# Neural Network-Informed Lotka-Volterra Dynamics for Cryptocurrency Market Analysis
## 4 Hybrid Neural Network--ODE Framework for Parameter Estimation and Forecasting
### Mathematical Formulation
In addition, the use of graph neural networks (GNNs) will be explored to represent the complex interdependencies between different cryptocurrencies [47]. Crypto assets often exhibit co-movements, influenced by trading volumes, blockchain correlations, or investor sentiment. Modeling these relationships as a graph enables more informed parameter estimation and can uncover hidden structures in the market. Another area of potential is neural ODEs [48], which integrate the strengths of neural networks with differential equations. This approach is particularly suitable for learning continuous-time representations, offering a natural fit for systems like Lotka-Volterra that evolve over time. Neural ODEs can adaptively model temporal changes in market competition and volatility with fewer parameters and better interpretability [49]. The integration of sentiment analysis from social media and blockchain news will also be considered. Since investor behavior in crypto is highly reactive to news, incorporating textual data through models like BERT or domain-specific transformer variants (e.g., FinBERT) could improve forecasting accuracy by providing context-aware inputs to dynamic models [50]. Finally, online learning techniques will be examined for their ability to continuously update model parameters as new data arrives. This is critical for maintaining prediction accuracy in a market characterized by rapid innovation, regulatory shifts, and speculative bubbles. Something important to note here is that, because our Lotka-Volterra model assumes constant coefficients, it cannot capture one-time shocks such as Bitcoin's May 2020 halving or regulatory crackdowns in 2021. As a result, the estimated parameters may partially reflect these exogenous events, rather than pure \"peer competition.\" Also, a limitation of our current study is the exclusion of external market covariates such as absolute price, trading volume, and macroeconomic factors. While our results focus on modeling relative competition within a closed system, future research should include control experiments that explicitly incorporate such exogenous variables to quantify their impact on market share evolution. Future work will incorporate time-varying coefficients, exogenous covariates, and the previously discussed methodologies to address this limitation, aiming to develop a more comprehensive, adaptive, and robust framework for modeling and forecasting market share evolution in the cryptocurrency domain. ## 7 Conclusions

The proposed framework demonstrates that integrating deep neural networks with Lotka-Volterra competitive dynamics provides a robust, data-driven means of capturing and forecasting the evolving structure of cryptocurrency market shares. By estimating interaction coefficients directly from observed BTC, ETH, and ALT time series and then simulating their joint evolution via a fourth-order Runge-Kutta scheme, the model not only outperforms traditional ARIMA benchmarks but also achieves lower prediction errors compared to a standard NN-ODE model. Our hybrid DNN-RK4 approach achieves a mean RMSE of 0.05001 and a mean MAE of 0.04099, significantly outperforming ARIMA (mean RMSE 0.10099, mean MAE 0.09420) and NN-ODE (mean RMSE 0.09190, mean MAE 0.07965) across all assets. This confirms the superior accuracy and practical effectiveness of our method for cryptocurrency market forecasting, indicating its practical potential for decision-making in portfolio management and market trend analysis. Furthermore, the Lotka-Volterra-NN hybrid approach yields interpretable insights into mutual competitive pressures (e.g., the asymmetric influence of BTC on ALTs and ETH on BTC). Moreover, a local stability analysis indicates that the estimated equilibrium is locally asymptotically stable for the best-fit parameters, explaining why any transient deviations decay in the absence of exogenous shocks.