EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 7020 ISSN 1307-5543 – ejpam.com Published by New York Business Global A Computational Guide to Stability Analysis of Nonlinear System: The Lotka-Volterra and SIR Models as Case Studies Suhaila Saidat1, Yanal Al-Shorman 2,∗, Ishak Hashim2,3, Obadah Said Solaiman2, Ahmad Al-Hammouri1, Mohammad Almousa1, Abdullah Alsoboh 4 1 Department of Mathematics, Faculty of Science and Technology, Irbid National University, Irbid, Jordan 2 Department of Mathematical Sciences, Faculty of Science & Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia 3 Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman, P.O. Box 346, United Arab Emirates 4 Department of Basic and Applied Sciences, College of Applied and Health Sciences, A’Sharqiyah University, Post Box No. 42, Post Code No. 400, Ibra, Sultanate of Oman Abstract. This paper provides a structured and reproducible guide for analyzing the local stability of nonlinear dynamical systems by systematically combining analytical linearization with computational phase portrait visualization. Although these techniques are standard, in- troductory materials often lack a unified code-based workflow that connects abstract theory to practical, visual interpretation. This guide bridges that gap using two canonical models from different scientific domains: the Lotka-Volterra predator-prey model and the Susceptible Infected Recovered (SIR) epidemic model. For each system, we identify equilibrium points, compute the Jacobian matrix, and use eigenvalue analysis to determine local stability, and perform a sensitiv- ity analysis to explore how dynamics are affected by key parameters. By comparing ecological and epidemiological models, we highlight how shared mathematical principles lead to distinct real-world dynamics, such as persistent oscillations versus threshold-based outbreaks. Python- generated phase portraits are used throughout to visually validate the analytical results, offering an intuitive complement to the theory. This work serves as a practical toolkit for students and researchers new to nonlinear modeling, emphasizing a clear step-by-step process essential for both educational settings and applied research. 2020 Mathematics Subject Classifications: 34D20, 92D25 Key Words and Phrases: Nonlinear dynamics, stability analysis, linearization, phase portrait, Lotka-Volterra model, SIR model ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.7020 Email addresses: P114391@siswa.ukm.edu.my (Y. Al-Shorman) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) S. Saidat et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7020 2 of 16 1. Introduction In many areas of science and engineering, from population dynamics to control theory, understanding the stability of nonlinear systems is essential [1, 2]. Analyzing how a system behaves near its equilibrium points provides invaluable insight into its overall structure and long-term evolution [3, 4]. The classical tools for this task are linearization and phase portrait analysis. The theoretical foundations for this approach were laid out in inspiring texts that remain fundamental to the field today [5]. Linearization provides a precise criterion for local stability by approximating a nonlinear system near an equilibrium, while phase portraits offer an intuitive geometric understanding of the dynamic of the system [6, 7]. These foundational methods continue to be the bedrock of modern mathematical modeling. The Lotka-Volterra model, for example, is still actively used and extended to explore complex ecological interactions such as [8] and [9]. Similarly, the SIR model forms the basis for a large body of epidemiological research, providing critical insights into the spread of infectious diseases from influenza to recent pandemics [10] and [11]. Alongside this research, there is a parallel conversation on the most effective ways to teach these concepts. Many educators have highlighted the power of computational tools such as Python and MATLAB to help students visualize abstract concepts [12, 13]. How- ever, while numerous resources focus on either the deep of theory [14–16] or advanced research applications [17], there remains a need for accessible, step by step guides that explicitly connect analytical calculations of stability with their computational visual- ization. This is particularly true for cross-disciplinary leaners who need to apply these methods without necessarily having a deep mathematical background. Linearization techniques can be used to