim is to extend the analysis developed in the previous section when τ 1 = τ 2 , τ 1 ≥ 0 and τ 2 ≥ 0, in system (5), and the equilibrium point E * is the Nash equilibrium E 3 . The Jacobian matrix evaluated at E 3 leads us to the following characteristic equation, namely, where Equation (14) reduces to In absence of delay, i.e., τ 2 = 0, E 3 is stable. With the time delay τ 2 varying, system (5) will lose the stability. To obtain such critical values of time delay, supposing λ = iω, ω > 0, is a purely imaginary root of (14), one has Taking the square, adding the equations and performing some simplification processes, and setting z = ω 2 , we have Obviously, if (17) has no positive solution for z, then (15) cannot have purely imaginary roots. Noticing that − det(J E 3 ) 2 < 0, it follows that Equation (17) has a unique positive root ω + , where Solving (16) for sin ωτ 2 and cos ωτ 2 , we get As sign (cos , one has the following sequence of critical delays where j = 0, 1, 2, ...
We next detect the stability switch at which the equilibrium loses stability. As λ is a function of delay τ 2 , we need the minimum solution of for which a derivative of λ(τ 2 ) is positive. By selecting τ 2 as the bifurcation parameter and differentiating the characteristic Equation (15), with respect, we get We now prove λ = iω + to be a simple root for (15). If this root is repeated, then (19) implies , a contradiction. Next, we can obtain that Therefore, we have This inequality implies that the real parts of complex eigenvalues of (15) turn to be positive from negative when crosses the imaginary axis as τ 2 increases. The previous analysis can be summarized as follows.

(2)
If Equation (22) presents at least two positive roots, then there exists some delayed interval sequence where the equilibrium E 3 of system (5) is locally asymptotically stable. The dynamical behaviour of system (5) near E 3 switches from stability to instability, and back again as time delays increase beyond the critical values, and Hopf bifurcations may occur.

Conclusions
This paper extends the discrete-time two-stage game of R&D competition between two high-tech enterprises of Zhou and Wang [29] to the case of continuous-time version with delays. The use of delay differential equations makes it possible to go beyond some limitations of other modelling approaches in a natural way. It is found that the boundary equilibrium points are always unstable and the Nash equilibrium loses its stability. The emergence of Hopf bifurcations is also characterised. Our findings, therefore, stress how the extent of time delays may be responsible for the existence of interesting dynamic outcomes, and underline the importance of the theoretical modelling framework used as a tool that may dramatically change the long-term findings of an economy.
Author Contributions: All the authors have equal contribution to this study. All authors have read and agreed to the published version of the manuscript.
Funding: This research received no external funding.