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Modeling the dynamics of nutrient–phytoplankton–zooplankton system with variable-order fractional derivatives
Chaos, Solitons and Fractals 116 (2018) 114–120
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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
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Modeling the dynamics of nutrient–phytoplankton–zooplankton system with variable-order fractional derivatives Behzad Ghanbari a, J.F. Gómez-Aguilar b,∗ a b
Department of Engineering Science, Kermanshah University of Technology, Kermanshah, Iran CONACyT-Tecnológico Nacional de México/CENIDET, Interior Internado Palmira S/N, Col. Palmira, C.P. 62490, Cuernavaca Morelos, México
a r t i c l e
i n f o
Article history: Received 31 August 2018 Revised 13 September 2018 Accepted 13 September 2018 Available online 22 September 2018 Keywords: Fractional calculus Variable-order fractional derivatives Nutrient–phytoplankton–zooplankton model Lagrange interpolation
a b s t r a c t We extended the nutrient–phytoplankton–zooplankton model involving variable-order fractional differential operators of Liouville–Caputo, Caputo–Fabrizio and Atangana–Baleanu. Variable-order fractional operators permits model and describe accurately real world problems, for example, diffusion or spread of nutrients or species in different states. Particularly, we model the interaction of nutrient phytoplankton and its predator zooplankton. The variable-order fractional numerical scheme based on the fundamental theorem of fractional calculus and the Lagrange polynomial interpolation was consider. Numerical simulation results are provided for illustrating the effectiveness and applicability of the algorithm to solve variable-order fractional differential equations.
1. Introduction Fractional derivatives become excellent instrument for the description of memory and hereditary properties of various materials and processes. Such effects are in fact neglected in models with classical integer-order. This can be viewed as the main advantage of fractional derivatives. It also plays a crucial role in the description of dynamics between two different points in many other fields [1–11]. Despite of the idea of fractional derivatives and integrals can be considered as a generalization of corresponding standard ones, it is still quite a strange topic, very hard to explain. Because, unlike commonly used differential operators, it is not related to some important geometrical meaning, such as the trend of functions or their convexity. So, sometimes this mathematical tool could be judged “far from reality”. But indeed many physical phenomena have “intrinsic” fractional order description and so fractional order calculus is necessary in order to explain them [1]. There are a several number of definitions of fractional derivatives. For instance, Riemann and Liouville introduced the concept of fractionalorder differentiation with power-law in [2,3]. Caputo and Fabrizio in [4], introduced a new derivative with fractional order based on the exponential-law and Atangana and Baleanu suggested another version of fractional-order derivative which uses the generalized Mittag-Leffler function with strong memory as non-local and nonsingular kernel in [5]. ∗
Corresponding author. E-mail address: [email protected] (J.F. Gómez-Aguilar).
https://doi.org/10.1016/j.chaos.2018.09.026 0960-0779/© 2018 Elsevier Ltd. All rights reserved.
© 2018 Elsevier Ltd. All rights reserved.
In most cases, exact solutions of differential equations with integer, non-integer order or variable-order derivatives are very difficult to obtain; this is the principal motivation to develop iterative methods or numerical techniques to solve these equations. For the iterative methods (Adomian decomposition method, the variational iteration method, fractional sub-equation method, homotopy perturbation techniques, among others) [12–17], the principal problem are the stability and the convergence. Traditionally, Adams– Bashforth method has been recognized as a great and powerful numerical method able to provide a numerical solution of fractional differential equations [18–25]. In [26], the authors approximated Liouville–Caputo fractional derivatives by Chebyshev polynomials. Rosenfeld and Dixon in [27] developed a numerical scheme based on scattered data interpolation via reproducing kernel Hilbert spaces to solved Liouville–Caputo fractional order differential equations. In [28], the authors proposed a new three-step fractional Adams–Bashforth scheme for solving linear and nonlinear fractional order differential equations involving the Caputo–Fabrizio operator. Shahbazi and Javidi considered 3/8 Simpsons rule to design a new high order predictor-corrector scheme [29]. Modifications combining the rectangle formula, trapezoid formula, polynomial interpolation or Gauss–Lobatto quadrature can be found in [30,31]. In [32-33], the authors developed a generalized version of Adams–Basforth method to partial differential equations involving Laplace transform, Lagrange polynomial interpolation and the forward-backward scheme. Recently, in [34–36], the authors developed a constant-order and variable-order numerical schemes that combines the fundamental theorem of fractional calculus and the two-step Lagrange polynomial.