approximate a linear system that closely resembles a non- linear system at its equilibrium points [18]. Standard linear algebraic methods can be employed to analyze the Jacobian matrix in these places to determine the nature and stability of the equilibria [19, 20]. The eigenvalues of the Jacobian indicate whether an equilibrium point is a saddle, node, or spiral, as well as its stability or instability [20]. Recent applications of these methods can be found in [21–37]. To enhance the analysis, we incorporate a parameter sensitivity study, in particular, the predation rate parameter b, to investigate how changes in interaction strength affect equilibrium configurations and stability of the system. Beyond predator-prey dynam- ics, nonlinear modeling is also fundamental in epidemiology. The Susceptible Infected Recovered model is a cornerstone of mathematical epidemiology, describing the spread of infectious diseases through compartments that represent different health states of the population [11]. The SIR model, like the Lotka-Volterra system, is nonlinear and admits equilibrium points whose stability can be analyzed using Jacobian linearization [38]. In this work, we include a comparative analysis between the Lotka-Volterra and SIR mod- els, highlighting how similar mathematical tools, such as local linearization, eigenvalue classification, and phase portraits, can be applied to problems from different domains. This cross-domain perspective enriches the learning process for students and emphasizes S. Saidat et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7020 3 of 16 the broad applicability of nonlinear stability analysis in applied mathematics. This pa- per aims to bridge the pedagogical gap by proposed a clear, step by step computational workflow that unifies these techniques. We demonstrate this integrated approach us- ing two celebrated models: the Lotka-Volterra predator-prey model and the SIR model, which are nonlinear systems widely used in ecology and theoretical biology. The goal is not to introduce new mathematical theory but rather to provide an accessible and introducible guide that empowers learners and practitioners to confidently apply stabil- ity analysis to nonlinear models. By placing these two distinct models’ side by side, we emphasize the universal power of these mathematical tools and explore the rich domain- specific insights they can reveal. The remainder of this paper is structured as follows. Section 2 details the mathematical methodology. Section 3 applies this methodology to the Lotka-Volterra model, including a parameter sensitivity analysis. In Section 4, we present a complete analysis of the SIR model, setting the stage for a direct comparison. Section 5 discussed the visual results of the phase portraits for both systems. Finally, Section 6 concludes with a summary of the findings and their implications. 2. Methods With our main goal set, this section is all about building our toolkit. To make sure our approach is perfectly clear, we have broken it down into two parts. First, in Section 2.1, we will walk gently through the essential mathematical theory behind stability analysis. Then, in Section 2.2, we will lay out the concrete step by step workflow we will follow for our case studies, showing exactly how we will put that theory into practice. 2.1. Mathematical Foundations of Stability Analysis We consider a general n-dimensional autonomous system of first-order ordinary dif- ferential equations (ODEs): dx dt = f(x). (1) Here, x(t) ∈ Rn and f : Rn → Rn is a continuously differentiable function. An equilib- rium point, denoted by x∗, is a point where the dynamics of the system stops, i.e., it satisfies f(x∗) = 0. To study the local stability near an equilibrium, we linearize the system by approxi- mating f(x) with its first-order Taylor expansion around x∗. The result is a linear system governed by the Jacobian matrix J , which consists of all first-order partial derivatives of f : J(x) =  ∂f1 ∂x1 ∂f1 ∂x2 · · · ∂f1 ∂xn ∂f2 ∂x1 ∂f2 ∂x2 · · · ∂f2 ∂xn ... ... . . . ... ∂fn ∂x1 ∂fn ∂x2 · · · ∂fn ∂xn  . (2) S. Saidat et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7020 4 of 16 The linearized system that describes the evolution of a small perturbation u = x− x∗ is given by du dt = J(x∗)u (3) The stability of the equilibrium point x∗ is determined by the eigenvalues λ of the Jacobian matrix evaluated at the point J(x∗): J(x∗)v = λv, where v is the corresponding eigenvector. • If all eigenvalues have negative real parts, then x∗ is locally asymptotically stable. • If at least one eigenvalue has a positive real part, then x∗ is unstable. • If all eigenvalues have non-positive real parts with at least one having a zero real part, the linear analysis is inconclusive, and higher-order terms are needed. For two-dimensional systems, the trace tr(J) and the determinant det(J) of the Jacobian matrix can be used to classify the type of equilibrium (e.g. node, saddle, spiral, or center). 