Comparative Analysis of Runge-Kutta Fehlberg and Finite Difference Methods in Solving the Lotka-Volterra Competitive Model

First Author\[1\]<sup>\*</sup>, Second Author<sup>2</sup>  
(Double Blind Review: Please do not type or edit anything here until final-camera ready submissions)

*<sup>1</sup>Universiti Teknologi MARA, Perlis, Malaysia (Please do not type or edit anything here, until final camera-ready paper submission)*

*<sup>2</sup>Universiti Teknologi MARA, Perlis, Malaysia (Please do not type or edit anything here, until final camera-ready paper submission)*

<table>
<tbody>
<tr class="odd">
<td>ARTICLE INFO</td>
<td></td>
<td>ABSTRACT</td>
</tr>
<tr class="even">
<td><p><em>Article history:</em></p>
<p>Received</p>
<p>Revised</p>
<p>Accepted</p>
<p>Online first</p>
<p>Published 1 March 2024</p></td>
<td></td>
<td>This paper primarily compares numerical methods, namely Runge-Kutta Fehlberg (RKF) and Finite Difference (FDM) methods, with the exact solution. These methods are chosen for their ability to effectively and accurately address the nonlinearity present in the model. The findings indicate that both RKF and FDM methods provide precise solutions, with RKF proving to be the most accurate compared to the exact solution of the Lotka-Volterra competitive model, as demonstrated through Mathematica 13.2 software. The research utilizes data from Gause's experiment involving two species, <em>Saccharomyces Cerevisiae</em> and <em>Schizosaccharomyces Kephir</em>. The populations of these species exhibit intraspecific interactions, increasing until reaching a constant level. <em>Saccharomyces</em> population increases hourly until the 57th hour, reaching 12 cells, while <em>Schizosaccharomyces</em> population increases hourly until the 99th hour, reaching 5 cells. Equilibrium and stability analyses provide crucial insights into the system's long-term behavior and perturbation response. There are 4 cases in this research, which are Case I, case II, case III, and Case IV. In case I and case II, a mixed population scenario, if the carrying capacities of one species are lower than the other species' carrying capacity divided by the competition coefficient, stable coexistence is observed. However, in case III, if the carrying capacities are lower than the other species' carrying capacity divided by the competition coefficient, one species undergoes competitive exclusion, leading to an unstable equilibrium or the extinction of the weaker species. Finally, in case IV, a stable equilibrium point is reached at the intersection of both isoclines when intraspecific competition exceeds interspecific competition, allowing both species to coexist without facing extinction.</td>
</tr>
<tr class="odd">
<td><p><em>Keywords:</em></p>
<p>Lotka-Volterra Competitive Model</p>
<p>RKF</p>
<p>FDM</p>
<p>Stability</p>
<p><em>DOI:</em></p>
<p>10.24191/jcrinn.v9i1</p></td>
<td></td>
<td></td>
</tr>
</tbody>
</table>

# 

# INTRODUCTION

Competition among multiple species for limited resources, such as food and habitat, or to suppress the growth of other species is a prevalent phenomenon in nature. Therefore, it is essential to examine models that capture the dynamics of competitive interactions among multiple species (Li & Cao, 2023). The study of ecological dynamics and competitive interactions among different species is fundamental for comprehending the behavior and stability of ecosystems. Within the field of ecology, the Lotka-Volterra model has been widely employed to describe the dynamics of interacting populations in ecological systems, including competitive interactions between species. This research introduces a model that explores the population dynamics when two or more species coexist in the same environment and engage in mutual interactions. Nevertheless, in the natural world, where organisms frequently inhabit close proximity, population densities eventually reach a threshold, and space becomes a limiting factor at higher population levels.

The Lotka-Volterra model has been widely employed to describe the dynamics of interacting populations in ecological systems, including competitive interactions between species. The model provides a mathematical framework that simulates how populations of different species change over time based on factors such as birth rates, death rates, and interactions with other species. By solving a set of coupled differential equations, researchers can explore how factors such as predation, resource competition, and mutualism influence the long-term dynamics of ecological communities. The model's ability to predict outcomes such as competitive exclusion or coexistence has made it a valuable tool for understanding complex ecological systems. It has applications in conservation biology, ecosystem management, and population ecology. The calculation of the RKF method in a different order led to the development of the RKF method. FDM is commonly employed in contemporary numerical analysis because of the efficiency with which modern computers can perform these linear algebra operations and the relative ease of their implementation.

