Research on Competition and Cooperation of Central Ports in the Guangdong-Hong Kong-Macao Greater Bay Area Based on the Multi-group Lotka-Volterra Model

： The construction of the Guangdong-Hong Kong-Macao Greater Bay Area requires the establishment of a comprehensive port cooperation mechanism. Coordination is key to port integration. This paper establishes a multi-group Lotka-Volterra model based on the theory of competition and cooperation to study the evolution of the relationships among Hong Kong Port, Shenzhen Port, and Guangzhou Port. The results show that the three major ports have predator-prey roles and form a symbiotic competition and cooperation ecological pattern. The extended Hotelling model is established to study the central ports’ spatial differences; the res ults show that whether it is the central port cooperation body or the entire central port group, the gains from Shenzhen Port – Guangzhou Port cooperation are greater than between either and Hong Kong Port for Guangdong-Hong Kong- Macao Greater Bay Area’s con tainer business. The biggest competitor to Hong Kong Port in the container business is Guangzhou Port. Hong Kong Port will gain more from cooperation with Shenzhen Port and should strengthen such cooperation. The study provides quantitative support and reference for positioning the strategic development of the high-end shipping business in Hong Kong Port, the pilot free-trade port in Shenzhen Port, and the exploration of the Shenzhen- Hong Kong composite port, Guangzhou Port’s comprehensive shipping hub.


Introduction
As a strategic hub node and infrastructure for foreign exchanges, ports are a barometer of economic development and play an extremely important role in maintaining the stability of the global industrial supply chain [1]. After years of construction and development, the port cluster in the Guangdong-Hong Kong-Macao Greater Bay Area has formed a coastal trade pattern centred on Hong Kong Port, Shenzhen Port, and Guangzhou Port and gradually improved the development of a global-oriented port, which belongs to the busiest and most intensive area in the global port range [2].

Literature review
In light of the current trend of port integration and collaborative development, the research on ports in the management field is mainly based on the perspective of competition and cooperation to establish and maintain a dynamic game relationship between participants [5]. To shift from pure confrontational competition to some degree of cooperation, the interaction between members must be adjusted from the perspective of the participants' own development and social resource optimisation, and full play must be given to their own advantages in such cooperation, facilitating achievement of maximum gains [6][7][8]. Therefore, the research on the competition and cooperation game has become a hot topic in academic circles in China and abroad. The existing research results regarding the port competition and cooperation game primarily include longitudinal and cross-sectional studies. The longitudinal research direction considers industry, while the cross-sectional research direction considers space. In terms of longitudinal studies of ports, Álvarez-SanJaime et al. [9] and Wan [10] examined the impact on regional port competition and cooperation and social welfare arising from the cooperation between ports and inland service providers. Pauline et al. [11] qualitatively analysed the contribution of port cooperation to social welfare, discussing the role of a central government in promoting port cooperation. For cross-sectional studies of ports, Zhao et al. [12][13] probed the competition and cooperation between ports along the 21st Century Maritime Silk Road and explored the stability of the port strategic alliance. Owing to the special spatial features of port competition, Hotelling models are widely applied. Yu Min et al. [14] based their study on a Hotelling model to analyse the income status of adjacent ports in inland river ports under different types of competition strategies. Considering the particularity in port competition for market demand and coastal resources, Fan et al. [15] built a game model to examine the competition and cooperation among ports along the Yellow Sea. Kaselimi et al. [16] built a wharf duopoly game model with a Hotelling demand structure based on customer choice to investigate the impact that exclusive wharfs have on competition among port clusters. Along with economic downturn, port integration has accelerated in China. Song [17] carried out research on port integration based on the perspective of global supply chain and analysed the measures and impact of maritime logistics, while Kuang et al. [18] argued that supply-side reform in port-city separation can be used as a breakthrough point for port reform in China. From the perspective of maximised social welfare for internal transport in an outward transport system, Guo et al. [19] discussed port integration in a multi-port region, as exemplified by the ports in north-eastern China. In terms of port cooperation gains, Lai et al. [20][21] discussed the gains of the port competition and cooperation game as well as the coordination mechanism of the port supply chain. In addition, research on port competition and cooperation can also be found in Notteboom [22], Yu et al. [23], and Zongdaga et al. [24].
There is little research on the ports of the Guangdong-Hong Kong-Macao Greater Bay Area, and most are qualitative analysis. Few of the existing related quantitative studies were conducted primarily from the perspective of the Pearl River Delta before the proposal of the Guangdong-Hong Kong-Macao Greater Bay Area construction strategy. Considering local governments and container terminals in a port cluster, Yu et al. [25] built a mathematical model to examine the competition and cooperation game in a port cluster, as exemplified by the Hong Kong Port and Shenzhen Port in the Pearl River Delta. Using a two-stage game model, Luo et al. [26] investigated wharf price competition between the Port of Hong Kong and Shenzhen Port as well as port scale expansion. In terms of port cooperation research, Wang et al. [27] used a game-theoretical model to examine the influencing factors and port conditions in the Pearl River Delta, noting the importance of regional port alliances. In terms of port cooperation income research, Xia [28] conducted modelling analysis based on the container port alliance and obtained the distribution of cooperative game income and its functional solution. In addition, as exemplified by Shenzhen and Hong Kong, Michael et al. [29] quantitatively analysed the convenience of port trade.
In summary, there are few quantitative research results regarding the competition and cooperation game in the port integration of the Guangdong-Hong Kong-Macao Greater Bay Area. In view of the important position of the central port in the construction of the Guangdong-Hong Kong-Macao Greater Bay Area, based on the competition and cooperation game theory, this study models and analyses the central ports of the Guangdong-Hong Kong-Macao Greater Bay Area, represented by Hong Kong Port, Shenzhen Port, and Guangzhou Port. The study then explores the evolution of competition and cooperation relations among different central ports as well as the impact of cooperation strategies on port partners and the overall gains of the port, with the aim of providing quantitative support and a decision-making reference for the strategic development of central ports as well as the integration of ports in the Guangdong-Hong Kong-Macao Greater Bay Area.