2.2. Applied Workflow In this study, we apply this mathematical framework to our case studies using the following structured workflow: (i) Identify Equilibrium Points: Analytically solve f(x) = 0 to find all equilibria. (ii) Perform Linearization: Compute the Jacobian matrix J(x) and evaluate it at each equilibrium point x∗. (iii) Classify Stability: Calculate the eigenvalues of each J(x∗) to determine the local stability and type of each equilibrium. (iv) Visualize Dynamics: Generate phase portraits using Python to visually confirm the analytical predictions. (v) Conduct Sensitivity Analysis: For the Lotka–Volterra model, investigate how varying a key parameter affects the system’s dynamics. We identify the equilibrium points, compute the Jacobian matrix at each point, and perform an eigenvalue analysis to classify their stability. Phase portraits are generated to visualize the local behavior near each equilibrium, confirming the analytical predictions. S. Saidat et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7020 5 of 16 3. Case Study: Lotka-Volterra Predator-Prey Model Our first case study is the classical Lotka–Volterra model, which describes the dy- namic relationship between a predator and its prey. For this model to be biologically meaningful, we assume that all parameters (a, b, c, d), representing birth, death, and interaction rates, are positive real constants: dx dt = ax− bxy, dy dt = −cy + dxy, (4) where x(t) and y(t) represent the prey and predator populations, respectively, and a, b, c, d > 0 are constant parameters. 3.1. Equilibrium Points The first step is to find the equilibrium points of the system by setting both deriva- tives to zero.  ax− bxy = 0, −cy + dxy = 0. Factoring both equations, we obtain the following: x(a− by) = 0, y(−c + dx) = 0. This yields two biologically relevant equilibrium points: • Trivial Equilibrium: E1 = (0, 0). This represents the complete absence of both prey and predators. • Co-existence Equilibrium: E2 = ( c d , a b ) . This represents a state where both populations co-exist with stable average populations. 3.2. Stability Analysis With the equilibria identified, our next step is to use linearization to understand the stability of each point. We first compute the general Jacobian matrix for the system: J(x, y) = [ a− by −bx dy −c + dx ] . We now evaluate this Jacobian at each equilibrium point: S. Saidat et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7020 6 of 16 (i) Analysis in the Trivial Equilibrium E1 = (0, 0). Evaluating the Jacobian at (0, 0) gives: J(0, 0) = [ a 0 0 −c ] . The eigenvalues are the diagonal entries: λ1 = a and λ2 = −c. Since a > 0 and −c < 0, the eigenvalues are real and have opposite signs. Thus, the origin is an unstable saddle point. Trajectories near the origin will be pushed away, indi- cating that if even a few prey and predators are present, the populations will not die out. (ii) Analysis in the Co-existence Equilibrium E2 = ( c d , a b ) . Evaluating the Jacobian at this point gives: J ( c d , a b ) = [ 0 − bc d ad b 0 ] . To find the eigenvalues, we solve the characteristic equation det(J − λI) = 0 ⇒ λ2 + ac = 0. This yields the purely imaginary eigenvalues λ = ±i √ ac. Purely imaginary eigenvalues correspond to a center. Thus, the analytical result predicts that, in the vicinity of the co-existence point, the predator and prey populations will oscillate in endless, stable cycles. 3.3. Parameter Sensitivity Analysis A powerful feature of a good model is understanding how its predictions change when its parameters are tweaked. This is known as a sensitivity analysis. In our Lotka– Volterra model, a particularly interesting parameter is b, which represents the rate at which prey is consumed by predators, or the impact of each predation event. What happens to the system if this rate changes? Let us consider the co-existence equilibrium E2 = ( c d , a b ) . We can immediately see that the equilibrium predator population, y∗ = a b , depends directly on b. If the predation rate (b) increases, the steady-state predator pop- ulation (y∗) required to maintain equilibrium