Various numerical methods have been employed in the existing literature to solve the Lotka-Volterra competitive model. A study by Paul et al. (2016a) concentrated on the RKF approach, which offers approximative solutions for systems of nonlinear ordinary differential equations, such as models of insect populations and Lotka-Volterra models for single species. Observing the nonlinear systems, Voroshilova and Wafubwa (2020) have significant potential for studying population dynamics and social phenomena through simulation and offer a discrete model based on the competitive Lotka-Volterra equations with configurable phase-space volume. The research of Baines and Christou (2021) shows a moving-mesh finite-difference approach to propose a solution to the Lotka-Volterra competition-diffusion model in theoretical ecology. Lin et al. (2022) represented the difficult challenge of forecasting the Lotka-Volterra equations, which are renowned for their complex dynamics, as the subject of this research.

The research by Chiang (2012) employed the Lotka-Volterra model to stimulate their growth and analyzed the dynamic competitive relationship between the two wafer types. Ma et al. (2013) examine the Lotka-Volterra predator-prey model by adding a prey refuge and interspecific competition among predator species. Paul et al. (2016b) propose this paper to compare the efficiency and accuracy of the RKF and the LADM for solving the Lotka-Volterra prey-predator model, which is a dynamic ecological system describing predator-prey interactions. **The research by** Adamu **(2018) represented** the interactions between populations that interact and affect one another's growth rates are the subject of this paper. Yuzbasi (2021), this study analyzes the Lotka-Volterra predator-prey model with discrete delays using the operational matrix method. Adeniji et al. (2022) presented the work and proposed two numerical techniques, RKF and RK4, for the Predator-Prey-Scavenger Model.

This research aims to compare these methods' efficacy in solving the Lotka-Volterra competitive model and discern which yields the best results. The study employs numerical methods like the FDM and RKF to unravel mathematical models like the Lotka-Volterra competitive model. The researchers will develop a Lotka-Volterra competitive model to observe intraspecific interactions and establish an exact solution using sample data. The focus will be on determining the stability and equilibrium, investigating whether the species will achieve a stable equilibrium or undergo competitive exclusion. The researchers will utilize sample data from Gause's experiment involving *Saccharomyces Cerevisiae* and *Schizosaccharomyces Kephir.*

# METHODOLOGY

# 

The data used in this research is from Gause’s experiment on two species, *Saccharomyces Cerevisiae and Schizosaccharomyces Kephir* (Gause, 1934). The logistic equation of both species is used to find the exact solution. It will be compared to numerical methods, such as the RKF and the FDM. The calculation of the RKF method in a different order led to the development of the RKF method. Without requiring more complex derivative calculations, the RKF approach can approximate an FDM with the same level of precision. *Saccharomyces Cerevisiae*, a versatile yeast strain, finds wide-ranging applications across diverse industries, including brewing, baking, and biofuel production. Its robustness and fermentative capabilities make it an ideal choice for these processes. On the other hand, *SchizoSaccharomyces Kephir*, another yeast species, has gained popularity as a model organism in research due to its genetic manipulability and distinctive cellular characteristics.

# Exact solution

# 

The logistic equation from the Lotka-Volterra model is used to model the growth of an isolated population:

|  |  |     |
|  |  | --- |
|  |  | (1) |

where is the volume (in volume of yeast) at the time (in hours), is the instantaneous rate of increase (births/deaths), and is the carrying capacity. Assume constant and linear density dependence; no time lags, migration, age structure, or limited resources. Now, a straightforward iterative method for calculating and is demonstrated for both *Saccharomyces Cerevisiae* and *Schizoszaccharomyces Kephir.*

By solving the equation (1), the solution for the initial condition can be obtained, which gives:

|  |  |     |
|  |  | --- |
|  |  | (2) |

Equation (2) shows the good fit value, which can be obtained using curve fitting. The fit is reasonable for:

<table>
<tbody>
<tr class="odd">
<td><p><em>Saccharomyces</em> in isolation where and :</p>
<p><img src="668281688624f_media/media/image10.PNG" style="width:2.9in;height:1.51416in" /></p>
<p><strong>Figure 1</strong>: Curve fitting for growth of <em>Saccahromyces</em> in isolation</p></td>
<td><p><em>Schizosaccharomyces</em> in isolation where and :</p>
<p><img src="668281688624f_media/media/image13.PNG" style="width:2.9in;height:1.46565in" /></p>
<p><strong>Figure 2</strong>: Curve fitting for growth of <em>Schizosaccharomyces</em> in isolation</p></td>
</tr>
</tbody>
</table>

Based on Figures 1 and 2, the value of and For both species, this can be obtained by using the best curve fitting. This is done with the help of *Mathematica 13.2* software. Based on Figures 1 and 2, *Saccharomyces* produces a population of 13.0 in an isolated case, while *Schizosaccharomyces* only produces a population of 5.8. *Schizosaccharomyces* consumes 1⁄5.8 = 0.17241 food, while *Saccahromyces* 1⁄13.0 = 0.07692 food in a single unit. In other words, *Schizosaccharomyces* consumes 2.24 times as much food per unit as *Saccahromyces* \[0.17241⁄0.07692 = 2.24\] and *Saccahromyces* consumes just 1⁄2.24 = 0.446 times as much food per unit as *Schizosaccharomyces*. These factors can make it recalculate the volume of one species into its equal in terms of the volume consumed by another species of food.