Construction and analysis of a multi-centre port competition and cooperation game model
The interaction relationships among ports in a certain area have similar characteristics to the concept of population in ecology, and the main manifestation is that the number of the individual population will eventually reach the environmental capacity, which may lead to the competition of that population for resources, a predator-prey relationship between different populations, and mutually beneficial symbiosis [30]. The comprehensive influence of environmental and social factors, such as geographical location, the natural conditions of a given port, and government governance differences, leads to dynamic changes in port scale and port competitiveness in competition and cooperation. Therefore, the population-related theory can be applied to the study of the port competition and cooperation game.
The Lotka-Volterra model was first proposed to simulate the relationship between biological populations and was then gradually applied to the study of economic and social systems [31][32]. In view of the fact that the ports in the Guangdong-Hong Kong-Macao Greater Bay Area form a multi-centre port group, this paper introduces the multi-group Lotka-Volterra model into the study of the competition and cooperation game of central ports in the Guangdong-Hong Kong-Macao Greater Bay Area and establishes a multi-group competition and cooperation game model to discuss the game relationship among these ports.

Expression of the interaction relationship among central ports
According to the mapping relationship between the grey derivatives and even logarithms, we obtained the following: 3  3  1  1  3  3  3  3  3  3  3 3 33 31 32 , and    A symbiotic ecosystem is formed among the three central ports, but has predator-prey relationships, and this interaction forms an unstable equilibrium, as shown in Figure 3  In this case, the competition between central port i and central port j cannot form a stable state, and both are likely to win in the competition. At this time, the interests of the central ports are not entirely consistent. This unstable balance may lead to the consequences of excessive competition and potentially develop into a situation of vicious and extreme competition. In view of the influence of the institutional environment, economic hinterland, comprehensive strength, and other factors relevant to the three central ports of the Guangdong-Hong Kong-Macao Greater Bay Area, it is necessary to intervene at the national level to promote port integration. The integration should be based on market-oriented orientation and supplemented by government-level macro guidance. The competition faced by ports is primarily market demand competition, and is due to competition in spatial location and route arrangements.
In view of this, this paper primarily discusses the competition and cooperation game caused by the difference of port spatial location, and a Hotelling model is a classic game model for studying spatial difference competition. Considering that this study examines the multi-port competition and cooperation game, the model is extended to a two-dimensional space to better reflect to the actual problem.

Construction of an extended Hotelling model
Taking the three central ports in the Guangdong-Hong Kong-Macao Greater Bay Area as an example, this paper develops abstract positioning based on the natural geographical location of the central port, assuming that Hong Kong Port is located at   1, 0 , Shenzhen Port is located at   ,, i F a a a is the demand function of that central port. In view of the close geographical location and similar natural conditions of the three central ports, we consider there to be no significant difference in the service cost of each central port, which means that for cost c , the following exist: 1 (9) According to the Hotelling model, we obtain the following:  (11) In view of the fact that this paper focuses on the cooperation behaviour based on the above competition, which is primarily reflected in the cooperation gains among different central ports in the Guangdong-Hong Kong-Macao Greater Bay Area and the overall changes in the gains of a given central port when different central ports cooperate. Considering the actual situation of the current development of the three central ports, this paper primarily discusses the change in gains when Shenzhen Port competes with Hong Kong Port on the basis of cooperation between Shenzhen Port and Guangzhou Port, the change in gains when Shenzhen Port competes with Guangzhou Port on the basis of cooperation between Shenzhen Port and Hong Kong Port, and the change in overall gains when different central ports cooperate.
Assuming that Shenzhen Port and Guangzhou Port compete with Hong Kong Port on the basis of cooperating in pricing( 23   ), at this time, the market demands of the central ports are as follows: The total market demand of Shenzhen Port and Guangzhou Port is:  (14) By deriving the above gains function, we obtain the following results: