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title: "A Comparative Study of Backward Euler and Adams-Moulton Methods for Lotka-Volterra Prey-Predator Model"
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Journal of Computing Research and Innovation 10(1) 2025
Journal of ResearchComputingand Innovation
# A Comparative Study of Backward Euler and AdamsMoulton Methods for Lotka-Volterra Prey-Predator Model
Nadia Sajidah Mohd Sobree!, Nurizatul Syarfinas Ahmad Bakhtiar?”, Nur Fatihah Fauzi*, Nur Izzati Khairudin*, Huda Zuhrah Ab. Halim*
12343College ofComputing, Informatics and Mathematics,
Universiti Teknologi MARA (UiTM) Perlis Branch, Arau Campus, 02600 Arau, Perlis, Malaysia
ARTICLE INFO ABSTRACT Article history: This research investigates the comparison between the backward Euler Received 24 October 2024 and Adams-Moulton methods in solving the Lotka-Volterra prey Revised 29 December 2024 predator model, specifically analyzing the interaction between wolf and Accepted 10 January 2025 moose populations. The study aims to identify which numerical method Publishednine first1 March 2025 provides a more accurateF approximation: of the model’s solutions. Data collected from the interactions between wolves and moose on Isle Royale from 1959 to 2019 was used, determining a carrying capacity of Keywords: 21.24 for wolves and 948.15 for moose. When the initial population is Lotka-Volterra prey predator model below this carrying capacity, it tends to grow due to the availability of Adams Moulton Method adequate resources. The comparison results revealed that the AdamsCarine Cat acity eto Moulton method provided the most accurate approximation, Stability successfully achieving the primary objective of the research. The equilibrium and stability of the system were assessed by defining its DOL dynamics through mathematical equations and evaluating the 10.24191 /jerinn.v10i1.503 eigenvalues of the Jacobian matrix, resulting in a growth rate value of 0.5017. The system was found to be stable when the populations of moose and wolves oscillated with consistent amplitude, influenced by the growth rate. The findings emphasize the importance of carrying capacity and initial conditions in understanding equilibrium and stability in prey-predator interactions, contributing to population dynamics. This research aids in the development of effective conservation and management strategies for maintaining ecosystem balance.
### 1.
## INTRODUCTION
The Lotka-Volterra model, established in the 1920s, describes the dynamics of prey predator interactions through differential equations (Anisiu & Academy, 2014). This research aims to compare the backward Euler and Adams-Moulton numerical methods in modelling these interactions, specifically focusing on the cyclical behavior, equilibrium, and stability of the prey predator relationship. The primary objectives are to investigate the interactions using these methods and to determine which method provides more accurate and efficient solutions. The novelty of this research lies in the comparative analysis of these
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two implicit numerical methods, providing insights into their effectiveness in ecological modelling, which can aid in developing better conservation strategies.
Zayernouri and Matzavinos (2016) highlighted that the Adams-Moulton method, an implicit technique used to solve ordinary differential equations, is particularly effective for modeling dynamic systems like the Keller-Segel chemotaxis system. They noted that this method produces more precise outcomes and is more efficient in terms of computational resources—such as time and memory—compared to conventional methods when applied to systems described by fractional differential equations.
The Backward Euler method is a numerical technique used to solve ordinary differential equations (ODEs) by using an initial value to predict the system's future state. This method is particularly useful for analyzing how systems change over time by focusing on the current rate of change. It also one of several methods available for solving differential equations.
The combination of these methods allows for a comprehensive analysis of the dynamics of predatorprey interactions, enabling the evaluation of equilibrium points and stability, which are crucial for understanding population dynamics and developing effective conservation strategies.
# 2. LITERATURE
Files should be in MS Word format only and should be formatted for direct printing. Figures and tables should be embedded and not supplied separately. Numerical approximation methods are commonly used to solve the Lotka-Volterra equations for studying prey-predator dynamics (Laham et al., 2012; Elsadany & Matouk, 2014). Paul et al. (2016) compared the Runge-Kutta-Fehlberg (RKF) method and the Laplace Adomian Decomposition method (LADM) by plotting their numerical solutions, which highlighted differences and accuracies in approximating ecological behaviors. Similarly, Manaf et al. (2023) and Rahaman et al. (2024) investigated the effectiveness of the Euler method, Taylor Series method and RKF method for the Lotka-Volterra competitive model, using graphical representations to show that RKF provided more accurate results compared to the Taylor Series method. Both studies utilized visual comparisons to assess the performance of different numerical methods in approximating solutions to the Lotka-Volterra model, enhancing the understanding of population dynamics in ecological systems.
