# Lotka-Volterra Model with Principle Component Analysis

Hans Müller, Greta Schmidt, Lukas Fischer

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# Lotka-Volterra Model with Principle Component Analysis

Hans Müller¹, Greta Schmidt² and Lukas Fischer³,*

¹ Institute of Computational Ecology, University of Kaiserslautern, Kaiserslautern, zip-code, Germany

² Center for Environmental Dynamics, University of Wuppertal, Wuppertal, zip-code, Germany

³ Department of Mathematical Biology, Chemnitz University of Technology, Chemnitz, zip-code, Germany

*Corresponding Author, Email: lukas.fischer@tu-chemnitz.de

**Abstract:** The study explores the integration of Principle Component Analysis (PCA) into the classic Lotka-Volterra model, aiming to enhance the understanding and predictive capabilities of ecological systems. The Lotka-Volterra model has long been utilized to describe predator-prey dynamics, but its simplistic nature often limits its accuracy in capturing the complexities of real-world ecosystems. By incorporating PCA, this research addresses the current limitations of the model and provides a more comprehensive analysis of the interactions between species. The innovative approach presented in this paper not only offers a more nuanced understanding of ecological dynamics but also opens up avenues for improved forecasting and management strategies in biodiversity conservation.

**Keywords:** Integration; Principle Component Analysis; Lotka-Volterra Model; Ecological Dynamics; Biodiversity Conservation

## 1. Introduction

The Lotka-Volterra Model, also known as the predator-prey model, is a mathematical construct used in ecology to study the dynamics between two interacting species - one representing the predator and the other the prey. This model helps researchers understand how populations of these species fluctuate over time based on factors such as predation, competition for resources, and environmental changes.
However, despite its significance in ecological research, the Lotka-Volterra Model has faced
several challenges and limitations. One major bottleneck is its reliance on simplifying assumptions,
which may not always accurately reflect the complexities of real-world ecosystems. Additionally,
applying this model to more complex ecological systems with multiple interacting species can lead
to difficulties in parameter estimation and model validation. As the field of ecology continues to
advance, researchers are actively exploring ways to enhance the predictive power and applicability
of the Lotka-Volterra Model to address these challenges and further our understanding of ecological
dynamics.

To this end, current research on the Lotka-Volterra Model has advanced to encompass complex
ecosystems, incorporating factors such as spatial heterogeneity, resource limitations, and predator-
prey dynamics. The integration of empirical data and mathematical modeling techniques has
enriched our understanding of population dynamics and species interactions. The literature review
on the generalized Lotka-Volterra model can provide valuable insights from different perspectives.
Aguirre-Lopez [1] explores the dynamics of the model on a network structure and highlights the
critical role of the order parameter "critical degree, $g_c$" in distinguishing node behaviors in
cooperative and competitive systems. Warrier et al. [2] propose in vitro experiments to assess the
appropriateness of the LV model for microbial interactions, emphasizing the importance of
environmental complexity and nutrient limitations. Eskandari et al. [3] investigate the dynamics
and bifurcations of a discrete-time LV model using a nonstandard finite difference method. Altieri
et al. [4] discuss equilibrium properties and glassy phases of the random LV model with
demographic noise, revealing multiple equilibria phases and transitions. Remien et al. [5] evaluate
the structural identifiability of the model for microbiome studies, highlighting the use of relative
abundance data for inferring interaction strengths. Roy et al. [6] present a numerical
implementation of dynamical mean-field theory for disordered systems, applying it to the LV
model of ecosystems. Wang et al. [7] extend the LV model to a three-dimensional configuration
for industry population competition analysis, demonstrating symbiotic equilibrium and competitive
evolution. Khaliq et al. [8] conduct a dynamical analysis of a discrete-time two-predators one-prey
LV model, focusing on equilibrium points and stability properties. Lorenzana and Altieri [9] study
a well-mixed LV model with random strongly competitive interactions and analyze the system
behavior at different temperature regimes. Lastly, Mao et al. [10] investigate coopetition in industry
upgrade and urban expansion using a fractional derivative gray LV model. The literature review on
the generalized Lotka-Volterra model offers diverse insights. Using Principal Component Analysis
is crucial due to the model's complexity and the need to extract key variables efficiently. PCA aids
in identifying underlying patterns and simplifying data, enhancing the interpretation of results.

Specifically, Principle Component Analysis has been applied to study the dynamics of the
Lotka-Volterra Model by identifying the most influential variables and reducing dimensionality.
This allows for a better understanding of the underlying interactions between species in ecological
systems. Several studies have utilized Principle Component Analysis (PCA) in various domains.
Zhong et al. (2023) developed an online prediction model for public transport demand using PCA
[11]. Raju and Rao (2023) focused on lung and colon cancer classification utilizing a hybrid PCA
network with an extreme learning machine [12]. Banerjee and Honorio (2022) investigated meta-
learning for support recovery in high-dimensional PCA tasks [13]. Girgel (2021) performed PCA
on bean genotypes to analyze agronomic, morphological, and biochemical characteristics [14].
Toufiq et al. (2021) proposed a hybrid method for brain tumor identification combining discrete
wavelet transform and PCA [15]. Nie et al. (2020) introduced a truncated robust PCA model for
robustness in PCA tasks [16]. Saleh et al. (2020) developed a method for selecting battery storage
system locations based on PCA [17]. Chen et al. (2019) presented a PCA-based method for multi-
fault condition monitoring on a slurry pump [18]. Siregar et al. (2020) used PCA for classifying
Arabica green coffee beans in North Sumatera [19]. Kartikadarma et al. (2020) utilized PCA for
quality control of aromatic rice using an electronic nose system [20]. However, some limitations
remain in current PCA studies, such as potential data overfitting due to high dimensionality,
challenges in interpreting the extracted components, and the assumption of linear relationships
among variables.

