# Neural Network-Informed Lotka-Volterra Dynamics for Cryptocurrency Market Analysis
## 4 Hybrid Neural Network--ODE Framework for Parameter Estimation and Forecasting
### Mathematical Formulation
By setting the time derivatives to zero,

\[\frac{du_{i}}{dt}=0, \tag{21}\]

we solve the resulting algebraic system and find a candidate equilibrium of approximately 41% for BTC, 20% for ETH, and 39% for ALTs:

\[u^{*}=(0.4150,\ 0.1953,\ 0.3897).\]

This represents the predicted steady-state market shares under the estimated parameters. To understand the local behavior near this equilibrium, we linearize the nonlinear system (12) around \(u^{*}\). With deviations from the equilibrium denoted as

\[U=u_{1}-u_{1}^{*},\quad V=u_{2}-u_{2}^{*},\quad W=u_{3}-u_{3}^{*}, \tag{22}\]

the linearized dynamics satisfy

\[\frac{d}{dt} \begin{pmatrix}U\\ V\\ W\end{pmatrix} =J(u^{*}) \begin{pmatrix}U\\ V\\ W\end{pmatrix} , \tag{23}\]

where \(J(u^{*})\) is the Jacobian matrix of partial derivatives evaluated at the equilibrium. The general solution near equilibrium is

\[ \begin{pmatrix}U\\ V\\ W\end{pmatrix} =c_{1}\mathbf{v}_{1}e^{\lambda_{1}t}+c_{2}\mathbf{v}_{2}e^{\lambda_ {2}t}+c_{3}\mathbf{v}_{3}e^{\lambda_{3}t}, \tag{24}\]where \(\lambda_{i}\) and \(\mathbf{v}_{i}\) are the real eigenvalues and eigenvectors of \(J(u^{*})\), and the coefficients \(c_{i}\) are obtained by projecting the initial deviation onto the eigenvector basis:

\[c_{1} =0.1583,\] \[c_{2} =0.1082, \tag{25}\] \[c_{3} =-0.5636.\]

Analyzing the eigenvalues of \(J(u^{*})\), we find

\[\lambda_{1}=-0.4850,\quad\lambda_{2}=-0.2072,\quad\lambda_{3}=-0.0013, \tag{26}\]

all strictly negative, suggesting that \(u^{*}\) is locally asymptotically stable under the current parameter estimates. The absence of imaginary parts indicates monotonic convergence without oscillations. However, given the presence of uncertainty in some parameters, the precise location and stability properties of \(u^{*}\) may vary in practice. This analysis assumes time-invariant parameters and neglects exogenous shocks, which are important factors in real cryptocurrency markets. The phase portrait shown in Figure 3 displays trajectories initiated from diverse initial conditions. All trajectories converge smoothly to the equilibrium \(u^{*}\), supporting the model's predictive validity within the assumptions and current parameter estimates. Overall, this approach--combining parameter estimation with Lotka-Volterra equilibrium and stability analysis--provides a useful framework to characterize the system's local dynamics, though further work is needed to incorporate parameter uncertainty, time variability, and external influences for more robust market predictions. Figure 3: Trajectories from various initial market shares converging to the unique stable equilibrium \(u^{*}\) (Blue Star Point). ### Case Study Results

In this section, we compare the effectiveness of our hybrid DNN-RK4 method against a range of ARIMA models and a neural ODE (NN-ODE) approach for predicting the market shares of BTC, ETH, and ALTs. The DNN-RK4 approach demonstrates the closest alignment to the real-world market share data, consistently producing the lowest prediction errors across the tested period. The ARIMA (autoregressive integrated moving average) model is a widely used statistical method for analyzing and forecasting time-series data. It combines three components: the autoregressive (AR) part models the relationship between current and past values; the integrated (I) part handles non-stationarity by differencing the data; and the moving average (MA) part captures dependencies between residual errors. By adjusting its parameters (p, d, and q), ARIMA can effectively model and forecast trends and patterns in sequential financial data, including cryptocurrency prices. In the ARIMA(\(p,d,q\)) model, \(p\) denotes the order of the autoregressive component (i.e., the number of lagged values of the series included), \(d\) represents the degree of differencing applied to make the series stationary (i.e., the number of times the series is replaced by the difference between consecutive observations), and \(q\) specifies the order of the moving-average component (i.e., the number of lagged forecast errors incorporated).