The Lotka-Volterra competition model describes the dynamics of two species competing for limited resources, characterized by oscillations in their populations. This model provides insights into interspecific and intraspecific competition within ecological communities, influenced by factors like competition coefficients and carrying capacity (Razali & Abdullah, 2013). Numerical methods such as the Taylor Series and Runge-Kutta methods are employed to approximate solutions to the differential equations governing these population dynamics. The Taylor Series method uses derivatives to estimate function values iteratively, while the Runge-Kutta method adjusts step sizes based on truncation errors to provide accurate approximations without requiring higher-order derivatives.
The logistic equation models carrying capacity explained by Al-Mogbali et al. (2018) stated that the sigmoidal growth of populations influenced by environmental changes. Their study examines models with variable carrying capacity and Holling type I and II functional responses, revealing that variable carrying capacity significantly affects prey-predator dynamics, leading to damped oscillations and stable equilibria where both populations coexist. The article highlights the importance of incorporating variable carrying capacity in ecological models to understand population dynamics better.
Vaidyanathan (2015) explores the Lotka-Volterra model with negative feedback, noting that in the absence of predators, prey populations grow to a stable carrying capacity. Conversely, without prey, predators decline and face extinction. However, both populations can coexist stably if certain conditions are met, specifically if the prey’s growth rate exceeds the predator's death rate and the predator’s growth rate surpasses the prey’s death rate. Other than that, prey refuges are also important for system stability. The research indicates that under certain conditions, the system can exhibit globally asymptotic stability at ‘tps. org/10.2419 1 jrinn v10i1.503
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specific equilibrium points, particularly when the intrinsic growth rate of prey is low and the prey refuge is sufficiently high (Majeed & Ghafel, 2022). This suggests that prey fear can enhance stability at positive equilibrium points, leading to a more resilient ecosystem.
# 3. METHODOLOGY
The research used data from Vucetich and Peterson (2011) to study the interactions between moose and wolves on Isle Royale, Lake Superior, from 1959 to 2019. It focuses on how wolves, as predators, might affect moose populations and their distribution. The initial populations were 788 moose and 50 wolves. Population growth was tracked through annual counts of living moose and wolves in the study area during this period.
The Lotka-Volterra model examines prey-predator dynamics, proposing that both species can achieve a dynamic equilibrium with their populations fluctuating in response to each other. The model includes two key equations: one for prey growth and predation, and one for predator growth and mortality. Parameters in these equations represent the prey’s growth rate, the predation rate, the predator’s death rate, and the growth rate of predators due to consuming prey.
The equations are generally expressed as follows:
Prey Equation:
Predator Equation:
where:
x represents the prey population,
y represents the predator population,
t is time,
a, B,y, and 6 are rates that determine the interactions between the prey and predator.
The equation’s terms can be interpreted as follows:
a@ represents the prey’s growth rate,
B represents the rate that predators capture and consume prey,
y represents the predator’s death rate,
6 represents the rate that predators grow by consuming prey.
The backward Euler and Adams-Moulton methods will be used to simulate prey-predator interactions, and these results will be compared to simulation data. Using these numerical methods helps systematically analyze and understand the dynamics of prey and predator populations over time.
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# 3.1 Exact Solution
The logistic model describes population growth through a logistic equation, while the exponential growth function is expressed as follows:
The answer for the initial condition can be derived by solving equation (3), as follows:
The best fit can be achieved by using curve fitting with the parameters r and k, which ensures the logistic equation closely matches the observed data for both wolves and moose. The results are shown in Fig. 1 and Fig. 2.
The best fit for wolve is r = 0.9 and k = 21.24: The best fit for moose is r = 0.7 and k = 948.15;
population
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Fig. 1. Curve fitting for growth of wolve (left) and moose (right)
Based on observations of Fig. 1 and Fig. 2 above, wolves produce a carrying capacity of 21.24, while moose produce a carrying capacity of 948.15. From the best fit curve fitting, the logistic equation was utilized to model the growth of the wolf and moose populations.