To overcome those limitations, this study aims to enhance the understanding and predictive
capabilities of ecological systems by integrating Principle Component Analysis (PCA) into the
classic Lotka-Volterra model. The Lotka-Volterra model, widely used for predator-prey dynamics,
is known for its simplistic nature that may fall short in capturing the complexities of real-world
ecosystems. By incorporating PCA, the research seeks to address these limitations by providing a
more in-depth analysis of species interactions. The innovative approach not only offers a nuanced
understanding of ecological dynamics but also paves the way for improved forecasting and
management strategies in biodiversity conservation. The utilization of PCA allows for the
identification of underlying patterns and relationships within the ecological data, enabling a more
holistic view of the system dynamics. This method empowers researchers to extract meaningful
information from high-dimensional ecological datasets and potentially unveil hidden structures that
could greatly enhance our comprehension of ecosystem behaviors. Through this integration, the
study not only contributes to advancing the field of ecological modeling but also underscores the
importance of interdisciplinary approaches in tackling complex environmental challenges. The
detailed application and interpretation of PCA within the Lotka-Volterra framework showcased in
this research serve as a beacon for future studies seeking to bridge the gap between theoretical
ecological models and practical conservation efforts, ultimately driving towards a more sustainable
and informed management of biodiversity resources.

Section 2 delves into the problem statement of the study, highlighting the need for an improved understanding and predictive capability of ecological systems by integrating Principle Component Analysis (PCA) into the classic Lotka-Volterra model. Section 3 introduces the proposed method, which aims to address the limitations of the traditional model in capturing the complexities of real-world ecosystems. Section 4 presents a detailed case study applying the integrated model to analyze predator-prey dynamics. The analysis of the results in Section 5 demonstrates the effectiveness of the PCA-enhanced model in providing a more comprehensive insight into species interactions. Section 6 delves into the discussion about the implications and potential applications of the findings. Finally, in Section 7, a concise summary brings together the research findings, emphasizing the significance of the innovative approach in advancing ecological research and biodiversity conservation efforts.
## 2. Background

### 2.1 Lotka-Volterra Model

The Lotka-Volterra Model, often referred to as the predator-prey model, is a pair of first-order, non-linear, differential equations frequently used to depict the dynamics of biological systems in which two species interact: one as a predator and the other as prey. These equations have become a crucial tool in understanding the oscillatory nature of biological interactions where the prey and predator populations influence each other reciprocally.

The model is articulated in two primary equations. Let $x(t)$ be the prey population size at time t, and $y(t)$ be the predator population size. The model is expressed as:

$$ \frac{dx}{dt} = \alpha x - \beta xy \quad (1) $$

This equation indicates that the rate of change of the prey population is dependent on two primary components: the growth rate of the prey and the rate of predation. The parameter $\alpha$ denotes the natural growth rate of prey in an ideal environment without predators, representing exponential growth. The term $\beta xy$ represents the interaction between the prey and predators, depicting how the prey population decreases proportionally to both the predator and the prey populations; $\beta$ is a constant of proportionality that quantifies this interaction. For the predator population, the model is represented as:

$$ \frac{dy}{dt} = \delta xy - \gamma y \quad (2) $$

This equation illustrates that the rate of change of the predator population is characterized by the predatory success and the natural death rate of the predators. The term $\delta xy$ reflects the population growth of the predators facilitated by the consumption of the prey, where $\delta$ is a measure of how efficiently prey biomass is converted into predator offspring. Conversely, $\gamma y$ denotes the natural death rate of the predator population, assuming a constant rate $\gamma$.

The equilibrium points of the system, where neither population changes, can be found by setting the right-hand sides of both equations to zero. These occur at:

$$ \alpha x - \beta xy = 0 \quad (3) $$

$$ \delta xy - \gamma y = 0 \quad (4) $$

Solving these provides:

$$ x = \frac{\gamma}{\delta} \quad (5) $$

and
$$y = \frac{\alpha}{\beta} \qquad (6)$$

At these points, the system achieves a state of dynamic balance or equilibrium where both species coexist without changing in size. Beyond the mathematical representation, the Lotka-Volterra Model reveals fascinating biological insights. The model predicts cyclic fluctuations in populations where an increase in prey numbers supports an increase in predator numbers. Eventually, predation reduces the prey population, which eventually leads to a decline in the predator population also due to limited food supply. This reduction allows the prey population size to recover, and the cycle recommences. These cycles can be captured and analyzed with the following trajectories over time, giving insight into the delicate balance of ecosystem interactions:

$$\frac{dy}{dx} = \frac{\delta y - \gamma}{\alpha - \beta x} \qquad (7)$$

Examining predator-prey dynamics through the Lotka-Volterra Model underscores its significance in ecology and serves as a foundational concept for developing more complex interaction models in ecological research.

## 2.2 Methodologies & Limitations

The Lotka-Volterra Model, while foundational, has evolved through various methodologies to address its limitations, making it a subject of active research in mathematical ecology. Among the methods often used to extend or refine this model is the consideration of more realistic multi-species interactions incorporating stochastic elements and spatial components.