B. Ghanbari, J.F. Gómez-Aguilar / Chaos, Solitons and Fractals 116 (2018) 114–120
In the biological models, interaction networks can be visualized as food-chains of species linked by trophic interactions. Phytoplankton provide food for marine life, oxygen for human being and purify the atmosphere by consuming carbon dioxide. Nevertheless, the rapid growth of phytoplankton may reduce the required amount of oxygen needed for the growth of other aquatic plants and animals. In the literature we found mathematical models to consider the interaction nutrient–phytoplankton–zooplankton. A mathematical model that describes three species food chain model consisting of toxin producing phytoplankton, zooplankton and fish population has been developed in [37]. In [38], the authors described a nutrient phytoplankton model by a couple of reactiondiffusion equations with delay. Biological systems presents longrange temporal memory or long-range space interactions, for this reason, the use of fractional derivatives can handle efficiently the dynamics of a disease model and also gives information on each point of the model. In [11] a fractional mathematical model for the interaction of nutrient phytoplankton and its predator zooplankton was investigated numerically. The fractional derivative of Liouville– Caputo type was used to obtain the generalization of the model. For solving the result fractional equations, a new numerical algorithm based on the polynomial interpolation was proposed. In this paper, we consider a variable-order fractional nutrient– phytoplankton–zooplankton system [11] via Liouville–Caputo, Caputo–Fabrizio–Caputo and Atangana–Baleanu–Caputo fractional derivatives. 2. Mathematical model The fractional nutrient–phytoplankton–zooplankton system [11] is generalized by replacing the classical derivative by the operator 0 Dtα (t ) α (t )
0 Dt
α (t )
0 Dt
α (t )
0 Dt
x1 (t ) = α0 − ax1 (t ) − b1 x1 (t )x2 (t ) + c1 x2 (t ) + c2 x3 (t ), (1a) x2 (t ) = b2 x1 (t )x2 (t ) − c3 x2 (t ) − x3 (t ) =
d1 x2 (t )x3 (t ) , e + x2 (t )
d2 x2 (t )x3 (t ) − f x2 (t )x3 (t ) − c4 x3 (t ), e + x2 (t )
(1b)
C α (t ) 0 Dt
f (t )=
x2 ( 0 ) = x2,0 > 0 ,
t
0
(t − τ )−α (t ) f˙ (τ )dτ ,
0 < α (t ) ≤ 1. (2)
CF C α (t ) 0 Dt
f (t ) =
(2 − α (t ))M (α (t )) t 2(1 − α (t )) 0 (t − τ ) ˙ × exp − α (t ) f ( τ )d τ , 1 − α (t )
0 < α (t ) < 1, (3)
where M (α (t )) = 2−α2 (t ) is a normalization function. The variable-order Atangana–Baleanu–Caputo fractional derivative with Mittag-Leffler (ABC) is defined as follows [5,6] ABC α (t ) 0 Dt
f (t ) =
B(α (t )) t (t − τ )α (t ) Eα (t ) − α (t ) 1 − α (t ) 0 1 − α (t ) × f˙ (τ )dτ , 0 < α (t ) ≤ 1,
where B(α (t )) = 1 − α (t ) +
(4)
α (t ) is a normalization function. (α (t ))
Now considering the numerical scheme developed in [36], we obtain numerical simulations for the nutrient–phytoplankton– zooplankton model in Liouville–Caputo; Caputo–Fabrizio–Caputo; and Atangana–Baleanu–Caputo fractional derivatives with variable order α (t). 3. Numerical schemes 3.1. Numerical scheme in Liouville–Caputo sense with variable-order A fractional ordinary differential equation of Liouville–Caputo type with variable-order can be expressed as follows: C α (t ) y 0 Dt
(t ) = f (t , y(t )).
