# Neural Network-Informed Lotka-Volterra Dynamics for Cryptocurrency Market Analysis
## 4 Hybrid Neural Network--ODE Framework for Parameter Estimation and Forecasting
### Mathematical Formulation
Intraspecies (self-interaction) terms and uncertainty:

* The self-interaction parameters \(p_{2}=-0.4521\) (BTC, low uncertainty), \(p_{7}=-1.6997\) (ETH, high uncertainty), and \(p_{12}=-0.1015\) (ALTs, high uncertainty) are negative, consistent with self-limitation effects. However, only the BTC self-limitation is statistically well identified; ETH and ALT self-regulation remain uncertain due to wide confidence intervals and large standard errors. Interspecies interactions and uncertainty:

* Parameters reflecting competition or inhibition between asset classes--such as \(p_{3}=0.7681\), \(p_{4}=-1.3624\), \(p_{6}=-1.2830\), \(p_{8}=-1.7748\), \(p_{10}=-0.1602\), and \(p_{11}=-0.5119\)--vary in both sign and certainty. BTC-related cross-interactions ( \begin{table}
\begin{tabular}{l l l l l} \hline \hline
**Parameter** & **Estimate** & **Std (Uncertainty)** & **95\% CI Range** & **Uncertainty Level** \\ \hline \(p_{1}\) & \(0.5358\) & \(0.0995\) & [\(0.3409\), \(0.7308\)] & Low \\ \(p_{2}\) & \(-0.4521\) & \(0.0826\) & [\(-0.6139\), \(-0.2903\)] & Low \\ \(p_{3}\) & \(0.7681\) & \(0.1440\) & [\(0.4859\), \(1.0502\)] & Medium \\ \(p_{4}\) & \(-1.3624\) & \(0.2519\) & [\(-1.8562\), \(-0.8687\)] & Medium \\ \(p_{5}\) & \(1.4664\) & \(2.1373\) & [\(-2.7227\), \(5.6554\)] & High \\ \(p_{6}\) & \(-1.2830\) & \(1.7878\) & [\(-4.7871\), \(2.2210\)] & High \\ \(p_{7}\) & \(-1.6997\) & \(2.7787\) & [\(-7.1458\), \(3.7465\)] & High \\ \(p_{8}\) & \(-1.7748\) & \(5.2371\) & [\(-12.0395\), \(8.4899\)] & High \\ \(p_{9}\) & \(0.1941\) & \(0.4939\) & [\(-0.7739\), \(1.1622\)] & Medium \\ \(p_{10}\) & \(-0.1602\) & \(0.4119\) & [\(-0.9675\), \(0.64771\)] & Medium \\ \(p_{11}\) & \(-0.5119\) & \(0.6752\) & [\(-1.8352\), \(0.8115\)] & High \\ \(p_{12}\) & \(-0.1015\) & \(1.2280\) & [\(-2.5085\), \(2.3054\)] & High \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameter estimates with uncertainty quantification (using standard error and 95% confidence interval). and \(p_{4}\)) are estimated with medium uncertainty, while most others, particularly those involving ETH and ALTs, have high uncertainty, meaning the direction and strength of these effects cannot be robustly established from the current data. Summary and implications:

Overall, the model reveals that some self-regulation and mutual inhibition mechanisms--especially for BTC--are reliably identified, while the interactions involving ETH and ALTs remain uncertain, as shown by their wide confidence intervals and high standard errors. This uncertainty is likely due to intrinsic market volatility, data limitations, and the complexity of the cryptocurrency ecosystem. The parameter values should, therefore, be regarded as indicative, rather than definitive. Future research leveraging more extensive data, models allowing for time-variation in parameters, or introducing regularization techniques could help reduce uncertainty and yield more robust and interpretable estimates. Stability analyses and further predictions should explicitly account for these uncertainty levels. ### System Stability Testing

To investigate the local dynamics of the crypto market share model, we first estimate the parameters \(p_{i}\) and \(p_{ij}\) from observed market data using a neural network combined with optimization techniques. These parameters define the nonlinear Lotka-Volterra system (12), where \(u_{i}\) represents the market shares of BTC, ETH, and ALTs, respectively. It is important to note that these parameter estimates exhibit a degree of uncertainty, as quantified using bootstrap confidence intervals. Therefore, the following equilibrium and stability analyses [40] should be interpreted as conditional on the current parameter estimates and their variability.