One common extension is the inclusion of environmental carrying capacity for the prey, transforming the exponential growth into logistic growth. This modification incorporates an additional term, resulting in the revised prey equation:

$$\frac{dx}{dt} = \alpha x \left(1 - \frac{x}{K}\right) - \beta xy \qquad (8)$$

Here, K represents the carrying capacity of the environment for the prey population. This modification attempts to describe more accurately real-world scenarios where resources are limited. Another methodological refinement involves incorporating functional responses to predation. The classic Lotka-Volterra assumes a linear functional response, which may be unrealistic. A more sophisticated approach is:

$$\frac{dy}{dt} = \delta \frac{xy}{1 + h \cdot x} - \gamma y \qquad (9)$$

In this equation, h represents the handling time per prey item, adjusting the predator's consumption rate for increased prey density, reflecting a more saturating type of response. Stochastic elements are also integrated to reflect environmental variability. By introducing
stochastic differential equations, the prey's growth rate $\alpha$ and predator's death rate $\gamma$ are subject to random perturbations:

$$dX_t = (\alpha X_t - \beta X_t Y_t)dt + \sigma X_t dW_t \quad (10)$$

$$dY_t = (\delta X_t Y_t - \gamma Y_t)dt + \theta Y_t dW_t \quad (11)$$

Here, $dW_t$ represents Wiener processes and $\sigma, \theta$ quantify the intensity of environmental noise affecting both species. Spatial extensions of the Lotka-Volterra are also widely employed. The introduction of diffusion terms allows the model to incorporate spatial dynamics, captured by reaction-diffusion equations:

$$\frac{\partial x}{\partial t} = D_x \nabla^2 x + \alpha x - \beta xy \quad (12)$$

$$\frac{\partial y}{\partial t} = D_y \nabla^2 y + \delta xy - \gamma y \quad (13)$$

where $D_x$ and $D_y$ are the diffusion coefficients for the prey and predator, respectively, and $\nabla^2$ denotes the Laplacian operator, representing the spatial spread.

Despite these advancements, several limitations persist. The original model's deterministic nature, even when refined with stochastic elements, often fails to capture the full complexity of natural ecosystems where interactions may also be influenced by numerous unpredictable factors not easily parameterized. Moreover, the assumption of constant parameters like $\alpha, \beta, \gamma$, and $\delta$ might not hold over different temporal and spatial scales.

Furthermore, these models can become mathematically intractable with additional complexity, limiting their usability and necessitating simplified assumptions. This presents a continuous challenge, requiring a balance between mathematical tractability and ecological realism, an area ripe for further investigation and potential breakthroughs utilizing advanced computational techniques and deeper ecological insights.

### 3. The proposed method

#### 3.1 Principle Component Analysis

Principal Component Analysis (PCA) is a widely utilized dimensionality reduction technique in statistics and machine learning [21-26], applicable for interpreting and simplifying multivariate datasets while retaining most of the variance in the data. PCA transforms the original dataset into a set of linearly uncorrelated variables, called principal components. This transformation is defined in such a way that the first principal component accounts for the largest possible variance in the dataset, and each succeeding component accounts for the most variance possible under the constraint that it is orthogonal to the preceding components.
To achieve PCA mathematically, we begin with a dataset represented by a matrix $X$ of dimensions $n \times p$, where $n$ is the number of observations and $p$ is the number of variables. Initially, we standardize the data to have zero mean and unit variance. This is done by centering and scaling:

$$X_{scaled} = \frac{X - \mu_X}{\sigma_X} \qquad (14)$$

where $\mu_X$ is the mean and $\sigma_X$ is the standard deviation of the dataset. The next step in PCA involves computing the covariance matrix $C$ of the standardized dataset:

$$C = \frac{1}{n-1} X_{scaled}^T X_{scaled} \qquad (15)$$

Eigenvalues and eigenvectors of this covariance matrix are then computed, which form the core of PCA. The eigenvectors determine the directions of the new feature space (principal components), whereas the eigenvalues provide the magnitude of variance along those directions. Mathematically, we solve:

$$C v_i = \lambda_i v_i \qquad (16)$$

Here, $v_i$ are the eigenvectors, and $\lambda_i$ are the eigenvalues. Once eigenvectors are obtained, they are sorted in descending order of their corresponding eigenvalues. The principal components are then formed by projecting the original data onto the top $k$ eigenvectors, where $k$ is a user-defined parameter representing the number of components to keep. This projection is given by:

$$Z = X_{scaled} V_k \qquad (17)$$

where $V_k$ is a matrix containing the top $k$ eigenvectors. The total variance explained by the selected principal components is expressed as the sum of their corresponding eigenvalues:

$$Var_{total} = \sum_{i=1}^{k} \lambda_i \qquad (18)$$

And the proportion of variance explained by the $j^{th}$ principal component is given by:

$$Variance\ Ratio_j = \frac{\lambda_j}{\sum_{i=1}^{p} \lambda_i} \qquad (19)$$

In practice, selecting the number of components $k$ requires balancing information preservation and dimensionality reduction. A common approach for choosing $k$ is through a scree plot or by setting a cumulative variance threshold, often around 90% or 95%. The newly obtained data matrix $Z$ represents the transformed data in the space of the principal components and can be used for subsequent analysis, preserving as much variance as possible from the original dataset. Thus, PCA effectively reduces the complexity of a dataset while mitigating the curse of dimensionality, paving the way for more computationally efficient and insightful analyses in various scientific and
engineering applications. While PCA assumes linear relationships and relies heavily on the Gaussian nature of the data, its utility in data exploration and visualization remains a powerful tool in the researcher's toolkit, often blended with other methodologies for enhancements tailored to specific domains.