The value of and will be used to solve the exact solution in equation (2). The value will be used and

substituted into equation (1) to solve the numerical methods, the RKF and FDM.

method.

# Runge-Kutta Fehlberg (rkf) method 

The fourth-order Runge-Kutta (RK4) technique is one of the most well-known constant-step procedures. The Runge-Kutta technique can approximate a Taylor Series approximation with reasonable accuracy without requiring higher derivative computations. To some extent, this technique may be seen as the basis upon which other techniques are built (Paul et al., 2016).

Consider the initial value problem:

|  |  |     |
|  |  | --- |
|  |  | (3) |

The RKF is one way to try to resolve this problem. The problem is solving the initial value problem in the above equation using the Runge-Kutta methods of order 4 and order 5.

First, the definitions are:

|  |  |     |
|  |  | --- |
|  |  | (4) |

A better value for the solution is determined using a Runge Kutta method of order 5:

|  |  |     |
|  |  | --- |
|  |  | (5) |

Repeat the method and step to obtain a better approximation.

# Finite Difference Method (fdm)

To use the FDM, first, essential parameters such as the intrinsic rate of increase , carrying capacity , and initial population are defined.

|  |                     |  |
|  | ------------------- |  |
|  | totalTime, timeStep |  |

To find the populations for *Saccharomyces*:

|  |          |     |
|  | -------- | --- |
|  | timestep | (6) |

# numerical methods for competitive equation

Given is the Lotka-Volterra competition model, or is also known as the coupled equation:

|  |  |     |
|  |  | --- |
|  |  | (7) |

where represents the volume of *Saccharomyces* and represents the volume of *Schizosaccharomyces*. The term represents the instantaneous rate of increase of *Saccharomyces*, meanwhile represents the instantaneous rate of increase of *Schizosaccharomyces*, and represents the carrying capacity of *Saccharomyces*, meanwhile represents the carrying capacity of *Schizosaccharomyces*. The parameter represents the per capita effect of *Schizosaccharomyces* on the population growth of *Saccharomyces*, and represents the per capita effect of *Saccharomyces* on the population growth of *Schizosaccharomyces*.

Since the coupled equation consumes a lot of time if it is solved using an exact analytical solution, it must use the *Mathematica 13.2* software to solve it numerically.

# results and discussions

This research compares numerical methods, such as the RKF and Finite Difference methods, with the exact solution.

<table>
<tbody>
<tr class="odd">
<td><p><img src="668281688624f_media/media/image39.PNG" style="width:2.7in;height:1.62287in" /></p>
<p><strong>Figure 3</strong>: Numerical approximation graphs for <em>Saccharomyces</em> logistic curve</p></td>
<td><p><img src="668281688624f_media/media/image40.PNG" style="width:2.9in;height:1.77187in" /></p>
<p><strong>Figure 4</strong>: Numerical approximation graphs for <em>Schizosaccharomyces</em> logistic curve</p></td>
</tr>
</tbody>
</table>

Based on Figures 1 and 2, the graph shows that the RKF method values are the best approximation compared to the Finite Difference method, as the RKF method's curve closely resembles the precise solution curve. It can be acknowledged that, in comparison to the exact solution for solving the Lotka-Volterra competitive model for *Saccharomyces*, the RKF approach functions as the more dependable approximation method than the Finite Difference method.

| **Table 1:** Error Analyses            |                 |                       |
| -------------------------------------- | --------------- | --------------------- |
|                                        | *Saccharomyces* | *SchizoSaccharomyces* |
| Mean Absolute Error (MAE) for RKF:     | 0.00000000696   | 0.0000000000414       |
| Root Mean Square Error (RMSE) for RKF: | 0.0000000119989 | 0.000000000049359     |
| Maximum (Infinity) Norm for RKF:       | 0.0000000287    | 0.0000000000765       |
| L2 (Euclidean) Norm for RKF:           | 0.0000000343288 | 0.00000000059436      |
|                                        |                 |                       |
| Mean Absolute Error (MAE) for FDM:     | 0.0228          | 0.0109                |
| Root Mean Square Error (RMSE) for FDM: | 0.0322          | 0.02145136            |
| Maximum (Infinity) Norm for FDM:       | 0.0619          | 0.0247                |
| L2 (Euclidean) Norm for FDM:           | 0.271           | 0.258                 |