For wolve:
with yo = 20, where y is the mean density at time t (in years), then, the logistic equation as;
For moose:
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with yo = 538, where y is the mean density at time t (in years), then, the logistic equation as;
# 3.2. Backward Euler Method
The Backward Euler formula is expressed as Yn41 = Yn + Af (%n+1,Yn41)- For the initial values, x) = 0 and the initial population of wolf is yp = 20 with a step size h = 1, the general equation for the Backward Euler method is:
The same calculation will be continued for another species which is moose with initial value is y) = 538.
# 3.3, Adams-Moulton Method
The second-order Adams-Moulton method is well-suited for solving stiff differential equations.
The calculation as below; To find y, : Y= Yo K, = f0¥o) Y= Yo + hf, Ky = f 2) h h WF Yo tak + 7k To find y2 : K, = f0¥o) K, = fa) h bnew = 2 +5 (BK2 — Ki) Ks = f 2, Drew) h h AMnew = Yo = Vi + 7K + 7K
Since the problem in this case takes a long time to solve using a precise analytical solution, Wolfram Mathematica 13.2 software is used to solve it numerically.
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# 4. RESULTS AND DISCUSSIONS
This research aims to compare the Lotka-Volterra prey-predator interactions using the backward Euler and Adams-Moulton methods, focusing on the cyclical dynamics of two species over time and analyzing the equilibrium and stability of their relationship based on initial population values. This analysis is intended to enhance understanding of population dynamics and the effects of predation within ecological systems.
Parameter estimation for the Logistic model is done using curve fitting techniques to find the growth rate r and carrying capacity k that best match observed population data. This is achieved by minimizing the difference between observed data and model predictions. For the wolf population, the best fit values were r = 0.9 and k = 21.24, using the logistic equation:
where yp is the initial population size. The report doesn't mention the r? value, which measures how well the model fits the data, but it may be found in the curve fitting analysis sections.
The logistic model is used in this study to understand how wolf and moose populations grow over time, considering the environment's carrying capacity. It helps predict long-term population behavior and interactions between these species. For wolves, the growth rate is r = 0.9 and the carrying capacity is k = 21.24, while for moose, r = 0.7 and k = 948.15. This model is important for predicting future population sizes, assessing stability, and informing wildlife management and conservation efforts.
Table 1. Exact solution and the numerical solution for wolve
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Table 2. Exact solution and the numerical solution for moose
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Table 1 and Table 2 shows the comparison between the backward Euler and Adams-Moulton methods and the exact solution for wolves. Graphs were plotted from the table to make the comparison clearer.
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Fig. 2. Comparison between numerical methods and exact solution of wolve (left) and moose (right)
Fig. 3 and Fig. 4 presents a comparative analysis of the numerical methods used to approximate the moose and wolf population dynamics in the Lotka-Volterra model, respectively, showcasing the results from both the backward Euler and Adams-Moulton methods alongside the exact solution. This graph likely includes numerical values that highlight the accuracy of each method in predicting population changes over time. The Adams-Moulton method is deemed the best method in this research due to its superior accuracy, as it consistently yields results that are closer to the exact solution compared to the backward Euler method. Additionally, the Adams-Moulton method is recognized for its stability, particularly in handling nonlinear problems like those presented in the Lotka-Volterra equations, which is essential for maintaining realistic population values in oscillatory dynamics. Furthermore, it may also offer greater computational efficiency, requiring fewer steps to achieve a similar level of accuracy. Collectively, these advantages underscore the effectiveness of the Adams-Moulton method in modelling the interactions between wolves and moose, making it the preferred choice for this ecological study.
Understanding prey-predator interactions is important for ecological modeling and management. This research focuses on the dynamics between moose and wolves, aiming to compare the accuracy of two numerical methods: backward Euler and Adams-Moulton. To evaluate these methods, we not only compare their numerical results but also analyze the system's equilibrium points and stability. Stability analysis helps predict whether populations will stay constant or return to equilibrium after a disturbance. Using the Jacobian method, we assessed the stability of equilibrium points by checking the eigenvalues, which indicated whether the populations would return to equilibrium or move away from it.