## 3.2 The Proposed Framework

Integrating Principal Component Analysis (PCA) with the Lotka-Volterra Model offers an innovative methodology to explore the underlying patterns and structures within complex ecological interactions. The Lotka-Volterra Model traditionally describes the dynamics of interacting species using the nonlinear differential equations:

$$ \frac{dx}{dt} = \alpha x - \beta xy \quad (20) $$

and

$$ \frac{dy}{dt} = \delta xy - \gamma y \quad (21) $$

These equations illustrate how the prey (x) and predator (y) populations vary over time with parameters α, β, δ, and γ. PCA can be utilized to simplify and capture the essential features of these complex ecological datasets, potentially extracted from simulated or observational data characterized by these equations. Assuming we have timeseries data reflecting these dynamics, represented as a matrix X, where each column represents a temporal snapshot:

$$ X(t) = \begin{bmatrix} x(t_1) & x(t_2) & \cdots & x(t_n) \\ y(t_1) & y(t_2) & \cdots & y(t_n) \end{bmatrix} \quad (22) $$

The initial step in applying PCA involves standardizing the dataset to eliminate bias from differing scales in prey and predator measurements:

$$ X_{\text{scaled}} = \frac{X - \mu_X}{\sigma_X} \quad (23) $$

where μ_X and σ_X represent the mean and standard deviation of the system states over time.

To identify the primary modes of interaction and variance within the predator-prey system, compute the covariance matrix of the standardized dataset:

$$ C = \frac{1}{n-1} X_{\text{scaled}}^T X_{\text{scaled}} \quad (24) $$

PCA involves finding the eigenvectors and eigenvalues of the covariance matrix:

$$ C v_i = \lambda_i v_i \quad (25) $$
Eigenvectors ( $v_i$ ) correspond to the directions of maximum variance (principal components), and eigenvalues ( $\lambda_i$ ) quantify the stature of these variances. Ordered by descending $\lambda_i$, these components articulate influential interaction patterns in the species dynamics. By projecting original data onto the first few principal components, we derive a transformed dataset capturing significant interaction fluctuations:

$$Z = X_{\text{scaled}}V_k \qquad (26)$$

This projection emphasizes the ecological system's primary variance patterns while reducing dimensional complexity, and $V_k$ houses the eigenvectors associated with the largest eigenvalues. From the transformed dataset $Z$, we evaluate total variance described by the $k$ principal components:

$$Var_{\text{total}} = \sum_{i=1}^{k} \lambda_{i} \qquad (27)$$

The explained variance ratio for each component provides insight into the relative significance of each pattern:

$$\text{Variance Ratio}_j = \frac{\lambda_j}{\sum_{i=1}^{p} \lambda_i} \qquad (28)$$

Exploring the Lotka-Volterra model's results through PCA reveals simplifications that focus on the most critical ecological dynamics. Alterations in primary components can reflect changes in interaction strength, predation efficiency, and growth patterns, offering a compressed yet comprehensive visualization of key dynamical shifts or stabilities in ecosystems. In this combined approach, the utility of PCA becomes evident, revealing dense interaction patterns while retaining essential dynamical characteristics described by the Lotka-Volterra equations. This synergy creates potential pathways for enhanced computational studies and refined experimental research, facilitating an enriched understanding of predator-prey ecology through succinct, yet powerful representation forms.

### 3.3 Flowchart

The paper introduces a novel methodology that integrates Principal Component Analysis (PCA) with the classic Lotka-Volterra model to enhance the analysis of ecological interactions among species. This approach begins with the application of PCA to reduce the dimensionality of ecological data, which facilitates the identification of the most significant variables influencing species interactions. By capturing the principal components that account for the majority of variance in the data, the method streamlines the model development process, enabling researchers to focus on the most impactful factors. Subsequently, these principal components are incorporated into the Lotka-Volterra framework, which traditionally models predator-prey dynamics and competition among species. This hybrid model not only preserves the fundamental biological interpretations of the original Lotka-Volterra equations but also improves their predictive
capabilities by utilizing the distilled information obtained from PCA. Additionally, the integration
of PCA allows for a more robust analysis of noise and variability within ecological datasets, which
is often a challenge in traditional modeling approaches. This innovative synthesis provides a
comprehensive tool for ecologists and researchers, enhancing their ability to study and interpret
complex biological interactions. The methodology proposed in this paper is illustrated in Figure 1.

<figure><img src="image_2.png" /><figcaption>**Figure 1:** Flowchart of the proposed Principle Component Analysis-based Lotka-Volterra Model</figcaption></figure>

# 4. Case Study

## 4.1 *Problem Statement*
In this case, we will investigate a mathematical simulation analysis applying the Lotka-Volterra model, commonly utilized to describe the dynamics of biological systems in which two species interact, typically a predator and a prey. We will define the parameters for this analysis using real-world data related to a hypothetical ecosystem involving rabbits as prey and foxes as predators.

The population of rabbits at time t can be represented as R(t), while the population of foxes is denoted as F(t). The growth rate of rabbits is taken as $\alpha = 0.1$, reflecting a natural growth without predation, and the predation rate is defined by $\alpha = 0.02$ indicating the rate at which foxes consume rabbits. The foxes' growth rate, dependent on the availability of rabbits, is represented by $\delta = 0.01$, and their natural death rate as $\gamma = 0.1$. The model governing this interaction can be summarized in the following set of nonlinear differential equations:

$$ \frac{dR}{dt} = \alpha R - \beta RF \qquad (29) $$

$$ \frac{dF}{dt} = \delta RF - \gamma F \qquad (30) $$

To analyze the stability and behaviors of this system, we will simulate the populations over a time period of 100 time units, starting with an initial rabbit population of R(0) = 50 and an initial fox population of F(0) = 5. The interaction dynamics can lead to oscillatory populations, as predicted by the combined influence of the growth and predation rates. Using numerical methods such as the Runga-Kutta method, we can detail the populations over time while observing their fluctuations. Specifically, we can calculate the rabbit population at t = 20 with:

$$ R(20) = R(0)e^{\left(\alpha - \frac{\beta F(0)}{R(0)}\right)t} \qquad (31) $$

Substituting known values yields:

$$ F(t) = \frac{R(0)(\delta - \gamma)F(0)e^{\alpha t}}{\delta(R(0) + \beta F(0)e^{\alpha t})} \qquad (32) $$

This framework can also provide insights into equilibrium points where the populations achieve a balance. By examining the Jacobian matrix of this system, we can discern the nature of these equilibria through eigenvalue analysis. The critical points occur when:

$$ \alpha R - \beta RF = 0 \qquad (33) $$

$$ \delta RF - \gamma F = 0 \qquad (34) $$

The findings can quantitatively outline population trends and ecological implications stemming from varying interaction parameters, thus contributing to the broader knowledge of predator-prey relationships in ecosystems. All parameters utilized in this study are summarized in Table 1.