In addition to the graphical representation of the solutions obtained using both methods, Table 1 shows that comprehensive error analyses were conducted to evaluate their accuracy and efficiency. Similarly, the results of these analyses demonstrated that the Runge-Kutta-Fehlberg (RKF) method outperforms the Finite Difference Method in solving the Lotka-Volterra Competitive Model, confirming the superiority of RKF as the preferred approach for this type of problem. Combining both approaches provides a comprehensive view of how the Finite Difference and Runge-Kutta Fehlberg methods compare to the exact solution, catering to both graphical and quantitative preferences.

*Mathematica 13.2* software solves the coupled equation to observe the species-interspecific interaction. The findings can be observed in Table 2 below.

Table 2. Numerical solution for *Saccharomyces* and *Schizosaccharomyces*

<table>
<thead>
<tr class="header">
<th>Hours</th>
<th><p>Numerical solution</p>
<p>(<em>Saccharomyces</em> in mixed)</p></th>
<th><p>Numerical solution</p>
<p>(<em>Schizosaccharomyces</em> in mixed)</p></th>
<th>Hours</th>
<th><p>Numerical solution</p>
<p>(<em>Saccharomyces</em> in mixed)</p></th>
<th><p>Numerical solution</p>
<p>(<em>Schizosaccharomyces</em> in mixed)</p></th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td>0</td>
<td>0.45</td>
<td>0.45</td>
<td>42</td>
<td>4.25750112</td>
<td>1.27372096</td>
</tr>
<tr class="even">
<td>1</td>
<td>0.488326274</td>
<td>0.467214728</td>
<td>43</td>
<td>4.35041528</td>
<td>1.28495</td>
</tr>
<tr class="odd">
<td>2</td>
<td>0.529413546</td>
<td>0.484910821</td>
<td>44</td>
<td>4.44039795</td>
<td>1.2954743</td>
</tr>
<tr class="even">
<td>3</td>
<td>0.573390065</td>
<td>0.503083157</td>
<td>45</td>
<td>4.52740535</td>
<td>1.30530446</td>
</tr>
<tr class="odd">
<td>4</td>
<td>0.620380118</td>
<td>0.521724397</td>
<td>46</td>
<td>4.61141682</td>
<td>1.31445345</td>
</tr>
<tr class="even">
<td>5</td>
<td>0.670502328</td>
<td>0.540824793</td>
<td>47</td>
<td>4.69243007</td>
<td>1.32293598</td>
</tr>
<tr class="odd">
<td>6</td>
<td>0.723867976</td>
<td>0.560372082</td>
<td>48</td>
<td>4.77045969</td>
<td>1.33076832</td>
</tr>
<tr class="even">
<td>7</td>
<td>0.780578961</td>
<td>0.580351315</td>
<td>49</td>
<td>4.84553987</td>
<td>1.33796832</td>
</tr>
<tr class="odd">
<td>8</td>
<td>0.840725914</td>
<td>0.600744753</td>
<td>50</td>
<td>4.91771479</td>
<td>1.34455461</td>
</tr>
<tr class="even">
<td>9</td>
<td>0.904386067</td>
<td>0.621531754</td>
<td>51</td>
<td>4.98704402</td>
<td>1.35054679</td>
</tr>
<tr class="odd">
<td>10</td>
<td>0.971621206</td>
<td>0.642688712</td>
<td>52</td>
<td>5.05359758</td>
<td>1.3559651</td>
</tr>
<tr class="even">
<td>11</td>
<td>1.04247543</td>
<td>0.664188985</td>
<td>53</td>
<td>5.11745352</td>
<td>1.36083008</td>
</tr>
<tr class="odd">
<td>12</td>
<td>1.1169734</td>
<td>0.686002903</td>
<td>54</td>
<td>5.17869989</td>
<td>1.36516258</td>
</tr>
<tr class="even">
<td>13</td>
<td>1.19511738</td>
<td>0.708097696</td>
<td>55</td>
<td>5.23742837</td>
<td>1.36898346</td>
</tr>
<tr class="odd">
<td>14</td>
<td>1.2768878</td>
<td>0.730437823</td>
<td>56</td>
<td>5.29373794</td>
<td>1.37231345</td>
</tr>
<tr class="even">
<td>15</td>
<td>1.36223922</td>
<td>0.752984778</td>
<td>57</td>
<td>5.34772941</td>
<td>1.37517313</td>
</tr>
<tr class="odd">
<td>16</td>
<td>1.45110006</td>
<td>0.775697377</td>
<td>58</td>
<td>5.3995072</td>
<td>1.37758262</td>
</tr>
<tr class="even">
<td>17</td>
<td>1.54337179</td>
<td>0.79853197</td>