(5)
The approximate solution of Eq. (5) is obtained as [36]
(1c)
yn+1 (t ) = y(0 ) +
or ABC Dtα (t ) , called Liouville–Caputo, Caputo–Fabrizio– 0 Caputo or Atangana–Baleanu–Caputo fractional derivatives with variable-order α (t), respectively. The variable-order Liouville–Caputo fractional derivative with power-law (C) is defined as follows [7]:
n
hα (t ) f (tm , ym ) (α (t )) m=0 α (t )(α (t ) + 1 ) 1
((n + 1 − m )α (t ) (n − m + 2 + α (t )) − (n − m )α (t ) hα (t ) f (tm−1 , ym−1 ) ×(n − m + 2 + 2α (t ) )) − α (t )(α (t ) + 1 )
x3 ( 0 ) = x3,0 > 0 ,
where α 0 , a, b1 , b2 , c1 , c2 , c3 , c4 , d1 , d2 , e and f are positive constants. In the above model x1 (t) denote the concentration of nutrient, x2 (t) denotes the biomass of phytoplankton which also produces toxicant harmful to the zooplankton biomass and x3 (t) denote the concentration of zooplankton population, the parameters {a, b1 , b2 , c1 , c2 , c3 , c4 , d1 , d2 , e, f, α 0 } represents the rate of nutrient loss; nutrient uptake rate for the phytoplankton population; nutrient-phytoplankton conversion rate; nutrient recycling rate after the death of phytoplankton; nutrient recycling rate after the death of zooplankton; phytoplankton mortality rate; zooplankton death rate, maximal zooplankton ingestion rate; maximal phytoplankton–zooplankton conversion rate; half saturation constant for a Holling type II functional response; rate of zooplankton decay due to toxin producing phytoplankton and the constant input nutrient concentration, respectively. More details of this model can be found in [11]. The variable-order operator 0 Dtα (t ) can be of type C0 Dtα (t ) ,
CF C D α (t ) t 0
The variable-order Caputo–Fabrizio fractional derivative with exponential-law in Liouville–Caputo sense (CFC) is defined as follows [36]:
with initial conditions
x1 ( 0 ) = x1,0 > 0 ,
1 (1 − α (t ))
115
×((n + 1 − m )α (t )+1 − (n − m )α (t ) (n − m + 1 + α (t ))) . (6) 3.2. Numerical scheme in Caputo–Fabrizio–Caputo sense with variable-order Now, we have the following fractional differential equation with variable-order in Caputo–Fabrizio–Caputo sense CF C α (t ) y 0 Dt
(t ) = f (t , y(t )).
(7)
The numerical solution fo Eq. (7) is obtained by the following expression [36]
(2−α (t ))(1 − α (t )) 3h
α (t )(2−α (t )) f (tn , yn ) (2 − α (t ))(1 − α (t )) h − + α (t )(2 − α (t )) f (tn−1 , yn−1 ).
yn+1 = yn +
2
2
+
4
4
(8)
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B. Ghanbari, J.F. Gómez-Aguilar / Chaos, Solitons and Fractals 116 (2018) 114–120
3.3. Numerical scheme in Atangana–Baleanu–Caputo sense with variable-order A fractional ordinary differential equation of Atangana–Baleanu–Caputo type with variable-order can be expressed as follows: ABC α (t ) y 0 Dt
(t ) = f (t , y(t )).
(9)
and the numerical solution of Eq. (9) is obtained by the following expression [36]
(α (t ))(1 − α (t )) f (t , y(tn )) (α (t ))(1 − α (t )) + α (t ) n n
1 + hα (t ) f (tm , ym ) ((n + 1 − m )α (t ) (α (t ) + 1 )((1 − α (t ))(α (t )) + α (t ) m=0
yn+1 (t ) = y0 +
(10)
× (n − m + 2 + α (t )) − (n − m )α (t ) (n − m + 2 + 2α (t ))) − hα (t ) f (tm−1 , ym−1 )((n + 1 − m )α (t )+1 − (n − m )α (t ) (n − m + 1 + α (t )))).
4. Numerical results In this section, we present the numerical solutions of the nutrient–phytoplankton–zooplankton model with Liouville–Caputo, Caputo– Fabrizio and Atangana–Baleanu fractional derivatives. Numerical simulations are showed for different values of fractional order.
