Fractional gray Lotka-Volterra models with application to cryptocurrencies adoption.

The Fractional Gray Lotka-Volterra Model (FGLVM) is introduced and used for modeling the transaction counts of three cryptocurrencies, namely, Bitcoin, Litecoin, and Ripple. The 2-dimensional study is on Bitcoin and Litecoin, while the 3-dimensional study is on Bitcoin, Litecoin, and Ripple. Dataset from 28 April 2013 to 10 February 2018 provides forecasting values for Bitcoin and Litecoin through the 2-dimensional FGLVM study, while dataset from 7 August 2013 to 10 February 2018 provides forecasting values of Bitcoin, Litecoin, and Ripple through the 3-dimensional FGLVM study. Forecasting values of cryptocurrencies for the n-dimensional FGLVM study, n={2,3} along 100 days of study time, are displayed. The graph and Lyapunov exponents of the 2-dimensional Lotka-Volterra system using the results of FGLVM reveal that the system is a chaotic dynamical system, while the 3-dimensional Lotka-Volterra system displays parabolic patterns in spite of the chaos indicated by the Lyapunov exponents. The mean absolute percentage error indicates that 2-dimensional FGLVM has a good accuracy for the overall forecasting values of Bitcoin and a reasonable accuracy for the last 300 forecasting values of Litecoin, while the 3-dimensional FGLVM has a good accuracy for the overall forecasting values of Bitcoin and a reasonable accuracy for the last 300 forecasting values of both Litecoin and Ripple. Both 2- and 3-dimensional FGLVM analyses evoke a future constant trend in transacting Bitcoin and a future decreasing trend in transacting Litecoin and Ripple. Bitcoin will keep relatively higher transaction counts, with Litecoin transaction counts everywhere superior to that of Ripple.