At the critical point (0,0), the Jacobian matrix has eigenvalues r = a and r = —y , meaning this point is a saddle point, with one positive and one negative eigenvalue. At the critical point ( §). the eigenvalues are purely imaginary, given by r = +ay i, which indicates oscillatory behavior around this equilibrium point. Table 3 below displays the results of the Jacobian method using Wolfram Mathematica 13.2 software.
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Table 3. Numerical solution for wolve and moose
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Fig. 3. Phase diagram
The three phase diagrams illustrate the predator-prey interactions under different values of the parameter r, with the x-axis representing the prey population and the y-axis representing the predator population. The actual value r = 0.5017 was calculated, while r = 0.01 and r = 0.9 were assumed for comparison to understand how the system's behavior and stability change under different conditions. Based on the Fig. 3, where r = 0.01, the system shows unstable or non-cyclic behavior. The prey population remains relatively steady while the predator population exhibits significant fluctuations. This suggests that the system is unstable and sensitive to changes in initial conditions, leading to divergent behavior without stable oscillations.
In contrast, the actual value of r = 0.5017, shows neutral stability. The closed loops in this phase diagram represent limit cycles, indicating that the populations of prey and predators undergo regular, sustained oscillations. This behavior is characteristic of a stable predator-prey interaction where neither population grows uncontrollably or faces extinction.
The last diagram, with r = 0.9, also shows limit cycles, but the oscillations are tighter and more controlled compared to r = 0.5017. This suggests a more stable interaction between the prey and predator populations, with the system exhibiting regular, balanced cycles. The populations in this scenario remain closer to equilibrium, indicating a stronger level of stability within the system.
In summary, as 7 increases, the system moves from instability at r = 0.01 to neutral stability with sustained oscillations at r = 0.5017 and r = 0.9, with tighter and more controlled cycles as r approaches 0.9. The stability analysis using the Jacobian matrix revealed that the system remained stable, with consistent oscillations in population sizes when applying the Adams-Moulton method, with actual value of r = 0.5017. This indicates its superiority in maintaining equilibrium.
5. CONCLUSION AND RECOMMENDATIONS
This research compares the numerical solutions of Lotka-Volterra prey-predator interactions using the backward Euler and Adams-Moulton methods, focusing on wolves and moose on Isle Royale from 1959 to 2019. The study reveals that the carrying capacity for wolves is about 21.24 and for moose is 948.15. When initial populations are below these carrying capacities, they grow until they reach these limits, given ample resources. The Adams-Moulton method provided the best approximation compared to the backward Euler method. The research also analyzed the equilibrium and stability of the system using the Jacobian matrix and eigenvalues, showing that the system's stability depends on the parameters' influence.
The findings highlight the importance of carrying capacity and initial conditions for understanding prey-predator dynamics, contributing valuable insights for developing conservation and management strategies to maintain ecosystem balance. For future research, it's advisable to extend data collection beyond
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2019 to better understand recent changes in wolf and moose populations on Isle Royale, offering insights into long-term ecological impacts and prey-predator dynamics. Including interactions with other species could provide a fuller picture of ecological balance and species interconnections.
Additionally, studying environmental factors like climate change and habitat alterations would improve understanding of their effects on population dynamics, leading to more accurate models. Utilizing advanced numerical methods and machine learning could enhance model precision, while incorporating spatial and temporal dynamics would offer a clearer view of species distribution and movement. Applying these findings to conservation and wildlife management could guide effective policy-making and maintain ecosystem balance.
# 6. ACKNOWLEDGEMENT
The authors express their gratitude to the reviewers for their valuable contributions in enhancing the quality of the research. They also thank the Symposium on Computer Sciences and Mathematics 2024 (SCOMATICS 2024) and Journal of Computing Research and Innovation (JCRINN) for allowing this work to be published.
# 7. CONFLICT OF INTEREST STATEMENT
The authors agree that this research was conducted in the absence of any self-benefits, commercial or financial conflicts.
## 8. AUTHORS’ CONTRIBUTIONS
Nurizatul Syarfinas Ahmad Bakhtiar, Nur Nadia Sajidah and Nur Fatihah Fauzi devised the research concept and designed the technique. Nur Izzati Khairudin and Huda Zuhrah Ab. Halim collected and assessed the data, respectively. The results were analyzed, and the manuscript written by Nurizatul Syarfinas Ahmad Bakhtiar and Nur Nadia Sajidah. The final version was reviewed and authorised for submission by all authors.