decreases. This reflects a more “efficient” system where fewer predators are needed to control the prey. But does this change the stability of the system? We find that the eigenvalues in this equilibrium are λ = ±i √ ac. S. Saidat et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7020 7 of 16 Notice that the parameter b does not appear in this expression at all. This leads to a crucial insight: while the predation rate b changes the equilibrium level of the predator population, it does not change the fundamental nature of the co- existence equilibrium. The equilibrium remains a center, and therefore no bifurcation occurs with respect to this parameter. The tendency of the system to oscillate is a robust feature. This will be clearly illustrated in the phase portraits presented in Section 5. To demonstrate this effect, we simulate the system for three representative values of the predation rate, b ∈ {0.2, 0.5, 0.8}, while keeping the other parameters fixed. These simulations will show how changing b alters the shape of the orbital cycles but does not break their closed-loop structure. 4. Comparative Case Study: Susceptible-Infected-Recovered Epidemic Model In this section, we extend our mathematical framework to analyze the classical Susceptible-Infected-Recovered (SIR) epidemic model. This model captures the spread of infectious diseases within a closed population by dividing the population into three compartments: S(t): number of susceptible individuals, I(t): number of infected individuals, and R(t): number of recovered (or removed) individuals. The dynamics of the model are governed by the following system of differential equa- tions:  dS dt = −βSI, dI dt = βSI − γI, dR dt = γI, where β > 0 is the transmission rate and γ > 0 is the recovery rate. We will identify the equilibrium points, perform a linear stability analysis using the Jacobian matrix, and illustrate the system behavior using phase portraits. Beyond its biological relevance, the SIR model serves as an effective pedagogical bridge between biological systems and general dynamical systems theory. By comparing this model with other biological and epidemiological frameworks, learners can better under- stand how concepts such as equilibrium, stability, and bifurcation transcend disciplinary S. Saidat et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7020 8 of 16 boundaries. This comparison reinforces the universality of mathematical tools used to describe population dynamics, reaction kinetics, and even ecological or biochemical in- teractions. In particular, analyzing the stability of disease-free and endemic equilibria provides a concrete, intuitive context through which students can grasp abstract notions of local and global stability, eigenvalue behavior, and nonlinear feedback mechanisms. Thus, the SIR model not only exemplifies epidemiological reasoning but also deepens the conceptual understanding of dynamical stability in life sciences and applied mathe- matics. 4.1. Equilibrium Points Setting dS dt = 0 and dI dt = 0, we find two biologically relevant equilibria: • Disease-Free Equilibrium (DFE): EDFE = (S, I) = (N, 0), which corresponds to a state without disease in the population . • Endemic Equilibrium (EE): EEE = (S∗, I∗) = ( γ β ,N − γ β ) . This equilibrium is only physically meaningful if S∗ < N , which requires γ β < N or equivalently βN γ > 1. This introduces the basic reproduction number R0 = βN γ , which is the key threshold parameter. The endemic equilibrium exists only if R0 > 1. 4.2. Stability Analysis The Jacobian matrix for the SIR system is: J(S, I) = [ −βI −βS βI βS − γ ] . Stability of the Disease-Free Equilibrium (DFE): Evaluating the Jacobian at EDFE = (N, 0) gives: J(N, 0) = 0 −βN 0 βN − γ  . S. Saidat et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7020 9 of 16 The eigenvalues are the diagonal entries: λ1 = 0, λ2 = βN − γ = γ(R0 − 1). The stability is determined by λ2: • If R0 < 1, then λ2 < 0 and the DFE is stable; the disease dies. • If R0 > 1, then λ2 > 0 and the DFE is unstable; an outbreak can occur. Stability of the endistic Equilibrium (EE, R0 > 1): Evaluating the Jacobian at EEE = ( γ β , N − γ β ) gives: J(EEE) =  −β ( N − γ β ) −γ β ( N − γ β ) 0  . The trace and determinant are: tr(J) = −β ( N − γ β ) = −βI∗ < 0, det(J) = γβI∗ > 0. Since tr(J) < 0 and det(J) > 0, both eigenvalues have negative real parts. Therefore, the endemic equilibrium is locally asymptotically stable whenever it exists. This means that if a disease becomes established (R0 > 1), the system will converge to a steady state where the disease persists in the population. 