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Stability analysis of a fractional-order two-species facultative mutualism model with harvesting
- Nattakan Supajaidee^{1} and
- Sompop Moonchai^{2, 3}Email authorView ORCID ID profile
https://doi.org/10.1186/s13662-017-1430-9
© The Author(s) 2017
- Received: 10 August 2017
- Accepted: 20 November 2017
- Published: 28 November 2017
Abstract
We present a fractional-order model of two-species facultative mutualism with harvesting. We investigate the stability of the equilibrium points of the model by using the linearization method for noncoexistence of equilibrium points and the Lyapunov direct method for the positive coexistence of an equilibrium point. In addition, we obtain sufficient conditions to ensure the local asymptotic stability and global uniform asymptotic stability for the model. Finally, we provide illustrated numerical examples to verify the stability results obtained in this study.
Keywords
- fractional differential equations
- stability analysis
- mutualism model
- Caputo fractional derivative
- Lyapunov function
1 Introduction
A population model is a mathematical model used to describe population dynamics. Equations describing population models can take different forms such as difference equations [1, 2], differential equations [1, 2], or delay differential equations [3]. There are several types of population models such as interspecific competition models, predator-prey models, and facultative mutualism models [4].
This system is an illustration of the situation in which the effects of mutualism have the most impact when the recipient population is at high density (Wolin and Lawlor [13]). If one species is missing, then the dynamics of the other species is characterized by logistic growth [14].
In recent years, fractional calculus has become an intriguing field. There are several definitions of the fractional-order derivative. Among them, the definitions commonly used are the Riemann-Liouville definition, the Grünwald-Letnikov definition [15], and the Caputo definition [16]. The Caputo derivative is reformulated from the more classic Riemann-Liouville derivative, and the initial conditions for Caputo fractional differential equations are expressed in the same manner as for integer-order differential equations [15]. For this reason, in this paper, we use the Caputo derivative. This is likely to be a good choice to solve problems in the dynamics of complex systems because the order of derivatives can be any real or complex number. Especially, the results may more closely resemble realistic dynamics than integer-order systems [17]. The advantages of fractional calculus support its use in applied mathematics, physics, chemistry, engineering, and even finance and social sciences [18]. Some examples include data-fitting problems for blood alcohol level, video tape counter readings and models for world population growth [19], obesity epidemics [20], Ebola epidemics [21], and bovine babesiosis disease and tick populations [22].
Stability analysis is an important tool to understand system dynamics. Recently, there have been many studies concerning the stability and asymptotic behavior of models in the form of differential equations [23–26], delay differential equations [27–30], and fractional differential equations [31–35].
In the case of fractional differential equations, linearization and Lyapunov methods have been a popular technique to analyze the stability of a system without visibly solving the equations [36–38]. The linearization method is applied to approximate the nonlinear system by using the linearized model [36, 39]. On the other hand, the Lyapunov method is used to investigate the stability of an equilibrium point of a system based on a Lyapunov function. In recent years, many researchers proposed a Lyapunov direct method to analyze the stability of fractional differential equations [37, 40–44]. Moreover, some Lyapunov functions have been constructed to study the stability of fractional differential equations [45–48]. However, the construction of a Lyapunov function and calculation of the fractional derivatives is complicated [42, 44].
In this paper, we present a fractional model for two-species facultative mutualism. Moreover, we analyze the stability of the equilibrium points for this model to obtain sufficient conditions for the local asymptotic stability of the noncoexistence equilibrium points using the linearization method. Conversely, we obtain sufficient conditions for the global uniformly asymptotic stability of the coexistence equilibrium point using the Lyapunov method.
We organize the paper as follows. In Section 2, we give some necessary definitions and some known properties. In Section 3, we propose a fractional-order two-species facultative mutualism model. In Section 4, we investigate the local stability of the three noncoexistence equilibrium points and the global stability of the coexistence equilibrium point in the sense of Lyapunov for the proposed model. Some numerical simulations to validate the theoretical results are shown in Section 5. In the last section, we present the conclusions of this paper.
2 Preliminaries
In this section, we introduce some definitions of fractional calculus and several important theorems of stability analysis.
Definition 2.1
([15])
Remark 2.2
([15])
Under natural conditions on the function \(f(t)\), as \(\alpha \rightarrow n\), the Caputo derivative becomes the conventional nth derivative of the function \(f(t)\).
