COMPARATIVE ANALYSIS OF EULER AND RUNGE-KUTTA FEHLBERG METHODS IN SOLVING THE LOTKA-VOLTERRA COMPETITIVE MODEL

Abstract

Problem:

Based on articles written by Razali and Abdullah (2013), exclusion often occurs at a slow rate. Therefore, it was difficult to determine if equilibrium and stability exist. Equilibrium is the state of an ecosystem in which all components are in balance with one another. In addition, equilibrium species supposedly survive through a competitive survival strategy rather than through reproduction or dispersal to survive in harsh conditions where food resources are scarce. Unfortunately, based on the principle of competitive exclusion proposed by Gause (1934) which states that two species competing for the same resources cannot coexist, one will drive out.

Aims/Objectives:

1) To analyse the competitive interactions between lions (Panthera leo) and leopards (Panthera pardus) in the Sabi Sand Game Reserve, South Africa by comparing among exact solutions using Euler and RKF methods on the Lotka–Volterra Competitive model.

2) To discuss the impact of carrying capacity (capability survival) of dynamic behaviors in competition between two different species.

3) To analyze equilibrium and stability of the competition between two species depending on the initial conditions.

Methodology/approach:

This study used data from Balme et al. (2017) to examine competitive interactions between two top predators, lions (Panthera leo) and leopards (Panthera pardus) from 2010 to 2015 in the Sabi Sand Game Reserve, South Africa. This study will test whether lions, as the dominant competitor, will limit the distribution and abundance of leopards by using dietary data collected concurrently on the 2 species. Dietary overlap between lions and leopards was initially limited, with lions targeting large- to very large-sized prey and leopards small- to medium-sized prey. The Euler and Runge-Kutta Fehlberg (RKF) methods will be applied for solving this model and subsequently compare the simulation results with their exact solutions. It is anticipated that these numerical approximations will undertake to find out approximate solutions to nonlinear problems. Numerical approximation tests the reliability and precision of both methods applied to the logistic equation. This result is supported by computations carried out using the Mathematica 13.2 software.

Results/finding:

The findings indicate that the RKF method provides an accurate approximation, outperforming the Euler method. The results demonstrate that a larger carrying capacity is associated with a greater ability for a species to thrive and survive in competitive environments. Carrying capacity refers to the maximum population size a species can sustain indefinitely, considering factors like food, habitat, water, and others. During the curve-fitting procedure, it was observed that leopards have a higher carrying capacity compared to lions. Therefore, it can be inferred that the population of leopards is not significantly affected by the presence of lions.

Implication/impact:

The study will be beneficial to future researchers who engage with nonlinear differential equations, mathematical biology, and numerical techniques. By utilizing data stimulation to mimic real-world problems. As suggestion, this study can provide insights and solutions for related issues faced by the Department of Wildlife and National Malaysia. For example, it can aid in the restoration of large carnivores to support struggling ecosystems in Malaysia (Ripple, 2014). Furthermore, this study will contribute to the verification of the theory of exclusion and its applicability in various competitive scenarios. It will help determine whether both predator species, lions, and leopards, can coexist while undergoing competition. Additionally, the study will demonstrate the practical application of numerical approximation as a predictive tool, enabling researchers to determine the outcome of interspecific competition without the need for extended experimental periods. Moreover, these findings will enhance our understanding and provide illustrations of how numerical approximation can be utilized to predict competition outcomes, aiding researchers in determining which species may prevail in different interspecific competition scenarios.