5. Results and Discussion: A Visual Comparison In this section, we will demonstrate how the analytical predication of our systems should behave, and show how this behavior looks like, thus, we will bring the theory to life by generating phase portraits for both of our models. We will first visualize each system on its own to confirm our findings, and then place them side by side to draw out the crucial insights that only a direct comparison can reveal. 5.1. Lotka-Volterra Dynamics: Persistent Oscillations The phase portrait for the Lotka–Volterra model visually confirms the analytical findings. The equilibrium at the origin (0, 0) is a saddle point, while the interior equilibrium ( c d , a b ) is a center, characterized by a family of closed periodic orbits. This neutrally stable structure reflects the endless cycle of predator-prey populations in this idealized model, as depicted in Figure 1. S. Saidat et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7020 10 of 16 Figure 1: Phase portraits of the Lotka-Volterra predator-prey system To further investigate the behavior of the system, we examined the sensitivity of the Lotka–Volterra model to variations in the interaction rate parameter b. This parameter directly influences the predator’s efficiency in capturing prey. We considered three rep- resentative values: b = 0.2, 0.5, and 0.8, while keeping a = 1, c = 1.5, and d = 0.075 fixed. The resulting trajectories reveal that while a higher interaction rate (b) alters the position and shape of the orbits, it does not break the underlying oscillatory dynamics. This sensitivity analysis is illustrated in Figure 2. Figure 2: Sensitivity analysis 5.2. Susceptible-Infected-Recovered Dynamics: Threshold Behavior The SIR epidemic model displays a fundamentally different behavior, governed by the threshold R0. When R0 < 1, all trajectories converge to the stable disease-free equilibrium (N, 0), indicating that the disease does not spread or die out, as shown in Figure 3. S. Saidat et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7020 11 of 16 Figure 3: Phase portrait of the SIR model for R0 < 1, showing convergence of all trajectories to the disease- free equilibrium (N, 0). In stark contrast, when R0 > 1, the DFE becomes unstable. As shown in the phase portrait, trajectories starting near the DFE are initially repelled, leading to an outbreak where the number of infected individuals increases. These trajectories do not grow indefinitely, instead, they spiral towards the stable endemic equilibrium, which was shown to be a stable node or spiral. This shows that the disease will persist in the population at a constant level. As shown in Figure 4. S. Saidat et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7020 12 of 16 Figure 4: SIR System with R0 > 1. 5.3. Comparative Analysis of Stability The side-by-side analysis reveals a crucial distinction rooted in the stability of the non-trivial equilibria. In the Lotka–Volterra model, the interior equilibrium is a neutrally stable center, leading to persistent oscillations whose amplitudes depend strongly on initial conditions. This cyclical dynamic reflects the endless predator–prey cycles of the idealized system. In contrast, the SIR epidemic model demonstrates a qualitatively different form of stability behavior. The Disease-Free Equilibrium (DFE) corresponds to the state (S, I,R) = (1, 0, 0), whose Jacobian matrix yields one zero eigenvalue λ1 = 0. The presence of a zero eigenvalue implies that the linear stability analysis is inconclusive in this direction. Formally, confirming the local stability of the DFE when R0 < 1 requires a nonlinear analysis, such as through the Center Manifold Theorem or a sufficiently constructed Lyapunov function. This observation underscores that, while linearization provides valuable insight, it may not fully capture the dynamical behavior of the system near critical thresholds. Moreover, when R0 crosses the critical value R0 = 1, the SIR model undergoes a tran- scritical bifurcation. At this point, the DFE loses stability and an endemic equilibrium emerges, exchanging stability properties with the DFE. This bifurcation marks the fun- damental epidemiological transition between disease elimination (R0 < 1) and sustained infection (R0 > 1). Thus, while both models share a common analytical framework, their stability properties highlight profoundly different biological realities—endless oscil- S. Saidat et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7020 13 of 16 lations in predator–prey dynamics versus a bifurcation-driven transition and asymptotic convergence in epidemic dynamics. 