Remark 2.3
([45])
Theorem 2.4
([49])
Theorem 2.5
([50])
Theorem 2.6
([37])
Definition 2.7
([51])
- (i)
The trivial solution is said to be stable if for any \(t_{0} \in \mathbb{R}\) and any \(\varepsilon>0\), there exists \(\delta=\delta (t_{0},\varepsilon)>0\) such that \(\Vert x({{t}_{0}}) \Vert <\delta\) implies \(\Vert x(t) \Vert <\varepsilon\) for all \(t>t_{0}\).
- (ii)
The trivial solution is said to be asymptotically stable if it is stable and for any \(t_{0} \in\mathbb{R}\) and any \(\varepsilon>0\), there exists \(\delta_{a}=\delta_{a}(t_{0},\varepsilon)>0\) such that \(\Vert x({{t}_{0}}) \Vert <\delta_{a} \) implies \(\lim_{t\to\infty }\Vert x(t) \Vert =0\).
- (iii)
The trivial solution is said to be uniformly stable if it is stable and \(\delta=\delta(\varepsilon)>0\) can be chosen independently of \(t_{0}\).
- (iv)
The trivial solution is uniformly asymptotically stable if it is uniformly stable and there exists \(\delta_{a}>0\), independent of \(t_{0}\), such that if \(\Vert x({{t}_{0}}) \Vert <\delta_{a} \), then \(\lim_{t\to\infty} \Vert x(t) \Vert =0\).
- (v)
The trivial solution is globally (uniformly) asymptotically stable if it is (uniformly) asymptotically stable and \(\delta_{a}\) can be an arbitrary large finite number.
Theorem 2.8
([34])
The equilibrium points \(x^{*}\) of system (2.3) are locally asymptotically stable if all eigenvalues \(\lambda_{i}\) of the Jacobian matrix \(J=\frac{\partial f}{\partial x}\) evaluated at the equilibrium points satisfy \(\vert \arg({{\lambda}_{i}}) \vert >\frac{\alpha\pi}{2}\).
Theorem 2.9
(Uniform Asymptotic Stability Theorem [42])
Remark 2.10
([37])
If \(x=x^{*}\) is the equilibrium point of system (2.3) and satisfies the conditions of Theorem 2.9, then \(x=x^{*}\) is uniformly asymptotically stable.
Theorem 2.11
([45])
3 Model description
3.1 Existence and uniqueness
3.2 Nonnegative solution
Theorem 3.1
If \(x_{1}(t_{0}) \geq0\) and \(x_{2}(t_{0}) \geq0\), then there is a unique solution \(x(t)\) to the Caputo fractional-order model (3.3) on \(t\geq t_{0}\), and the solution remains in \(\mathbb{R}_{+}^{2}\).
Proof
In Section 3.1, the uniqueness of a solution of \(x(t)\) to system (3.3) is obtained. Thus, we only need to prove that the solution \(x(t)=(x_{1}(t),x_{2}(t))\) remains in \(\mathbb{R}_{+}^{2}\).
Let \(x(t_{0})=(x_{1}(t_{0}),x_{2}(t_{0}))\) in \(\mathbb{R}_{+}^{2}\) be the initial solution of system (3.3). By contradiction, suppose that there exists a solution \(x(t)\) that lies outside of \(\mathbb {R}_{+}^{2}\). The consequence is that \(x(t)\) crosses the \(x_{1}\) axis or \(x_{2}\) axis. Now we have to consider two cases.
Case 1: If the solution \(x(t)\) passes through the \(x_{2}\) axis, then there exists \(t^{*}\) such that \(t^{*} \geq t_{0}\) and \(x_{1}(t^{*})=0\), and there exists \(t_{1}\) sufficiently close to \(t^{*}\) such that \(t_{1}> t^{*}\) and \(x_{1}(t)<0\) for all \(t \in(t^{*},t_{1}]\). By the previous conclusion there are two possibilities.
Since \(x_{1}(t_{0}) \geq0\), we have \(x_{1}(t)\geq0\), which contradicts the assumption. So \(x_{1}(t)\geq0\) for any \(t\geq t_{0}\).
Case 2: Suppose that the solution \(x(t)\) passes through the \(x_{1}\) axis. Because the second equation of system (3.3) has the same form as the first one, the proof for case 2 is similar to the proof in the previous case.
Therefore, we can conclude that the solution \(x(t)\) of system (3.3) lies within \(\mathbb{R}_{+}^{2}\). □
4 Stability analysis of equilibrium points
- 1.
The origin \(E_{0}(0,0)\), which represents extinction of both species.