<td>59</td>
<td>5.4491768</td>
<td>1.37956171</td>
</tr>
<tr class="odd">
<td>18</td>
<td>1.63892843</td>
<td>0.821442694</td>
<td>60</td>
<td>5.49684431</td>
<td>1.38112961</td>
</tr>
<tr class="even">
<td>19</td>
<td>1.73761639</td>
<td>0.84438175</td>
<td>61</td>
<td>5.54261575</td>
<td>1.38230493</td>
</tr>
<tr class="odd">
<td>20</td>
<td>1.83925455</td>
<td>0.86729974</td>
<td>62</td>
<td>5.58659634</td>
<td>1.38310575</td>
</tr>
<tr class="even">
<td>21</td>
<td>1.94363525</td>
<td>0.89014606</td>
<td>63</td>
<td>5.62888945</td>
<td>1.38354926</td>
</tr>
<tr class="odd">
<td>22</td>
<td>2.05052591</td>
<td>0.912869324</td>
<td>64</td>
<td>5.66959732</td>
<td>1.38365226</td>
</tr>
<tr class="even">
<td>23</td>
<td>2.15967086</td>
<td>0.935417806</td>
<td>65</td>
<td>5.70881867</td>
<td>1.38343046</td>
</tr>
<tr class="odd">
<td>24</td>
<td>2.2707936</td>
<td>0.957739883</td>
<td>66</td>
<td>5.7466509</td>
<td>1.3828992</td>
</tr>
<tr class="even">
<td>25</td>
<td>2.38359947</td>
<td>0.979784521</td>
<td>67</td>
<td>5.78318718</td>
<td>1.38207276</td>
</tr>
<tr class="odd">
<td>26</td>
<td>2.49777871</td>
<td>1.00150179</td>
<td>68</td>
<td>5.81851854</td>
<td>1.38096493</td>
</tr>
<tr class="even">
<td>27</td>
<td>2.61301061</td>
<td>1.02284327</td>
<td>69</td>
<td>5.85273219</td>
<td>1.37958864</td>
</tr>
<tr class="odd">
<td>28</td>
<td>2.7289667</td>
<td>1.04376246</td>
<td>70</td>
<td>5.88591193</td>
<td>1.37795611</td>
</tr>
<tr class="even">
<td>29</td>
<td>2.84531453</td>
<td>1.06421537</td>
<td>71</td>
<td>5.91813877</td>
<td>1.37607899</td>
</tr>
<tr class="odd">
<td>30</td>
<td>2.96172201</td>
<td>1.08416074</td>
<td>72</td>
<td>5.9494891</td>
<td>1.37396802</td>
</tr>
<tr class="even">
<td>31</td>
<td>3.07786071</td>
<td>1.10356036</td>
<td>73</td>
<td>5.98003739</td>
<td>1.37163358</td>
</tr>
<tr class="odd">
<td>32</td>
<td>3.19341023</td>
<td>1.12237951</td>
<td>74</td>
<td>6.00985303</td>
<td>1.36908511</td>
</tr>
<tr class="even">
<td>33</td>
<td>3.30806155</td>
<td>1.14058714</td>
<td>75</td>
<td>6.03900365</td>
<td>1.36633172</td>
</tr>
<tr class="odd">
<td>34</td>
<td>3.42151959</td>
<td>1.15815586</td>
<td>76</td>
<td>6.06755232</td>
<td>1.36338172</td>
</tr>
<tr class="even">
<td>35</td>
<td>3.53350673</td>
<td>1.17506225</td>
<td>77</td>
<td>6.09555974</td>
<td>1.360243</td>
</tr>
<tr class="odd">
<td>36</td>
<td>3.64376628</td>
<td>1.19128708</td>
<td>78</td>
<td>6.12308326</td>
<td>1.3569229</td>
</tr>
<tr class="even">
<td>37</td>
<td>3.75206119</td>
<td>1.20681466</td>
<td>79</td>
<td>6.15017697</td>
<td>1.35342817</td>
</tr>
<tr class="odd">
<td>38</td>
<td>3.85817963</td>
<td>1.22163359</td>
<td>80</td>
<td>6.17689285</td>
<td>1.34976524</td>
</tr>
<tr class="even">
<td>39</td>
<td>3.96193372</td>
<td>1.23573614</td>
<td>81</td>
<td>6.20327904</td>
<td>1.34593991</td>
</tr>
<tr class="odd">
<td>40</td>
<td>4.0631592</td>
<td>1.24911804</td>
<td>82</td>
<td>6.22938232</td>
<td>1.34195776</td>
</tr>
<tr class="even">
<td>41</td>
<td>4.1617196</td>
<td>1.26177883</td>
<td>83</td>
<td>6.25524567</td>
<td>1.33782375</td>
</tr>
<tr class="odd">
<td>84</td>
<td>6.28091079</td>
<td>1.33354267</td>
<td>115</td>
<td>7.12461049</td>
<td>1.14231846</td>
</tr>
<tr class="even">
<td>85</td>
<td>6.30641634</td>
<td>1.32911881</td>
<td>116</td>
<td>7.15636962</td>
<td>1.13451949</td>
</tr>
<tr class="odd">
<td>86</td>
<td>6.33179918</td>
<td>1.3245562</td>
<td>117</td>
<td>7.18852595</td>
<td>1.12662855</td>
</tr>
<tr class="even">
<td>87</td>
<td>6.35709417</td>