## 9. REFERENCES
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- Anisiu, M.-C., & Academy, R. (2014). Lotka, Volterra and their model, 32 (01). https://tinyurl.com/y63wbb75
- Elsadany, A. A. & Matouk, A. E. (2014). Dynamical behaviors of fractional-order Lotka—Volterra predator-prey model and its discretization. Journal ofApplied Mathematics and Computing, 49(1-2), 269-283. https://tinyurl.com/Sacmwthy
- Laham, M. F., Krishnarajah, L., & Jumaat, A. K. (2012). A numerical study on predator prey model. In International Journal of Modern Physics: Conference Series (Vol. 9, pp. 347-353). World Scientific Publishing Company. https://doi.org/10.1142/S2010194512005417
Majeed,of, S. J., &adultGhafel, S.predator.F. (2022). StabilityIraqianalysis ofJournalaprey-predatorof modelScience, with prey refuge43744387. and fear https://www.iasj.net/iasj/download/2868ff3c9c60b840
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- Manaf, M. N. A., Fauzi, N. F., Bakhtiar, N. S. A., Khairudin, N. I., & Halim, H. Z. A. (2023). Comparative analysis of Taylor Series and Runge-Kutta Fehlberg methods in solving the Lotka-Volterra competitive model. Applied Mathematics and Computational Intelligence (AMCI), 12(3), 91-103. https://doi.org/10.58915/amcei.v12i3.323
- Paul, S., Mondal, S. P., & Bhattacharya, P. (2016). Numerical solution of Lotka Volterra prey predator model by using Runge-Kutta-Fehlberg method and Laplace Adomian decomposition method. Alexandria Engineering Journal, 55(1), 613-617. https://doi.org/10.1016/j.aej.2015.12.026
- Rahaman, N. H. A., Bakhtiar, N. S. A., Hajimia, H., Fauzi, N. F., & Khairudin, N. I. (2024) Comparative analysis of Euler and Runge-Kutta Fehlberg methods in solving the Lotka-Volterra competitive model. Mathematical Scieneces and Informatic Journals (MIJ), — 5(2), 95-104. https://ir.uitm.edu.my/id/eprint/106661/1/106661.pdf
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- © 2025 by the authors. Submitted for possible open access publication under the terms and
- SY conditions of the Creative Commons Attribution (CC BY) _ license (http://creativecommons.org/licenses/by/4.0/).
# About the Authors
Nadia Sajidah Mohd Sobree with Bachelor of Science (Hons.) Management Mathematics from College of Computing, Informatics and Mathematics, Universiti Teknologi MARA Perlis. She can be reached through her email at ndiasjidh5@gmail.com
Nurizatul Syarfinas Ahmad Bakhtiar obtained her first degree in Pure Mathematics from the University Sains Malaysia, Malaysia in 2011. She received a master’s and PhD degree in Mathematical Modeling from the same university in 2012 and 2020 respectively. Currently, she is a senior lecturer at the Universiti Teknologi MARA Perlis. Her research interests include dissipative solitons and dynamical systems and analysis. She can be reached through her email at nurizatul@uitm.edu.my
Nur Fatihah Fauzi obtained her first degree in Pure Mathematics from the University Sains Malaysia, Malaysia in 2011. She received a master’s and PhD degree in Mathematical Modeling from the same university in 2012 and 2018 respectively. Currently, she is a senior lecturer at the Universiti Teknologi
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MARA Perlis. Her research interests include boundary layer flow and heat transfer. She can be reached through her email at fatihah@uitm.edu.my
Nur Izzati Khairudin obtained her first degree in Pure Mathematics from the University Sains Malaysia, Malaysia in 2011. She received a master’s and PhD degree in Mathematical Modeling from the same university in 2012 and 2017 respectively. Currently, she is a senior lecturer at the Universiti Teknologi MARA Perlis. Her research interests include dissipative solitons and dynamical systems and analysis. She can be reached through her email at zat.khairudin@uitm.edu.my
Huda Zuhrah Ab Halim is currently a senior lecturer in Universiti Teknologi MARA, Perlis. Her PhD degree is in the field of Operations Research. Her research interest includes Metaheuristic, Network Routing, Mathematical Programming and Mathematical Modelling. She can be reached through her email at hudazuhrah@uitm.edu.my
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