- 2.
\(E_{1}({K_{1}}{A_{1}},0)\), where \(A_{1}=1-\frac{e_{1}}{r_{1}}\), which represents extinction of the second species. The existence condition of \(E_{1}\) is \(0\le e_{1}< r_{1}\).
- 3.
\(E_{2}(0,{K_{2}}{A_{2}})\), where \(A_{2}=1-\frac{e_{2}}{r_{2}}\), which represents extinction of the first species. The existence condition of \(E_{2}\) is \(0\le e_{2}< r_{2}\).
- 4.
\(E_{3}(x^{*}_{1},x^{*}_{2})\), which is called the coexistence equilibrium point.
As for the equilibrium points \(E_{0}, E_{1}\), and \(E_{2}\), they are called the noncoexistence equilibrium points. The existence condition of \(E_{3}\) is presented in the next proposition.
Proposition 4.1
Proof
The proof for this proposition is the same as that for Proposition 1 of [54], which is the proof that there exists a unique coexisting equilibrium point for integer-order system (1.1). □
4.1 Local stability of equilibrium points
Theorem 4.2
If \(r_{1}< e_{1}\) and \(r_{2}< e_{2}\), then the trivial solution \(E_{0}\) of the fractional-order system (3.3) is locally asymptotically stable.
Proof
Theorem 4.3
If \(e_{1}< r_{1}\) and \(r_{2}< e_{2}\), then the second species extinction equilibrium point \(E_{1}({K_{1}}{A_{1}},0)\) of the fractional-order system (3.3) is locally asymptotically stable.
Proof
Theorem 4.4
If \(r_{1}< e_{1}\) and \(e_{2}< r_{2}\), then the first species extinction equilibrium point \(E_{2}(0,{K_{2}}{A_{2}})\) of the fractional order system (3.3) is locally asymptotically stable.
Proof
The idea of the proof is similar to that of Theorem 4.3. □
4.2 Global stability of positive coexistence equilibrium
In this subsection, we investigate sufficient conditions for global uniform asymptotic stability of the positive coexisting equilibrium for the corresponding fractional-order system using the Lyapunov function.
Theorem 4.5
If \({A_{1}}{b_{12}}<2, {A_{2}}{b_{21}}<2, {A_{1}}{A_{2}}{b_{12}}{b_{21}}<1, 0\le e_{1}<r_{1}\), and \(0 \le e_{2}< r_{2}\), then the unique interior positive equilibrium \((x^{*}_{1},x^{*}_{2})\) of the fractional-order system (3.3) is globally uniformly asymptotically stable on \(\mathbb {R}_{+}^{2}\).
Proof
Remark 4.6
The Lyapunov function in Theorem 4.5 is modified from the function of integer-order differential systems presented in [14] and different from the functions presented in [45–48].
Remark 4.7
Theorem 4.5 can be applied to study a facultative mutualism of two species. In particular, the interaction between two species is assumed to be described by model (3.3) with parameters satisfying the conditions of the theorem. Subsequently, any solutions starting at a positive initial point eventually tend to the positive coexistence equilibrium of the model. This means biologically that the two species always coexist in the same habitat.
5 Numerical simulations
In this section, we present numerical examples to verify the theoretical results in Theorem 4.5.
We carried out numerical simulations on the fractional-order two-species facultative mutualism system (3.3) to show the stability of the positive equilibrium result. The parameters \(r_{1}=0.6\), \(r_{2}=0.4\), \(e_{1}=0.4\), \(e_{2}=0.3\), \(K_{1}=250\), \(K_{2}=250\), \(b_{12}=0.2\), and \(b_{21}=0.25\) satisfy the assumptions of Theorem 4.5. Calculating the solution, we obtain the positive coexistence equilibrium point \(x^{*}_{1}=87.8661\) and \(x^{*}_{2}=67.9916\).
Many numerical methods have been applied to solve nonlinear fractional differential equations such as the Adams method [55], the predictor corrector method [56], the Adomian decomposition method [57], the extrapolation method [58], and the variational iteration method [59]. However, among these methods, the Adams method is useful for studying the long-term behavior of the solutions of nonlinear fractional differential equations [60]. Thus, in this study, we use the Adams method for solving the model with the Matlab software.
The parameters in the model satisfy all the conditions of Theorem 4.5. Hence, the equilibrium point \((x^{*}_{1},x^{*}_{2})=(87.8661,67.9916)\) is globally uniformly asymptotically stable on \(\mathbb{R}_{2}^{+}\). From Figures 1 and 2 we can see that all the solutions of the system converge uniformly to this point.
