Abstract
With the development of the economy, developing high-technology industries and upgrading traditional industries have become a global focus. The relationship between the development of high-technology and traditional industries has also affected urban expansion. In this paper, a fractional derivative gray Lotka−Volterra model with time delay (FDGLV) is established by introducing fractional derivative and delay factor into the GLV model. The model is solved by using the fractional Adams−Bashforth−Moulton predictor−corrector algorithm. The gray correlation analysis is used to determine the time delay value of the model, and the parameters are optimized by using the whale algorithm. Then, the data of high-technology industry, traditional industry, and urban expansion in Chengdu, Changsha, and Chongqing are collected to verify the effectiveness of the model. Finally, the model is used to analyze the coopetition between industrial upgrading and urban expansion in Wuhan, and predict the development status in the next 5 years.
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All data included in this study are available upon request.
Abbreviations
- LV:
-
Lotka−Volterra
- GLV:
-
gray Lotka−Volterra
- FGLV:
-
fractional-order gray Lotka−Volterra
- FDGLV:
-
fractional derivative gray Lotka−Volterra
- ANN:
-
artificial intelligence model
- LSSVR:
-
least squares support vector regression
- ARIMA:
-
autoregressive integrated moving average
- GM (1,1):
-
gray (1,1) model
- WOA:
-
whale optimization algorithm
- \(x^{(0)}(k)\) :
-
original series
- \(x^{(r_{i})} (k)\) :
-
\(r_{i}\)-order accumulative generating
- p :
-
fractional derivative
- r :
-
accumulative order
- \({\hat{x}}^{(0)}(k)\) :
-
predicted value
- \(z^{(r_{i}) }(k)\) :
-
mean generation
- a, b, c :
-
coefficient of FDGLV
- \(\tau \) :
-
time delay
- k :
-
period time
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Acknowledgements
The authors are grateful to the editors and the anonymous reviewers for their insightful comments and suggestions, which have improved the quality of the paper immensely. The author would like to thank Yuannong Mao of the University of Waterloo, Canada, for his work on drawing and language modification. This research was partly supported by the National Natural Science Foundation of China (Project No. 51479151 ).
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YZ contributed to methodology, conceptualization, original draft, visualization. SM helped in formal analysis, funding acquisition, review, and editing, project administration. YK contributed to validation and data curation
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Mao, S., Zhang, Y., Kang, Y. et al. Coopetition analysis in industry upgrade and urban expansion based on fractional derivative gray Lotka–Volterra model. Soft Comput 25, 11485–11507 (2021). https://doi.org/10.1007/s00500-021-05878-z
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DOI: https://doi.org/10.1007/s00500-021-05878-z