<td>1.31985858</td>
<td>118</td>
<td>7.22108483</td>
<td>1.11864645</td>
</tr>
<tr class="odd">
<td>88</td>
<td>6.38233386</td>
<td>1.31502932</td>
<td>119</td>
<td>7.25405142</td>
<td>1.11057404</td>
</tr>
<tr class="even">
<td>89</td>
<td>6.40754958</td>
<td>1.31007164</td>
<td>120</td>
<td>7.28743053</td>
<td>1.10241215</td>
</tr>
<tr class="odd">
<td>90</td>
<td>6.43277019</td>
<td>1.30498839</td>
<td>121</td>
<td>7.32122623</td>
<td>1.09416172</td>
</tr>
<tr class="even">
<td>91</td>
<td>6.45802381</td>
<td>1.29978233</td>
<td>122</td>
<td>7.35544249</td>
<td>1.08582368</td>
</tr>
<tr class="odd">
<td>92</td>
<td>6.48333607</td>
<td>1.29445585</td>
<td>123</td>
<td>7.39008276</td>
<td>1.07739899</td>
</tr>
<tr class="even">
<td>93</td>
<td>6.50873152</td>
<td>1.28901122</td>
<td>124</td>
<td>7.42514991</td>
<td>1.06888871</td>
</tr>
<tr class="odd">
<td>94</td>
<td>6.53423407</td>
<td>1.28345061</td>
<td>125</td>
<td>7.46064672</td>
<td>1.0602939</td>
</tr>
<tr class="even">
<td>95</td>
<td>6.55986513</td>
<td>1.27777587</td>
<td>126</td>
<td>7.4965753</td>
<td>1.0516157</td>
</tr>
<tr class="odd">
<td>96</td>
<td>6.58564551</td>
<td>1.27198879</td>
<td>127</td>
<td>7.53293734</td>
<td>1.04285529</td>
</tr>
<tr class="even">
<td>97</td>
<td>6.61159531</td>
<td>1.26609105</td>
<td>128</td>
<td>7.5697344</td>
<td>1.03401391</td>
</tr>
<tr class="odd">
<td>98</td>
<td>6.63773257</td>
<td>1.26008409</td>
<td>129</td>
<td>7.60696725</td>
<td>1.02509286</td>
</tr>
<tr class="even">
<td>99</td>
<td>6.66407496</td>
<td>1.25396933</td>
<td>130</td>
<td>7.64463643</td>
<td>1.01609352</td>
</tr>
<tr class="odd">
<td>100</td>
<td>6.6906394</td>
<td>1.24774809</td>
<td>131</td>
<td>7.68274223</td>
<td>1.00701729</td>
</tr>
<tr class="even">
<td>101</td>
<td>6.71744114</td>
<td>1.24142152</td>
<td>132</td>
<td>7.72128405</td>
<td>0.997865693</td>
</tr>
<tr class="odd">
<td>102</td>
<td>6.74449518</td>
<td>1.23499075</td>
<td>133</td>
<td>7.76026113</td>
<td>0.988640296</td>
</tr>
<tr class="even">
<td>103</td>
<td>6.77181577</td>
<td>1.22845684</td>
<td>134</td>
<td>7.79967254</td>
<td>0.979342684</td>
</tr>
<tr class="odd">
<td>104</td>
<td>6.7994158</td>
<td>1.22182074</td>
<td>135</td>
<td>7.83951641</td>
<td>0.969974595</td>
</tr>
<tr class="even">
<td>105</td>
<td>6.82730805</td>
<td>1.21508338</td>
<td>136</td>
<td>7.87979053</td>
<td>0.960537823</td>
</tr>
<tr class="odd">
<td>106</td>
<td>6.85550447</td>
<td>1.20824564</td>
<td>137</td>
<td>7.92049263</td>
<td>0.951034171</td>
</tr>
<tr class="even">
<td>107</td>
<td>6.88401598</td>
<td>1.20130834</td>
<td>138</td>
<td>7.96161961</td>
<td>0.941465572</td>
</tr>
<tr class="odd">
<td>108</td>
<td>6.91285341</td>
<td>1.19427229</td>
<td>139</td>
<td>8.00316775</td>
<td>0.931834061</td>
</tr>
<tr class="even">
<td>109</td>
<td>6.94202676</td>
<td>1.18713827</td>
<td>140</td>
<td>8.04513332</td>
<td>0.922141677</td>
</tr>
<tr class="odd">
<td>110</td>
<td>6.97154526</td>
<td>1.17990703</td>
<td>141</td>
<td>8.0875119</td>
<td>0.912390565</td>
</tr>
<tr class="even">
<td>111</td>
<td>7.001418</td>
<td>1.17257932</td>
<td>142</td>
<td>8.13029848</td>
<td>0.902582964</td>
</tr>
<tr class="odd">
<td>112</td>
<td>7.03165332</td>
<td>1.16515586</td>
<td>143</td>
<td>8.17348775</td>
<td>0.892721158</td>
</tr>
<tr class="even">
<td>113</td>
<td>7.06225891</td>
<td>1.15763741</td>
<td>144</td>
<td>8.21707371</td>
<td>0.882807541</td>
</tr>
<tr class="odd">
<td>114</td>
<td>7.09324239</td>
<td>1.15002469</td>
<td></td>
<td></td>
<td></td>
</tr>
</tbody>
</table>