4.1. Numerical scheme in Liouville–Caputo sense with variable-order Applying the approximate solution given by Eq. (6), we obtain the following equations:
x1,n+1 (t ) = x1,0 +
1
n
(α (t )) m=0
hα (t ) f1 (tn , x1,m , x2,m , x3,m ) ((n + 1 − m )α (t ) (n − m + 2 + α (t )) α (t )(α (t ) + 1 )
− (n − m )α (t ) (n − m + 2 + 2α (t ))) −
hα (t ) f1 (tm−1 , x1,m−1 , x2,m−1 , x3,m−1 ) ( (n + 1 − m )α (t )+1 α (t )(α (t ) + 1 )
− (n − m )α (t ) (n − m + 1 + α (t ))) ,
x2,n+1 (t ) = x2,0 +
1
n
(α (t )) m=0
hα (t ) f2 (tn , x1,m , x2,m , x3,m ) ((n + 1 − m )α (t ) (n − m + 2 + α (t )) α (t )(α (t ) + 1 )
− (n − m )α (t ) (n − m + 2 + 2α (t ))) −
hα (t ) f2 (tm−1 , x1,m−1 , x2,m−1 , x3,m−1 ) ( (n + 1 − m )α (t )+1 α (t )(α (t ) + 1 )
(11)
− (n − m )α (t ) (n − m + 1 + α (t ))) ,
x3,n+1 (t ) = x3,0 +
1
n
(α (t )) m=0
hα (t ) f3 (tn , x1,m , x2,m , x3,m ) ((n + 1 − m )α (t ) (n − m + 2 + α (t )) α (t )(α (t ) + 1 )
− (n − m )α (t ) (n − m + 2 + 2α (t ))) −
hα (t ) f3 (tm−1 , x1,m−1 , x2,m−1 , x3,m−1 ) ( (n + 1 − m )α (t )+1 α (t )(α (t ) + 1 )
− (n − m )α (t ) (n − m + 1 + α (t ))) , where
α0 − ax1 (t ) − bx1 (t )x2 (t ) + c1 x2 (t ) + c2 x3 (t ), d1 x2 (t )x3 (t ) d2 x2 (t )x3 (t ) f2 (t, x1 (t ), x2 (t ), x3 (t )) := b2 x1 (t )x2 (t ) − c3 x2 (t ) − , f3 (t, x1 (t ), x2 (t ), x3 (t )) := − f x2 (t )x3 (t ) − c4 x3 (t ). (12) e + x2 (t ) e + x2 (t ) f1 (t, x1 (t ), x2 (t ), x3 (t )) :=
4.2. Numerical scheme in Caputo–Fabrizio–Caputo sense with variable-order The numerical solution of system (1) in Caputo–Fabrizio–Caputo sense is given by the following equations:
B. Ghanbari, J.F. Gómez-Aguilar / Chaos, Solitons and Fractals 116 (2018) 114–120
x1,n+1 (t ) = x1,n +
−
2
(2 − α (t ))(1 − α (t )) 2
x2,n+1 (t ) = x2,n +
−
2 2
+
3h α (t )(2 − α (t )) f2 (tn , x1,n , x2,n , x3,n ) 4
+
h α (t )(2 − α (t )) f2 (tn−1 , x1n−1 (t ), x2n−1 (t ), x3n−1 (t )), 4
(2 − α (t ))(1 − α (t ))
(2 − α (t ))(1 − α (t ))
3h α (t )(2 − α (t )) f1 (tn , x1,n , x2,n , x3,n ) 4
h α (t )(2 − α (t )) f1 (tn−1 , x1n−1 (t ), x2n−1 (t ), x3n−1 (t )), 4
2 2
+
+
(2 − α (t ))(1 − α (t ))
(2 − α (t ))(1 − α (t ))
x3,n+1 (t ) = x3,n + −
(2 − α (t ))(1 − α (t ))
117
(13)
+
3h α (t )(2 − α (t )) f3 (tn , x1,n , x2,n , x3,n ) 4
+
h α (t )(2 − α (t )) f3 (tn−1 , x1n−1 (t ), x2n−1 (t ), x3n−1 (t ) ), 4
where f1 (t, x1 (t), x2 (t), x3 (t)), f2 (t, x1 (t), x2 (t), x3 (t)) and f3 (t, x1 (t), x2 (t), x3 (t)) are given by Eq. (12).