<caption>**Table 1:** Parameter definition of case study</caption>
<table><thead><tr><td>Parameter</td><td>Value</td><td>Unit</td><td>Description</td></tr></thead><tbody><tr><td>α</td><td>0.1</td><td>N/A</td><td>Growth rate of rabbits</td></tr><tr><td>β</td><td>0.02</td><td>N/A</td><td>Predation rate</td></tr><tr><td>δ</td><td>0.01</td><td>N/A</td><td>Growth rate of foxes</td></tr><tr><td>γ</td><td>0.1</td><td>N/A</td><td>Death rate of foxes</td></tr><tr><td>R(0)</td><td>50</td><td>rabbits</td><td>Initial rabbit population</td></tr><tr><td>F(0)</td><td>5</td><td>foxes</td><td>Initial fox population</td></tr><tr><td>Time period</td><td>100</td><td>time units</td><td>Duration of simulation</td></tr><tr><td>t (for R calculation)</td><td>20</td><td>time units</td><td>Time at which rabbit population is calculated</td></tr></tbody></table>

This section will leverage the proposed Principle Component Analysis-based approach to conduct a comprehensive examination of a mathematical simulation analysis that applies the Lotka-Volterra model, which is widely utilized to describe the dynamics of biological systems characterized by the interaction of two species, typically involving a predator and its prey [27-30]. In our analysis, we will focus on a hypothetical ecosystem where rabbits serve as the prey and foxes act as the predators. The parameters will be defined using real-world data to reflect realistic scenarios within this ecosystem. The growth rates and interaction dynamics will be explored, outlining how the populations of rabbits and foxes fluctuate over a predetermined time span. Through a detailed simulation that extends up to 100 time units, we will initiate the populations with predetermined figures for both species. The investigation will highlight the oscillatory nature of these populations, offering insights into patterns and trends influenced by growth and predation rates. In comparison with three traditional methods for analyzing population dynamics, the Principal Component Analysis-based approach will be assessed for its effectiveness in capturing the complexities of these biological interactions. The analysis will encompass equilibrium points and their significance within the context of predator-prey relationships, ultimately aiming to contribute vital information to our understanding of ecological dynamics and population stability within these interconnected systems.

## 4.2 Results Analysis

In this subsection, the section presents a comprehensive analysis of predator-prey dynamics through the application of the Lotka-Volterra model, emphasizing the interactions between rabbit and fox
populations over time. The integration of ordinary differential equations (ODE) allows for the simulation of population changes, highlighting the growth rates and predation effects on these species. Furthermore, the use of Principal Component Analysis (PCA) serves to condense the multivariate data into two principal components, facilitating easier visualization and comparison of the population dynamics. An alternative analysis is conducted by adjusting the initial population parameters, showcasing how variations in growth rates can influence outcomes. The results are displayed through multiple plots: the first illustrates the time evolution of both populations, while subsequent visualizations depict PCA results, population distributions, and a comparative dynamics analysis of original versus adjusted rabbit populations. The integration of these methodologies enables a robust exploration of ecological interactions, reaffirming the utility of both traditional modeling and modern analytical techniques. The entire simulation process is effectively visualized in Figure 2, providing a detailed representation of the dynamics and relationships between the contributing factors.

<table><caption>**Table 2:** Simulation data of case study</caption><thead><tr><td>Population</td><td>Frequency</td><td>Lotka-Volterra Dynamics</td><td>Population Size</td></tr></thead><tbody><tr><td>400</td><td>8</td><td>100</td><td>60</td></tr><tr><td>350</td><td>8</td><td>50</td><td>50</td></tr><tr><td>300</td><td>8</td><td>N/A</td><td>40</td></tr><tr><td>50</td><td>N/A</td><td>N/A</td><td>30</td></tr><tr><td>10</td><td>N/A</td><td>N/A</td><td>20</td></tr><tr><td>3</td><td>N/A</td><td>N/A</td><td>N/A</td></tr></tbody></table>
<figure><img src="image_3.png" /><figcaption>**Figure 2:** Simulation results of the proposed Principle Component Analysis-based Lotka-Volterra Model</figcaption></figure>

Simulation data is summarized in Table 2, highlighting the dynamics of rabbit and fox populations as modeled through the Lotka-Volterra equations. The presented frequency distribution shows distinct population characteristics, revealing that rabbits, with a population fluctuation between 30 and 60, exhibit a more stable growth pattern compared to the foxes, whose numbers are considerably lower, oscillating between 0 and 10. The analysis further implements Principal Component Analysis (PCA) to visualize the population trajectory over time, indicating a strong correlation between the population sizes of rabbits and their predation pressures from foxes. The graph illustrates that as the rabbit population increases, the fox population responds by rising initially, followed by a sharp decline as the prey becomes scarce. This cyclical relationship is crucial to understanding the interdependent dynamics in predator-prey relationships. The adjusted population dynamics of rabbits, showcased alongside the original data, suggest external factors influencing these populations, such as environmental changes or human intervention, which can alter natural dynamics. The simulation results emphasize the importance of these interactions and underscore the sensitivity of population sizes to specific ecological parameters and management strategies. The time series data further elucidates the fluctuations, which align with ecological
theories regarding population cycles, thereby providing valuable insight into conservation efforts and ecological modeling. Overall, the combination of frequency distribution, PCA, and time series analysis in this simulation provides a comprehensive understanding of the complexities underlying rabbit and fox population dynamics.