6 Conclusion
In this study, we presented a Caputo fractional-order model of two-species facultative mutualism and analyzed the stability of the model. We investigated the local asymptotic stability of the three noncoexistence equilibrium points using the linearization method. For the coexistence equilibrium point, we analyzed the global uniform asymptotic stability via the Lyapunov method. These results provide sufficient conditions to ensure the local asymptotic stability and global uniform asymptotic stability of the model. Finally, we used some numerical simulations of the model to illustrate the stability results.
Declarations
Acknowledgements
This research is supported by Department of Mathematics, Faculty of Science, Chiang Mai University. Appreciation is extended toward the Centre of Excellence in Mathematics, CHE, for their financial support.
Authors’ contributions
Both authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Brauer, F, Castillo-Chavez, C: Mathematical Models in Population Biology and Epidemiology, vol. 40. Springer, Berlin (2001) MATHGoogle Scholar
- Fulford, G, Forrester, P, Jones, A: Modelling with Differential and Difference Equations, vol. 10. Cambridge University Press, Cambridge (1997) View ArticleMATHGoogle Scholar
- Kuang, Y: Delay Differential Equations: With Applications in Population Dynamics, vol. 191. Academic Press, San Diego (1993) MATHGoogle Scholar
- Salisbury, A: Mathematical models in population dynamics. Dissertation. New College of Florida (2011) Google Scholar
- Vanmeter, KC, Hubert, RJ: Microbiology for the Healthcare Professional, 2nd edn. Mosby Elsevier, Amsterdam (2015) Google Scholar
- Stachowicz, JJ, Hay, ME: Facultative mutualism between an herbivorous crab and a coralline alga: advantages of eating noxious seaweeds. Oecologia 105(3), 377-387 (1996) View ArticleGoogle Scholar
- Rockwood, LL: Introduction to Population Ecology. Blackwell, Oxford (2006) Google Scholar
- Gilbert, LE: Ecological consequences of a coevolved mutualism between butterflies and plants. In: Coevolution of Animals and Plants, pp. 210-240. University of Texas Press, Austin (1980) Google Scholar
- Handel, SN: The competitive relationship of three woodland sedges and its bearing on the evolution of ant-dispersal of Carex pedunculata. Evolution 32(1), 151-163 (1978) MathSciNetView ArticleGoogle Scholar
- Batra, LR: Insect-fungus symbiosis: nutrition, mutualism, and commensalism. In: International Mycological Congress (2nd: 27 Aug.-3 Set. 1977), Florida (1979) Google Scholar
- Roughgarden, J: Evolution of marine symbiosis - a simple cost-benefit model. Ecology 56(5), 1201-1208 (1975) View ArticleGoogle Scholar
- Legovic, T, Gecek, S: Impact of maximum sustainable yield on mutualistic communities. Ecol. Model. 230, 63-72 (2012) View ArticleMATHGoogle Scholar
- Wolin, C, Lawlor, L: Models of facultative mutualism: density effects. Am. Nat. 124, 843-862 (1984) View ArticleGoogle Scholar
- Georgescu, P, Zhang, H: Lyapunov functional for two-species mutualisms. Appl. Math. Comput. 229, 754-764 (2014) MathSciNetMATHGoogle Scholar
- Igor, P: Fractional Differential Equations. Mathematics in Science and Engineering, vol. 198. Academic Press, New York (1999) MATHGoogle Scholar
- Caputo, M: Linear models of dissipation whose Q almost frequency independent: II (reprint). Fract. Calc. Appl. Anal. 11(1), 3-14 (2008) MATHGoogle Scholar
- Luo, Y, Chen, YQ: Fractional Order Motion Controls. Wiley, Chichester (2012) View ArticleGoogle Scholar
- Machado, JT, Kiryakova, V, Mainardi, F: Recent history of fractional calculus. Commun. Nonlinear Sci. Numer. Simul. 16(3), 1140-1153 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Almeida, R, Bastos, NRO, Monteiro, MT: Modeling some real phenomena by fractional differential equations. Math. Methods Appl. Sci. 39(16), 4846-4855 (2016) MathSciNetView ArticleMATHGoogle Scholar