6. Conclusion This paper has presented a structured, computational guide to local stability analysis, a cornerstone of nonlinear dynamics. By combining Jacobian-based linearization with phase portrait visualization, we developed a cohesive workflow and applied it to two classical models: the Lotka–Volterra predator–prey system and the SIR epidemic model. Our analysis of the Lotka–Volterra system demonstrated the emergence of persistent predator–prey oscillations, corresponding to a neutrally stable center. In contrast, com- plete analysis of the SIR model revealed a critical threshold phenomenon governed by the basic reproduction number R0. When R0 < 1, the system converges to disease-free equilibrium, whereas for R0 > 1, trajectories approach a stable endemic equilibrium. The direct comparison between the two models proved particularly insightful, showing how subtle differences in nonlinear structure can lead to profoundly different long-term behaviors: persistent oscillations versus convergence to a steady state. The primary contribution of this work is pedagogical. It offers a clear and repro- ducible template for students, educators, and practitioners who are new to nonlinear modeling. By bridging abstract theory with computational practice, this guide fosters a deeper and more intuitive understanding of how mathematical tools can be used to explore complex systems in ecology, epidemiology, and beyond. Future work could extend this guide to include bifurcation analysis, systems ex- hibiting limit cycles, stochastic effects, or the dynamics of higher–dimensional models. Such extensions would further enrich the study of nonlinear systems and broaden the applicability of this framework to more realistic scenarios. References [1] Michael Ghil and Denise Sciamarella. Dynamical systems, algebraic topology and the climate sciences. Nonlinear Processes in Geophysics, 30(4):399–434, 2023. [2] Steven H. Strogatz. Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering. Chapman and Hall/CRC, 2024. [3] Miguel A. González-Casado, Antônio S. Teixeira, and Angel Sánchez. Evidence of equilibrium dynamics in human social networks evolving in time. Communications Physics, 8(1):227, 2025. [4] J. Huang et al. An overview of agent-based models for transport simulation and analysis. Journal of Advanced Transportation, 2022(1):1252534, 2022. [5] S. Nardean, M. Ferronato, and A. Abushaikha. Linear solvers for reservoir simula- tion problems: An overview and recent developments. Archives of Computational Methods in Engineering, 29(6):4341–4378, 2022. [6] C. Maczka, S. Skurativskyi, and V. Vladimirov. Study of linear and nonlinear models with Mathematica. World Scientific, 2022. S. Saidat et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7020 14 of 16 [7] H. Oersted and Y. Ma. Geometric and feedback linearization on UAV, 2023. arXiv preprint arXiv:2311.06784. [8] J. D. Davis et al. Methods of quantifying interactions among populations using Lotka–Volterra models. Frontiers in Systems Biology, 2:1021897, 2022. [9] M. Mahmoudi and Z. Eskandari. Fractional calculus in ecological systems: Bi- furcation analysis and continuation techniques for discrete Lotka–Volterra models. Mathematical Methods in the Applied Sciences, 2024. [10] J. Huang and J. S. Morris. Infectious disease modeling. Annual Review of Statistics and Its Application, 12, 2024. [11] S. Peddinti and Y. Sabbani. Mathematical modeling of infectious disease spread using differential equations and epidemiological insights. Journal of Nonlinear Anal- ysis and Optimization, 15:114–122, 2024. [12] D. Ahmed, F. Roshayanti, and M. Novita. Applying MATLAB to visualize col- lege physics concepts and enhance student understanding. International Journal of Innovative Science, Engineering and Technology, 11(12):52–60, 2024. [13] S. Niazai, A. A. Rahimzai, and H. Atifnigar. Applications of MATLAB in natural sciences: A comprehensive review. European Journal of Theoretical and Applied Sciences, 1(5):1006–1015, 2023. [14] A. Abbas. Mathematical modeling and qualitative analysis of dynamical systems. 2025. [15] D. Krawczyk, S. Kolomiiets, and Y. Chvankin. Nonlinear models of economic dy- namics as a tool for public health research. Health Economics and Management Review, 6(1):179–196, 2025. [16] B. Müller. Stability of linear dynamical systems and application to Lotka–Volterra equations and epidemiology, 2021. [17] R. Ganikhodzhaev et al. Some degenerate cases of discrete Lotka–Volterra dy- namical systems and their applications in epidemiology. Journal of Mathematical Sciences, pages 1–15, 2024. [18] H.