# Neural Network-Informed Lotka-Volterra Dynamics for Cryptocurrency Market Analysis
## 2 Dynamics of Interacting Populations Under Competition  
The hypothesis regarding population variation suggests that the rate of change is directly proportional to the current population size. The most widely accepted method for modeling species population growth in the absence of competition [19; 20] is represented in the following equation:  
\[\frac{dN(t)}{dt}=rN(t)\bigg{(}1-\frac{N(t)}{K}\bigg{)} \tag{1}\]  
Here, \(N(t)\) denotes the population size at time \(t\), \(r\) represents the intrinsic growth rate, and \(K\) is the environmental carrying capacity--the maximum population the environment can sustain. This logistic model forms the basis of many demand estimation and forecasting techniques, including the logistic growth family [21] and Gompertz models [22]. When multiple species coexist in a shared ecosystem, they inevitably compete for limited resources. Definitions and explanations of such interspecies competition are found in [19; 23], and can be summarized as follows: \"Competition arises when two or more species experience diminished fitness-reduced growth or saturation levels-due to their simultaneous presence\". In such a scenario, each species' access to resources diminishes due to the presence of others, resulting in decreased growth rates and reduced carrying capacities. More specific types of species [17] interaction have been categorized as follows:  
* Predator-prey: One population's growth decreases while the other's increases. * Competition: Growth rates of both populations decrease. * Mutualism (symbiosis): Growth rates of both populations increase. In the context of a closed and established cryptocurrency market, firms interact similarly: competition over limited market share can reduce the share of each participant, especially if they aim to maximize both market dominance and profits. This aligns with the second interaction type (competition), making it the most suitable analogy. The simplest way to reflect this competition in a growth model is by introducing terms that quantify mutual interference. The resulting formulation is the Lotka-Volterra model, developed by [PERSON] and [PERSON]. This model has been thoroughly explored in the literature [17; 20; 23] for scenarios involving two or more interacting species. For \(m\) competing participants (or entities) in a market [24; 25], the system dynamics of Equation (1) can be described by the following set of first-order nonlinear differential equations:  
\[\frac{dN_{i}}{dt}=N_{i}\bigg{(}a_{i}-\sum_{j=1}^{m}a_{ij}N_{j}\bigg{)},\quad i =1,2,\ldots,m \tag{2}\]  
In this expression, \(\frac{dN_{i}}{dt}\) represents the rate of change of the \(i\)-th entity, \(a_{i}\) is its individual growth coefficient, and \(a_{ij}\) measures the competitive effect of species \(j\) on species \(i\) (interspecies when \(i\
eq j\) and intraspecies when \(i=j\)). Note that these coefficients generally differ. Each equation in this system can be derived by transforming the logistic growth equation as follows:  
\[\frac{dN(t)}{dt}=rN(t)\bigg{(}1-\frac{N(t)}{K}\bigg{)}=N(t)\Big{(}r-\frac{r}{K }N(t)\Big{)}=N(t)(r-aN(t)) \tag{3}\]  
Additional terms are then included to represent the competitive interactions among entities in the market. For the case where \(m=3\), the generalized Lotka-Volterra system described in Equation (2) is expressed as follows:  
\[ \begin{split}\frac{dx}{dt}&=x(a_{10}+a_{11}x+a_{12}y+a _{13}z)\\ \frac{dy}{dt}&=y(a_{20}+a_{21}x+a_{22}y+a_{23}z)\\ \frac{dz}{dt}&=z(a_{30}+a_{31}x+a_{32}y+a_{33}z)\end{split}  \tag{4}\]  
Here \(x,y,\) and \(z\) represent the three competing entities. The Lotka-Volterra formulation is particularly well suited to capturing the nonlinear competitive dynamics among firms in a market. Its ability to incorporate interaction terms directly into the growth equations allows for a more realistic modeling [26] of interdependencies, which are often nonlinear and time-varying.