4.3. Numerical scheme in Atangana–Baleanu–Caputo sense with variable-order In the Atangana–Baleanu–Caputo sense, the numerical representation to the system (1) is obtained as follows:
(α (t ))(1 − α (t )) f (t , x , x , x ) (α (t ))(1 − α (t )) + α (t ) 1 n 1,n 2,n 3,n n
1 + hα (t ) f1 (tn , x1,m , x2,m , x3,m ) (α (t ) + 1 )((1 − α (t ))(α (t )) + α (t ) m=0
x1,n+1 (t ) = x1,0 +
× ((n + 1 − m )α (t ) (n − m + 2 + α (t )) − (n − m )α (t ) (n − m + 2 + 2α (t ))) − hα (t ) f1 (tm−1 , x1,m−1 , x2,m−1 , x3,m−1 ) × ((n + 1 − m )α (t )+1 − (n − m )α (t ) (n − m + 1 + α (t )))),
(α (t ))(1 − α (t )) f (t , x , x , x ) (α (t ))(1 − α (t )) + α (t ) 2 n 1,n 2,n 3,n n
1 + hα (t ) f2 (tn , x1,m , x2,m , x3,m ) (α (t ) + 1 )((1 − α (t ))(α (t )) + α (t ) m=0
x2,n+1 (t ) = x2,0 +
× ((n + 1 − m )α (t ) (n − m + 2 + α (t )) − (n − m )α (t ) (n − m + 2 + 2α (t ))) − hα (t ) f2 (tm−1 , x1,m−1 , x2,m−1 , x3,m−1 ) × ((n + 1 − m )α (t )+1 − (n − m )α (t ) (n − m + 1 + α (t )))),
(α (t ))(1 − α (t )) f (t , x , x , x ) (α (t ))(1 − α (t )) + α (t ) 3 n 1,n 2,n 3,n n
1 + hα (t ) f3 (tn , x1,m , x2,m , x3,m ) (α (t ) + 1 )((1 − α (t ))(α (t )) + α (t ) m=0
x3,n+1 (t ) = x3,0 +
× ((n + 1 − m )α (t ) (n − m + 2 + α (t )) − (n − m )α (t ) (n − m + 2 + 2α (t ))) − hα (t ) f3 (tm−1 , x1,m−1 , x2,m−1 , x3,m−1 )× × ((n + 1 − m )α (t )+1 − (n − m )α (t ) (n − k + 1 + α (t )))),
(14)
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Numerical solutions of different schemes for Example 1 with alpha(t)= 0.85
Fig. 3. Numerical simulations for example 1 in Atangana–Baleanu–Caputo sense. Fig. 1. Numerical simulations of example 1 for Liouville–Caputo, Caputo–Fabrizio– Caputo and Atangana–Baleanu fractional derivatives with α (t ) = 0.85.
Fig. 4. Numerical simulations of example 2 for Liouville–Caputo, Caputo–Fabrizio– Caputo and Atangana–Baleanu–Caputo fractional derivatives with α (t ) = 0.95.
Fig. 2. Numerical simulations for example 1 in Liouville–Caputo sense.
where f1 (t, x1 (t), x2 (t), x3 (t)), f2 (t, x1 (t), x2 (t), x3 (t)) and f3 (t, x1 (t), x2 (t), x3 (t)) are given by Eq. (12). Example 1. For the following set of parameters: α0 = 0.5, a = 1, b1 = 0.3, b2 = 0.25, c1 = 0.06, c2 = 0.06, c3 = 0.2, c4 = 0.5, d1 = 2.1, d2 = 0.2, e = 1, f = 0.1. Figs. 1–3, show the dynamics of the nutrient–phytoplankton–zooplankton model considering different fractional orders α (t) arbitrarily chosen. The time simulation was t = 60 [s] and step size h = 1 × 10−2 . Example 2. For the following set of parameters: α0 = 1.4, a = 0.7, b1 = 3.2, b2 = 2.4, c1 = 0.06, c2 = 0.06, c3 = 0.68, c4 = 0.21, d1 = 2.1, d2 = 1, e = 1, f = 0.1. Figs. 4–6, show the dynamics of the nutrient–phytoplankton–zooplankton model considering
Fig. 5. Numerical simulations of example 2 for Liouville–Caputo, Caputo–Fabrizio– Caputo and Atangana–Baleanu–Caputo fractional derivatives with α (t ) = 1/(1 + exp(−t ).
B. Ghanbari, J.F. Gómez-Aguilar / Chaos, Solitons and Fractals 116 (2018) 114–120
119
of fractional derivatives with singular power-law, based in this comments, we can conclude the index law is not valid in fractional differentiation. Competing interests The authors declare that there is no conflict of interests regarding the publication of this paper. Authors’ contributions All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript. Acknowledgments
Fig. 6. Numerical simulations of example 2 for Liouville–Caputo, Caputo–Fabrizio– Caputo and Atangana–Baleanu–Caputo fractional derivatives with α (t ) = 0.97 + 0.03 sin(t/10 ).