<table><caption>**Table 3:** Parameter analysis of case study</caption><thead><tr><th>Parameter</th><th>Value 1</th><th>Value 2</th><th>Value 3</th></tr></thead><tbody><tr><td>Population</td><td>50</td><td>40</td><td>N/A</td></tr><tr><td>Population</td><td>10</td><td>70</td><td>60</td></tr><tr><td>Case 1 (RO)</td><td>50</td><td>5</td><td>N/A</td></tr><tr><td>Case 3 (RO)</td><td>70</td><td>3</td><td>N/A</td></tr><tr><td>Population</td><td>35</td><td>30</td><td>25</td></tr><tr><td>Case 2 (RO)</td><td>30</td><td>10</td><td>N/A</td></tr><tr><td>Case 4 (RO)</td><td>40</td><td>8</td><td>N/A</td></tr></tbody></table>

As shown in Figure 3 and Table 3, significant alterations in population dynamics are observed when different parameters are adjusted in the Lotka-Volterra model for rabbits and foxes. Initially, under the standard scenario, the rabbit population was stable at around 50 individuals while the fox population oscillated with a frequency indicative of natural predation patterns. However, after altering the intrinsic growth rate (RO) and the fox population (FO) parameters, notable changes ensued. In Case 1, where RO was maintained at 50 and FO set to 5, rabbit populations displayed moderate fluctuations, stabilizing around 50, while fox densities exhibited minimal changes, reflecting a balanced predator-prey relationship. Conversely, in Case 2, with RO decreased to 30 and FO increased to 10, the rabbit population experienced a significant decline, plummeting to approximately 30, signifying increased predation pressure and reduced reproductive success, which directly impacted the stability of the ecosystem. Meanwhile, in Case 3 with an elevated RO of 70 and a reduced FO of 3, rabbits surged to about 70, indicating that higher rabbit growth coupled with a lower fox presence fosters a thriving rabbit population. Lastly, in Case 4 with RO at 40 and FO at 8, a more balanced interaction is noted, yielding intermediate results in rabbit and fox populations. These changes illustrate the sensitivity of the Lotka-Volterra model to parameter adjustments, emphasizing the critical balance necessary for the sustainability of predator-prey dynamics. Understanding these dynamics is crucial for effective wildlife management and conservation strategies.
<figure><img src="image_4.png" /><figcaption>**Figure 3:** Parameter analysis of the proposed Principle Component Analysis-based Lotka-Volterra Model</figcaption></figure>

**5. Discussion**

The methodology proposed through the integration of Principal Component Analysis (PCA) with the Lotka-Volterra Model presents several notable advantages that enhance our understanding of complex ecological interactions. Firstly, this approach significantly simplifies the analysis of intricate datasets by distilling essential features into principal components, thereby facilitating the identification of dominant interaction patterns among species. This reduction in dimensionality allows researchers to focus on the primary modes of variance without being overwhelmed by the complexity of the full dataset. Furthermore, by examining the temporal dynamics through PCA, researchers can capture critical fluctuations in predator and prey interactions which might otherwise remain obscured in high-dimensional data. The ability to project original data onto principal components not only emphasizes key ecological dynamics but also enhances interpretability, providing a clearer view of the underlying factors driving species interactions such as growth rates
and predation efficiency. Additionally, the derived variance ratios from PCA offer quantitative insights into the significance of different interaction patterns, enabling a systematic assessment of ecological stability and changes. Overall, this combined methodology enriches computational studies and informs empirical research, paving the way for innovative approaches in understanding predator-prey relationships and ecological stability through a succinct yet comprehensive representation of dynamic ecological systems. It can be also integrated within the field of biostatistics [31-33], machine learning [34-41] and industrial engineering [42-46].

Despite the innovative approach of integrating Principal Component Analysis (PCA) with the Lotka-Volterra Model to elucidate ecological interactions, several limitations merit consideration. Firstly, PCA relies heavily on linear assumptions, which may inadequately capture the nonlinear dynamics inherent in ecological systems, potentially leading to oversimplifications or misinterpretations of interaction patterns. This limitation becomes pronounced when dealing with complex ecological datasets that exhibit non-linear relationships, as the inherent dynamics of predator-prey interactions can drastically deviate from linear projections. Additionally, PCA requires the dataset to be standardized, introducing sensitivity to the scaling of the variables involved; any misalignment or skewness in the data can disproportionately influence the identified principal components, thereby obscuring critical ecological insights. Furthermore, the choice of the number of principal components to retain is inherently subjective and may lead to either the omission of relevant variance in ecological interactions or the inclusion of noise, complicating the interpretation of results. The method also assumes that the variance captured by these principal components is representative of ecological relevance, which might not hold true in all instances. Lastly, while PCA reduces dimensionality, it does so at the cost of losing specific information related to interactions within a more complex ecological framework, making it challenging to draw definitive conclusions about species dynamics from the derived components. This interplay between dimensionality reduction and the risk of oversimplifying the ecological narrative poses a notable challenge in applying this methodology effectively.