- Demirci, E: A fractional order model for obesity epidemic in a non-constant population. Adv. Differ. Equ. 2017, 79 (2017) MathSciNetView ArticleGoogle Scholar
- Area, I, Batarfi, H, Losada, J, Nieto, JJ, Shammakh, W, Torres, Á: On a fractional order Ebola epidemic model. Adv. Differ. Equ. 2015(1), 278 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Zafar, ZUA, Rehan, K, Mushtaq, M: Fractional-order scheme for bovine babesiosis disease and tick populations. Adv. Differ. Equ. 2017(1), 86 (2017) MathSciNetView ArticleGoogle Scholar
- Mairet, F, Ramírez, H, Rojas-Palma, A: Modelling and stability analysis of a microalgal pond with nitrification. Anual Sociedad de Matemática de Chile 51, 448-468 (2015) Google Scholar
- Menouer, MA, Moussaoui, A, Dads, EA: Existence and global asymptotic stability of positive almost periodic solution for a predator-prey system in an artificial lake. Chaos Solitons Fractals 103, 271-278 (2017) MathSciNetView ArticleMATHGoogle Scholar
- Lu, G, Lu, Z: Geometric approach to global asymptotic stability for the SEIRS models in epidemiology. Nonlinear Anal., Real World Appl. 36, 20-43 (2017) MathSciNetView ArticleMATHGoogle Scholar
- Guo, L, Chen, YQ: System stability analysis via a perturbation technique. Commun. Nonlinear Sci. Numer. Simul. 57, 111-124 (2017) MathSciNetView ArticleGoogle Scholar
- Berezansky, L, Braverman, E: A note on stability of Mackey-Glass equations with two delays. J. Math. Anal. Appl. 450(2), 1208-1228 (2017) MathSciNetView ArticleMATHGoogle Scholar
- Hou, Q, Wang, T: Global stability and a comparison of SVEIP and delayed SVIP epidemic models with indirect transmission. Commun. Nonlinear Sci. Numer. Simul. 43, 271-281 (2017) MathSciNetView ArticleGoogle Scholar
- Caetano, D, Faria, T: Stability and attractivity for Nicholson systems with time-dependent delays. Electron. J. Qual. Theory Differ. Equ. 2017, 63 (2017) MathSciNetView ArticleGoogle Scholar
- Liu, B: Asymptotic behavior of solutions to a class of non-autonomous delay differential equations. J. Math. Anal. Appl. 446(1), 580-590 (2017) MathSciNetView ArticleMATHGoogle Scholar
- Baleanu, D, Wu, GC, Zeng, SD: Chaos analysis and asymptotic stability of generalized Caputo fractional differential equations. Chaos Solitons Fractals 102, 99-105 (2017) MathSciNetView ArticleMATHGoogle Scholar
- Čermák, J, Došlá, Z, Kisela, T: Fractional differential equations with a constant delay: stability and asymptotics of solutions. Appl. Math. Comput. 298, 336-350 (2017) MathSciNetGoogle Scholar
- Yadav, VK, Das, S, Bhadauria, BS, Singh, AK, Srivastava, M: Stability analysis, chaos control of a fractional order chaotic chemical reactor system and its function projective synchronization with parametric uncertainties. Chin. J. Phys. 55(3), 594-605 (2017) View ArticleGoogle Scholar
- Ji, G, Ge, Q, Xu, J: Dynamic behavior of a fractional order two-species cooperative systems with harvesting. Chaos Solitons Fractals 92, 51-55 (2016) MathSciNetView ArticleMATHGoogle Scholar
- Huo, J, Zhao, H: Dynamical analysis of a fractional SIR model with birth and death on heterogeneous complex networks. Phys. A, Stat. Mech. Appl. 448, 41-56 (2016) MathSciNetView ArticleGoogle Scholar
- Ghaziani, RK, Alidousti, J, Eshkaftaki, AB: Stability and dynamics of a fractional order Leslie-Gower prey-predator model. Appl. Math. Model. 40(3), 2075-2086 (2016) MathSciNetView ArticleGoogle Scholar
- Li, Y, Chen, YQ, Igor, P: Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability. Comput. Math. Appl. 59(5), 1810-1821 (2010) MathSciNetView ArticleMATHGoogle Scholar
- Baranowski, J, Zagorowska, M, Bauer, W, Dziwinski, T, Piatek, P: Applications of direct Lyapunov method in Caputo non-integer order systems. Elektron. Elektrotech. 21(2), 10-13 (2015) Google Scholar