The authors are grateful to all of the anonymous reviewers for their valuable suggestions. José Francisco Gómez Aguilar acknowledges the support provided by CONACyT: Cátedras CONACyT para jóvenes investigadores 2014 and SNI-CONACyT. References
different fractional orders α (t) arbitrarily chosen. The time simulation was t = 60 [s] and step size h = 1 × 10−2 . The graphical representations show that the model depends notably to the fractional-order and the selected system parameters. In contrast to the constant-order nutrient–phytoplankton– zooplankton model, the memory effect change with time. This means that the memory rate of system is determined by the current time instant. For different time periods we have different memory abilities. In consequence, the constant-order systems permits characterizing the long memory of systems, while the variable-order systems can be used to characterize variable memory effect and capture the areas of transition between dynamic regimes of the physical phenomena. 5. Conclusions Based in the fundamental theorem of fractional calculus and the two-step Lagrange polynomial, we obtain numerical solutions for variable-order fractional nutrient–phytoplankton–zooplankton system. Numerical examples with different variable-orders have been presented to demonstrate the effectiveness of the method. The results obtained with the Liouville–Caputo derivative (based on the singular power-law) not completely describe the memory effects. The Caputo–Fabrizio and Atangana–Baleanu fractional derivatives consider a non-singular kernel based on the exponential-law and the Mittag-Leffler law (which of course the generalization of exponential function). These kernels permits describe the memory of the systems without the inclusion of artificial singularities into the mathematical models, that is to say, the history of the dynamical process are fully described. The generalized Mittag-Leffler function with strong memory involved in the Atangana–Baleanu fractional derivative is a best filter than the exponential and powerlaw functions. Furthermore, the Atangana–Baleanu fractional order derivative is at the same time Liouville–Caputo and Caputo– Fabrizio thus possesses Markovian and non-Markovian properties. These operators describe different waiting times distribution as such is observed in the ecological interactions. The Caputo–Fabrizio and Atangana–Baleanu derivatives not obeying the index law imposed in fractional calculus [39–41]. Due to this apparent limitation, these non-singular operators present crossover behavior and permits describe more appropriate real world problems than that
[1] Caponetto R, Dongola G, Fortuna L, Petrás I. World scientific series on nonlinear science series A. Fractional order systems modeling and control applications; 2010. [2] Samko SG, Kilbas AA, Marichev OI. Fractional integrals and derivatives: theory and applications. Yverdon: Gordon and Breach; 1993. [3] Podlubny I. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Fractional differential equations. San Diego, Calif, USA: Academic Press; 1999. [4] Caputo M, Fabrizio M. A new definition of fractional derivative without singular kernel. Progr Fract Differ Appl 2015;1:73–85. [5] Atangana A, Baleanu D. New fractional derivatives with nonlocal and non-singular kernel. Theory Appl Heat Transf Model Therm Sci 2016;20(2):763–9. [6] Yang XJ. Fractional derivatives of constant and variable orders applied to anomalous relaxation models in heat-transfer problems. Therm Sci 2017;21(3):1161–71. [7] Moghaddam BP, Yaghoobi S, Machado JT. An extended predictor-corrector algorithm for variable-order fractional delay differential equations. J Comput Nonlinear Dyn 2016;1:1–11. [8] Moghaddam BP, Yaghoobi S, Machado JT. An extended predictor-corrector algorithm for variable-order fractional delay differential equations. J Comput Nonlinear Dyn 2016;1:1–11. [9] Khare S, Misra O, Dhar J. Role of toxin producing phytoplankton on a plankton ecosystem. Nonlinear Anal Hybrid Syst 2010;4(3):496–502. [10] Chatterjee A, Pal S, Chatterjee S. Bottom up and top down effect on toxin producing phytoplankton and its consequence on the formation of plankton bloom. Appl Math Comput 2011;218(7):3387–98. [11] Asl MS, Javidi M. Novel algorithms to estimate nonlinear FDEs: applied to fractional order nutrient–phytoplankton–zooplankton system. J Comput Appl Math 2018;339:193–207. [12] Duan JS, Rach R, Wazwaz AM. Steady-state concentrations of carbon dioxide absorbed into phenyl glycidyl ether solutions by the adomian decomposition method. J Math Chem 2015;53(4):1054–67. [13] Lu L, Duan J, Fan L. Solution of the magnetohydrodynamics Jeffery–Hamel flow equations by the modified adomian decomposition method. Adv Appl Math Mech 2015;7(05):675–86. [14] Ma HC, Yao DD, Peng XF. Exact solutions of non-linear fractional partial differential equations by fractional sub-equation method. Thermal Sci 2015;19(4):1239–44. [15] Guo S, Mei L. Exact solutions of space-time fractional variant Boussinesq equations. Adv Sci Lett 2012;10(1):700–2. [16] Mohyud-Din ST, Nawaz T, Azhar E, Akbar MA. Fractional sub-equation method to space-time fractional Calogero–Degasperis and potential Kadomtsev–Petviashvili equations. J Taibah Univ Sci 2017;11(2):258–63. [17] Jafari H, Ghorbani M, Ghasempour S. A note on exact solutions for nonlinear integral equations by a modified homotopy perturbation method. New Trends Math Sci 2013;1(2):22–6. [18] Diethelm K, Ford NJ, Freed AD. A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn 2002;29(1–4):3–22. [19] Diethelm K, Ford NJ, Freed AD. Detailed error analysis for a fractional adams method. Numer Algorithms 2004;36(1):31–52. [20] Galeone L, Garrappa R. Fractional Adams–Moulton methods. Math Comput Simul 2008;79(4):1358–67. [21] Li C, Tao C. On the fractional adams method. Comput Math Appl 2009;58(8):1573–88.