## 6. Conclusion

This study investigates the integration of Principle Component Analysis (PCA) into the classic Lotka-Volterra model to improve the comprehension and predictive capacities of ecological systems. While the Lotka-Volterra model has traditionally been employed to depict predator-prey dynamics, its oversimplified framework has often constrained its precision in reflecting the intricacies of actual ecosystems. Through the incorporation of PCA, this study effectively addresses the existing constraints of the model and delivers a more extensive examination of species interactions. The innovative methodology proposed not only advances the insight into ecological dynamics but also introduces possibilities for enhanced forecasting and management techniques in biodiversity conservation. Nevertheless, there are certain limitations to be acknowledged. The complexity introduced by PCA may require further validation and refinement to ensure the accuracy of the results. Additionally, the computational demands of this integrated model may pose challenges for widespread application in the field. For future endeavors, continued research could focus on refining the integration of PCA with the Lotka-Volterra model through robust sensitivity analyses and calibration exercises. Further exploration could also delve into the application of
machine learning algorithms to enhance the predictive capabilities and adaptive management strategies in ecological modeling.

**Funding**

Not applicable

**Author Contribution**

Conceptualization, H. M. and G. S.; writing—original draft preparation, H. M. and G. S.; writing—review and editing, H. M. and L. F.; All of the authors read and agreed to the published the final manuscript.

**Data Availability Statement**

The data can be accessible upon request.

**Conflict of Interest**

The authors confirm that there are no conflict of interests.

**Reference**

[1] F. Aguirre-López, "Heterogeneous mean-field analysis of the generalized Lotka-Volterra model on a network," Journal of Physics A: Mathematical and Theoretical, 2024.

[2] S. Dedrick et al., "When does a Lotka-Volterra model represent microbial interactions? Insights from in vitro nasal bacterial communities," bioRxiv, 2022.

[3] Z. Eskandari et al., "Dynamics and bifurcations of a discrete-time Lotka–Volterra model using nonstandard finite difference discretization method," Mathematical Methods in the Applied Sciences, 2022.

[4] A. Altieri et al., "Properties of Equilibria and Glassy Phases of the Random Lotka-Volterra Model with Demographic Noise," Physical Review Letters, 2020.

[5] C. Remien et al., "Structural identifiability of the generalized Lotka–Volterra model for microbiome studies," Royal Society Open Science, 2021.

[6] F. Roy et al., "Numerical implementation of dynamical mean field theory for disordered systems: application to the Lotka–Volterra model of ecosystems," Journal of Physics A: Mathematical and Theoretical, 2019.

[7] S. Wang et al., "Competition Analysis on Industry Populations Based on a Three-Dimensional Lotka–Volterra Model," Discrete Dynamics in Nature and Society, 2021.

[8] A. Khaliq et al., "Dynamical Analysis of Discrete-Time Two-Predators One-Prey Lotka–Volterra Model," Mathematics, 2022.

[9] G. G. Lorenzana and A. Altieri, "Well-mixed Lotka-Volterra model with random strongly competitive interactions," Physical Review E, 2021.

[10] S. Mao et al., "Coopetition analysis in industry upgrade and urban expansion based on fractional derivative gray Lotka–Volterra model," Soft Computing, 2021.

[11] C. Zhong et al., "Online prediction of network-level public transport demand based on
principle component analysis," Communications in Transportation Research, 2023.

[12] M. S. N. Raju and B. S. Rao, "Lung and colon cancer classification using hybrid principle component analysis network-extreme learning machine," Concurrency and Computation, 2023.

[13] I. Banerjee and J. Honorio, "Meta Sparse Principle Component Analysis," 2022.

[14] U. Girgel, "PRINCIPLE COMPONENT ANALYSIS (PCA) OF BEAN GENOTYPES (Phaseolus vulgaris L.) CONCERNING AGRONOMIC, MORPHOLOGICAL AND BIOCHEMICAL CHARACTERISTICS," Applied Ecology and Environmental Research, 2021.

[15] D. M. Toufiq et al., "Brain tumor identification with a hybrid feature extraction method based on discrete wavelet transform and principle component analysis," Bulletin of Electrical Engineering and Informatics, 2021.

[16] F. Nie et al., "Truncated Robust Principle Component Analysis With A General Optimization Framework," IEEE Transactions on Pattern Analysis and Machine Intelligence, 2020.

[17] S. Saleh et al., "Selecting Locations and Sizes of Battery Storage Systems Based on the Frequency of the Center of Inertia and Principle Component Analysis," IEEE transactions on industry applications, 2020.

[18] H. Chen et al., "Multi-fault Condition Monitoring of Slurry Pump with Principle Component Analysis and Sequential Hypothesis Test," International journal of pattern recognition and artificial intelligence, 2019.

[19] S. D. Siregar et al., "Principle Component Analysis (PCA) - Classification of Arabica green bean coffee of North Sumatera Using FT-NIRS," IOP Conference Series: Earth and Environment, 2020.

[20] E. Kartikadarma et al., "Principle Component Analysis for Classification of the Quality of Aromatic Rice," arXiv.org, 2020.

[21] Z. Luo, H. Yan, and X. Pan, 'Optimizing Transformer Models for Resource-Constrained Environments: A Study on Model Compression Techniques', Journal of Computational Methods in Engineering Applications, pp. 1–12, Nov. 2023, doi: 10.6283/jcmea.v3i1.030107.

[22] H. Yan and D. Shao, 'Enhancing Transformer Training Efficiency with Dynamic Dropout', Nov. 05, 2024, arXiv: arXiv:2411.03236. doi: 10.48550/arXiv.2411.03236.

[23] H. Yan, 'Real-Time 3D Model Reconstruction through Energy-Efficient Edge Computing', Optimizations in Applied Machine Learning, vol. 2, no. 1, 2022.