Table 2 shows that the numbers of *Saccharomyces* in mixed populations continuously increase; meanwhile, the numbers of *Schizosaccharomyces* in mixed populations increase steadily, but after 65<sup>th</sup> hour, it starts to decrease. From 1 hour until 144 hours, *Saccharomyces* outnumbered *Schizosaccharomyces*.

Four different cases in the stability of competition are being studied: species 1 wins, species 2 wins, unstable equilibrium, and coexistence of both species.

<table>
<thead>
<tr class="header">
<th><p><img src="668281688624f_media/media/image41.PNG" style="width:2.2in;height:2.02697in" /></p>
<p>Figure 5. Case I (Species 1 win)</p></th>
<th><p><img src="668281688624f_media/media/image42.PNG" style="width:2.2in;height:2.01447in" /></p>
<p>Figure 6. Case II (Species 2 win)</p></th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td><p><img src="668281688624f_media/media/image43.PNG" style="width:2.6in;height:1.79255in" /></p>
<p><em>Figure 7. Case III (Unstable equilibrium)</em></p></td>
<td><p><img src="668281688624f_media/media/image44.PNG" style="width:2.6in;height:1.69807in" /></p>
<p><em>Figure 8. Case IV (Coexistence of both species)</em></p></td>
</tr>
</tbody>
</table>

Figure 5, *Saccharomyces*' isocline is situated above and to the right of *Schizosaccharomyces* in their competitive interaction. Considering the points below both isoclines, the populations of Saccharomyces and *Schizosaccharomyces* will increase. Conversely, points above both isoclines will result in a decrease in both populations. However, at points below *Saccharomyces*' isocline and above *Schizosaccharomyces*' isocline, *Saccharomyces* will thrive while *Schizosaccharomyces* will decline. This trajectory will continue until *Saccharomyces* can stabilize at its own carrying capacity while *Schizosaccharomyces* is driven to extinction. As seen in the sample data from Gause's experiment, the point in case I shows a stable equilibrium where *Saccharomyces* always outcompetes other species. This circumstance is referred to as the competitive exclusion of *Schizosaccharomyces* by *Scaccharomyces*. Figure 6 presents a different situation, with the *Schizosaccharomyces* isocline positioned higher and to the right of the *Saccharomyces* isocline. The point where both isoclines intersect remains the same as in case I. However, below *Schizosaccharomyces*'s isocline and above *Saccharomyces*'s isocline, *Saccharomyces* will grow while *Schizosaccharomyces* will decline. This trajectory will continue until *Schizosaccharomyces* can stabilize at its own carrying capacity while *Saccharomyces* is driven to extinction. *Saccharomyces* is competitively excluded by *Schizosaccharomyces* in this situation because *Schizosaccharomyces* consistently outcompetes *Saccharomyces*.