- Li, C, Ma, Y: Fractional dynamical system and its linearization theorem. Nonlinear Dyn. 71(4), 621-633 (2013) MathSciNetView ArticleMATHGoogle Scholar
- Li, Y, Chen, YQ, Igor, P: Mittag-Leffler stability of fractional order nonlinear dynamic systems. Automatica 45(8), 1965-1969 (2009) MathSciNetView ArticleMATHGoogle Scholar
- Zhang, F, Li, C, Chen, YQ: Asymptotical stability of nonlinear fractional differential system with Caputo derivative. Int. J. Differ. Equ. 2011, 12 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Delavari, H, Baleanu, D, Sadati, J: Stability analysis of Caputo fractional-order nonlinear systems revisited. Nonlinear Dyn. 67(4), 2433-2439 (2012) MathSciNetView ArticleMATHGoogle Scholar
- Gallegos, JA, Duarte-Mermoud, MA: On the Lyapunov theory for fractional order systems. Appl. Math. Comput. 287, 161-170 (2016) MathSciNetMATHGoogle Scholar
- Liu, S, Jiang, W, Li, X, Zhou, XF: Lyapunov stability analysis of fractional nonlinear systems. Appl. Math. Lett. 51, 13-19 (2016) MathSciNetView ArticleMATHGoogle Scholar
- Vargas-De-León, C: Volterra-type Lyapunov functions for fractional-order epidemic systems. Commun. Nonlinear Sci. Numer. Simul. 24(1-3), 75-85 (2015) MathSciNetView ArticleGoogle Scholar
- Duarte-Mermoud, MA, Aguila-Camacho, N, Gallegos, JA, Castro-Linares, R: Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems. Commun. Nonlinear Sci. Numer. Simul. 22(1), 650-659 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Aguila-Camacho, N, Duarte-Mermoud, MA, Gallegos, JA: Lyapunov functions for fractional order systems. Commun. Nonlinear Sci. Numer. Simul. 19(9), 2951-2957 (2014) MathSciNetView ArticleGoogle Scholar
- Zhou, XF, Hu, LG, Liu, S, Jiang, W: Stability criterion for a class of nonlinear fractional differential systems. Appl. Math. Lett. 28, 25-29 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Liang, S, Wu, R, Chen, L: Laplace transform of fractional order differential equations. Electron. J. Differ. Equ. 2015(139), 1 (2015) MathSciNetMATHGoogle Scholar
- Kexue, L, Jigen, P: Laplace transform and fractional differential equations. Appl. Math. Lett. 24(12), 2019-2023 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Abdeljawad, T, Gejji, V: Lyapunov-Krasovskii stability theorem for fractional systems with delay. Rom. J. Phys. 56(5-6), 636-643 (2011) MathSciNetMATHGoogle Scholar
- Ahmed, E, El-Sayed, AMA, El-Saka, HAA: On some Routh-Hurwitz conditions for fractional order differential equations and their applications in Lorenz, Rössler, Chua and Chen systems. Phys. Lett. A 358(1), 1-4 (2006) MathSciNetView ArticleMATHGoogle Scholar
- Walter, W: Ordinary Differential Equations. Springer Graduate Texts in Mathematics, vol. 182 (1991) Google Scholar
- Vargas-De-León, C: Lyapunov functions for two-species cooperative systems. Appl. Math. Comput. 219(5), 2493-2497 (2012) MathSciNetMATHGoogle Scholar
- Diethelm, K, Ford, NJ, Freed, AD: Detailed error analysis for a fractional Adams method. Numer. Algorithms 36(1), 31-52 (2004) MathSciNetView ArticleMATHGoogle Scholar
- Diethelm, K, Ford, NJ, Freed, AD: A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn. 29(1), 3-22 (2002) MathSciNetView ArticleMATHGoogle Scholar
- Adomian, G: Solving Frontier Problems of Physics: The Decomposition Method, vol. 60. Springer, Berlin (2013) MATHGoogle Scholar
- Diethelm, K, Walz, G: Numerical solution of fractional order differential equations by extrapolation. Numer. Algorithms 16(3), 231-253 (1997) MathSciNetView ArticleMATHGoogle Scholar
- Wu, G, Lee, EWM: Fractional variational iteration method and its application. Phys. Lett. A 374(25), 2506-2509 (2010) MathSciNetView ArticleMATHGoogle Scholar
- Deshpande, AS, Daftardar-Gejji, V, Sukale, YV: On Hopf bifurcation in fractional dynamical systems. Chaos Solitons Fractals 98, 189-198 (2017) MathSciNetView ArticleMATHGoogle Scholar