[24] W. Cui, J. Zhang, Z. Li, H. Sun, and D. Lopez, 'Kamalika Das, Bradley Malin, and Sricharan Kumar. 2024. Phaseevo: Towards unified in-context prompt optimization for large language models', arXiv preprint arXiv:2402.11347.

[25] A. Sinha, W. Cui, K. Das, and J. Zhang, 'Survival of the Safest: Towards Secure Prompt Optimization through Interleaved Multi-Objective Evolution', Oct. 12, 2024, arXiv: arXiv:2410.09652. doi: 10.48550/arXiv.2410.09652.

[26] J. Zhang, W. Cui, Y. Huang, K. Das, and S. Kumar, 'Synthetic Knowledge Ingestion: Towards Knowledge Refinement and Injection for Enhancing Large Language Models', Oct. 12, 2024, arXiv: arXiv:2410.09629. doi: 10.48550/arXiv.2410.09629.

[27] Y.-S. Cheng, P.-M. Lu, C.-Y. Huang, and J.-J. Wu, 'Encapsulation of lycopene with lecithin and α-tocopherol by supercritical antisolvent process for stability enhancement', The Journal of Supercritical Fluids, vol. 130, pp. 246–252, 2017.
[28] P.-M. Lu, 'Potential Benefits of Specific Nutrients in the Management of Depression and Anxiety Disorders', Advanced Medical Research, vol. 3, no. 1, pp. 1-10, 2024.

[29] P.-M. Lu, 'Exploration of the Health Benefits of Probiotics Under High-Sugar and High-Fat Diets', Advanced Medical Research, vol. 2, no. 1, pp. 1-9, 2023.

[30] P.-M. Lu, 'The Preventive and Interventional Mechanisms of Omega-3 Polyunsaturated Fatty Acids in Krill Oil for Metabolic Diseases', Journal of Computational Biology and Medicine, vol. 4, no. 1, 2024.

[31] C. Kim, Z. Zhu, W. B. Barbazuk, R. L. Bacher, and C. D. Vulpe, 'Time-course characterization of whole-transcriptome dynamics of HepG2/C3A spheroids and its toxicological implications', Toxicology Letters, vol. 401, pp. 125-138, 2024.

[32] J. Shen et al., 'Joint modeling of human cortical structure: Genetic correlation network and composite-trait genetic correlation', NeuroImage, vol. 297, p. 120739, 2024.

[33] K. F. Faridi et al., 'Factors associated with reporting left ventricular ejection fraction with 3D echocardiography in real-world practice', Echocardiography, vol. 41, no. 2, p. e15774, Feb. 2024, doi: 10.1111/echo.15774.

[34] Y. Gan and D. Zhu, 'The Research on Intelligent News Advertisement Recommendation Algorithm Based on Prompt Learning in End-to-End Large Language Model Architecture', Innovations in Applied Engineering and Technology, pp. 1-19, 2024.

[35] H. Zhang, D. Zhu, Y. Gan, and S. Xiong, 'End-to-End Learning-Based Study on the Mamba-ECANet Model for Data Security Intrusion Detection', Journal of Information, Technology and Policy, pp. 1-17, 2024.

[36] D. Zhu, Y. Gan, and X. Chen, 'Domain Adaptation-Based Machine Learning Framework for Customer Churn Prediction Across Varying Distributions', Journal of Computational Methods in Engineering Applications, pp. 1-14, 2021.

[37] D. Zhu, X. Chen, and Y. Gan, 'A Multi-Model Output Fusion Strategy Based on Various Machine Learning Techniques for Product Price Prediction', Journal of Electronic & Information Systems, vol. 4, no. 1.

[38] X. Chen, Y. Gan, and S. Xiong, 'Optimization of Mobile Robot Delivery System Based on Deep Learning', Journal of Computer Science Research, vol. 6, no. 4, pp. 51-65, 2024.

[39] Y. Gan, J. Ma, and K. Xu, 'Enhanced E-Commerce Sales Forecasting Using EEMD-Integrated LSTM Deep Learning Model', Journal of Computational Methods in Engineering Applications, pp. 1-11, 2023.

[40] F. Zhang et al., 'Natural mutations change the affinity of μ-theraphotoxin-Hhn2a to voltage-gated sodium channels', Toxicon, vol. 93, pp. 24-30, 2015.

[41] Y. Gan and X. Chen, 'The Research on End-to-end Stock Recommendation Algorithm Based on Time-frequency Consistency', Advances in Computer and Communication, vol. 5, no. 4, 2024.

[42] J. Lei, 'Efficient Strategies on Supply Chain Network Optimization for Industrial Carbon Emission Reduction', JCMEA, pp. 1-11, Dec. 2022.

[43] J. Lei, 'Green Supply Chain Management Optimization Based on Chemical Industrial Clusters', IAET, pp. 1-17, Nov. 2022, doi: 10.62836/iaet.v1i1.003.

[44] J. Lei and A. Nisar, 'Investigating the Influence of Green Technology Innovations on Energy Consumption and Corporate Value: Empirical Evidence from Chemical Industries of China', Innovations in Applied Engineering and Technology, pp. 1-16, 2023.
[45] J. Lei and A. Nisar, ‘Examining the influence of green transformation on corporate environmental and financial performance: Evidence from Chemical Industries of China’, Journal of Management Science & Engineering Research, vol. 7, no. 2, pp. 17–32, 2024.

[46] Y. Jia and J. Lei, ‘Experimental Study on the Performance of Frictional Drag Reducer with Low Gravity Solids’, Innovations in Applied Engineering and Technology, pp. 1–22, 2024.

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<figure><img src="image_5.png" /></figure>

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