Figure 7 shows that *the isoclines of both species intersect. Here, Saccharomyces's carrying capacity is greater than Schizosaccharomyces's carrying capacity divided by the competition coefficient, and Schizosaccharomyces's carrying capacity is greater than Saccharomyces's carrying capacity divided by the competition coefficient. The point identical to the other cases exists above and below both isoclines. Saccharomyces actively excludes competitors from locations below its isocline and above Schizosaccharomyces' isocline. Conversely, Saccharomyces is competitively excluded by Schizosaccharomyces at sites where Saccharomyces' isocline is above, and Schizosaccharomyces' isocline is below. This demonstrates an unstable equilibrium where the two isoclines converge and depend on initial concentrations of Schizosaccharomyces and Saccharomyces.* Figure 8 shows that the isoclines intersect, but the carrying capacities for both species are lower than the carrying capacity divided by the competition coefficient I. In such instances, all population trajectories lead both species' populations to the intersection of the isoclines. At this point, situated on both isoclines, neither population experiences further growth, stabilizing at this equilibrium. This unique case is the only scenario where competitive exclusion does not occur. When intraspecific competition outweighs interspecific competition, a stable equilibrium point is achieved at the intersection of both isoclines, allowing both species to coexist without facing extinction. There is a stable equilibrium point at the intersection of both isoclines where the two species can coexist without extinction when intraspecific competition is larger than interspecific competition.

# conclusion and recommendations

In summary, this paper offers valuable insights into the numerical approximation method, such as the RKF and FDM methods, and the behavior of the Lotka-Volterra competitive model. The data from Gause's experiment is used in the study, which focuses on *Saccharomyces Cerevisiae* (*Saccharomyces*) and *Schizosaccharomyces Kephir* (*Schizosaccharomyces*). This paper evaluates the accuracy and performance of the RKF and FDM in approximating the models. The results reveal that the RKF method surpasses the FDM in effectiveness for approximation, as demonstrated by the close resemblance of RKF method values to the exact solution curve. It proves to be the most reliable approach for solving the Lotka-Volterra competitive model. This research highlights the efficacy of the FDM and RKF methods for solving the Lotka-Volterra competitive model. It provides valuable insights into the dynamics of species competition. The findings underscore the importance of carrying capacity and initial conditions in understanding equilibrium and stability in competitive interactions. This research contributes to population dynamics and offers practical implications for wildlife management and conservation efforts. numerous recommendations can be proposed for the numerical resolution of the Lotka-Volterra competitive model. First and foremost, researchers and specialists in the field are encouraged to prioritize the RKF method over the Finite Difference approach. The RKF method consistently demonstrated superior precision and adaptability in estimating population behavior, particularly when dealing with the complexities of the Lotka-Volterra competitive model. Additionally, it is advisable to calibrate and validate the Lotka-Volterra model using specific experimental or observational data related to the populations under investigation. Performing sensitivity analyses to assess the impact of parameter changes on numerical solutions is also recommended. Furthermore, researchers evaluate the model's predictions and enhance its credibility by comparing numerical solutions with real-world data. Lastly, exploring the integration of environmental parameters into the model is essential to represent their influence on population behavior. This adjustment enhances the accuracy and realism of numerical solutions and contributes to a more comprehensive understanding of the model's applicability.

# acknowledgments

We want to express our deepest gratitude to all who made this article possible. First and foremost, we acknowledge our colleagues who provided their insights and expertise that greatly assisted the research. However, they may not agree with all the interpretations this article provides. We would also like to show our gratitude to the anonymous reviewers for their so-called insights. Their comments and feedback were innumerably insightful and constructive; we greatly appreciate their valuable contributions. Lastly, we thank our readers for their attention and hope our work will inspire and inform future research. Please note that any errors in the article are our own and should not tarnish the reputations of these esteemed people.

# CONFLICT OF INTEREST STATEMENT

# 

The authors agree that this research was conducted without any self-benefits or commercial or financial conflicts and declare the absence of conflicting interests with the funders.

# AUTHOR’S CONTRIBUTIONS

Jaya Subramaniam carried out the research and wrote and revised the article. Rahmah Hassan conceptualized the central research idea and provided the theoretical framework. Rahmah Hassan and Chong Ju Lian designed the research and supervised the research progress; Rahmah Hassan anchored the review and revisions and approved the article submission.

# references

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<table>
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<td><blockquote>
<p><img src="668281688624f_media/media/image45.png" style="width:1.01042in;height:0.36111in" alt="A picture containing text, clipart Description automatically generated" /></p>
</blockquote></td>
<td>© 2024 by the authors. Submitted for open access publication under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).</td>
</tr>
</tbody>
</table>

1.  <sup>\*</sup> Corresponding author. *E-mail address*: <donottypehere@email.com> (Please do not type or edit anything here, our editors will do the work for you)