Modeling cross-market capital flows using fractional competition dynamics

Introduction Interactions between the stock market and the rapidly expanding cryptocurrency market have become an important feature of modern financial systems. Compared with traditional equity markets, cryptocurrency markets are generally more volatile and more sensitive to investor sentiment, speculative behavior, and technological change. Classical integer-order models may not adequately capture the nonlinear interactions, delayed responses, and memory effects that characterize such cross-market capital dynamics. Methods This study develops a fractional-order Lotka-Volterra competition model to describe the dynamic competition between stock and cryptocurrency markets. The model employs Caputo fractional derivatives to incorporate memory effects and persistent shocks, and includes migration terms to represent capital reallocation between the two markets. The resulting system is investigated numerically using the fractional Adams-Bashforth-Moulton predictor-corrector scheme. Results Numerical simulations show that the fractional-order parameter has a substantial effect on market behavior. Lower fractional orders produce stronger memory effects, slower convergence to equilibrium, and smoother long-term transitions, whereas higher orders recover behavior closer to the classical integer-order case. The simulations further indicate that changes in competition coefficients and capital migration rates strongly affect long-term market dominance, coexistence, and equilibrium levels. Discussion The proposed fractional competition framework provides a more flexible and realistic description of stock-cryptocurrency interactions than classical models. By accounting for memory-driven dynamics, it offers useful insight into long-term capital allocation, market coexistence, and the possible influence of regulatory conditions and investor sentiment. These findings suggest that fractional-order modeling can serve as a valuable tool for studying evolving financial markets and cross-market capital flows.