-G. Lee. Linearization of nonlinear control systems. Springer, 2022. [19] J. Dagar. Applications of linear algebraic equations in engineering systems: A comprehensive study, 2024. [20] S. Saeedinia. Extending Jacobian matrix in proving stability for nonlinear sys- tems with one equilibrium point such as compressor, 2024. arXiv preprint arXiv:2410.22191. [21] Shadi Al-Ahmad, Mustafa Mamat, Nidal Anakira, and Rami Alahmad. Modified differential transformation method for solving classes of non-linear differential equa- tions. TWMS Journal of Applied and Engineering Mathematics, 2022. [22] Y. Al-Qudah, F. Al-Sharqi, M. Mishlish, and M. M. Rasheed. Hybrid integrated decision-making algorithm based on AO of possibility interval-valued neutrosophic soft settings. International Journal of Neutrosophic Science, 22(3):84–98, 2023. [23] Y. Al-Qudah, K. Alhazaymeh, N. Hassan, M. Almousa, and M. Alaroud. Transitive Closure of Vague Soft Set Relations and its Operators. International Journal of Fuzzy Logic and Intelligent Systems, 22(1):59–68, 2022. S. Saidat et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7020 15 of 16 [24] Y. Al-Shorman, A. Alsoboh, Obadah S. Solaiman, and I. Hashim. Derivative-Free Method for Nonlinear Systems with Some Real-life Applications. European Journal of Pure and Applied Mathematics, 18(3):6262, 2025. [25] A. Alsoboh, A. Amourah, M. Darus, and C. A. Rudder. Studying the Harmonic Functions Associated with Quantum Calculus. Mathematics, 11(10):2220, 2023. [26] Abdullah Alsoboh, Ala Amourah, Omar Alnajar, Mamoon Ahmed, and Tamer M Seoudy. Exploring q-Fibonacci Numbers in Geometric Function Theory: Univalence and Shell-like Starlike Curves. Mathematics, 13(8):1294, 2025. [27] Abdullah Alsoboh, Murat Çağlar, and Mucahit Buyankara. Fekete-Szegö Inequality for a Subclass of Bi-Univalent Functions Linked to q-Ultraspherical Polynomials. Contemporary Mathematics, pages 2531–2545, 2024. [28] Abdullah Alsoboh and Georgia Irina Oros. A class of bi-univalent functions in a leaf-like domain defined through subordination via q-calculus. Mathematics, 12(10):1594, 2024. [29] A. Amourah, A. Alsoboh, D. Breaz, and S. M. El-Deeb. A Bi-Starlike Class in a Leaf-like Domain Defined through Subordination via q-Calculus. Mathematics, 12(11):1735, 2024. [30] N Ratib Anakira, AK Alomari, and Ishak Hashim. Application of optimal homotopy asymptotic method for solving linear delay differential equations. In AIP Conference Proceedings, volume 1571, pages 1013–1019. American Institute of Physics, 2013. [31] A. A. Lupaş, A. S. Tayyah, and J. Sokó l. Sharp Bounds on Hankel Determinants for Starlike Functions Defined by Symmetry with Respect to Symmetric Domains. Symmetry, 17(8):1244, 2025. [32] H. Qoqazeh, Y. Al-Qudah, M. Almousa, and A. Jaradat. On D-Compact topological spaces. Journal of Applied Mathematics and Informatics, 39(5–6):883–894, 2021. [33] A. S. Tayyah, W. G. Atshan, and G. I. Oros. Third-Order Differential Subordination Results for Meromorphic Functions Associated with the Inverse of the Legendre Chi Function via the Mittag-Leffler Identity. Mathematics, 13(13):2089, 2025. [34] A. Al-Hammouri and A. Ashyralyev. A Note on the second order of accuracy differ- ence scheme for the elliptic-telegraph identification problem with dirichlet boundary condition. International Journal of Applied Mathematics, 38(1):1–13, 2025. [35] A. Ashyralyev and A. Al-Hammouri. A note on the elliptic-telegraph identification problem with non-local condition. AIP Conference Proceedings, 2334(1):060006, 2021. [36] A. Almousa, S. Saidat, A. Al-Hammouri, S. Alsaadi, and G. Banihani. Solutions of Nonlinear Integro-Differential Equations Using a Hybrid Adomian Decomposition Method with Modified Bernstein Polynomials. International Journal of Fuzzy Logic and Intelligent Systems, 24(3):271–279, 2024. [37] M. Almousa, A. Al-Hammouri, S. Saidat, S. Alsaadi, and G. Banihani. A hybrid Laplace transform-optimal homotopy asymptotic method (LT-OHAM) for solving integro-differential equations of the second kind. International Journal of Analysis and Applications, 22:17, 2024. [38] M. Ye et al. Applications of the Poincaré–Hopf theorem: Epidemic models and S. Saidat et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7020 16 of 16 Lotka–Volterra systems. IEEE Transactions on Automatic Control, 67(4):1609– 1624, 2021.