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[22] Daftardar-Gejji V, Jafari H. Analysis of a system of non autonomous fractional differential equations involving Caputo derivatives. J Math Anal Appl 2007;328:1026–33. [23] Daftardar-Gejji V, Sukale Y, Bhalekar S. A new predictor-corrector method for fractional differential equations. Appl Math Comput 2014;244:158–82. [24] Odibat ZM, Momani S. Application of variational iteration method to nonlinear differential equations of fractional order. Int J Nonlinear Sci Numer Simul 2006;7(1):27–34. [25] Lin R, Liu F, Anh V, Turner I. Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation. Appl Math Comput 2009;212(2):435–45. [26] Sugiura H, Hasegawa T. Quadrature rule for Abel’s equations: uniformly approximating fractional derivatives uniformly approximating fractional derivatives. J Comput Appl Math 2009;223:459–68. [27] Rosenfeld JA, Dixon WE. Approximating the Caputo fractional derivative through the Mittag–Leffler reproducing kernel hilbert space and the kernelized Adams–Bashforth–Moulton method. SIAM J Numer Anal 2017;55(3):1201–17. [28] Owolabi KM, Atangana A. Analysis and application of new fractional Adams–Bashforth scheme with Caputo–Fabrizio derivative. Chaos Solitons Fractal 2017;105:111–19. [29] Asl MS, Javidi M. An improved PC scheme for nonlinear fractional differential equations: error and stability analysis. J Comput Appl Math 2017;324:101–17. [30] Li C, Yi Q, Chen A. Finite difference methods with non-uniform meshes for nonlinear fractional differential equations. J Comput Phys 2016;316:614–31. [31] Zhao L, Deng W. Jacobian-predictor-corrector approach for fractional differential equations. Adv Comput Math 2014;40(1):137–65.
[32] Gnitchogna R, Atangana A. New two step laplace Adam–Bashforth method for integer a noninteger order partial differential equations. Numer Methods Partial Differ Equ 2017;34(5):1739–58. [33] Alkahtani BST. Atangana–Batogna numerical scheme applied on a linear and non-linear fractional differential equation. Eur Phys J Plus 2018;133(3):1–11. [34] Atangana A, Owolabi KM. New numerical approach for fractional differential equations. Math Model Nat Phenom 2018;13(1):1–17. [35] Toufik M, Atangana A. New numerical approximation of fractional derivative with non-local and non-singular kernel: application to chaotic models. Eur Phys J Plus 2017;132(10):1–16. [36] Solís-Pérez JE, Gómez-Aguilar JF, Atangana A. Novel numerical method for solving variable-order fractional differential equations with power, exponential and Mittag–Leffler laws. Chaos Solitons Fractals 2018;115:175–85. [37] Upadhyay RK, Chattopadhyay J. Chaos to order: role of toxin producing phytoplankton in aquatic systems. J Nonlinear Anal Model Control 2005;10(2):383–96. [38] Dai C, Zhao M, Yu H. Dynamics induced by delay in a nutrient-phytoplankton model with diffusion. Ecol Complex 2016;26:29–36. [39] Atangana A. Non validity of index law in fractional calculus: a fractional differential operator with Markovian and non-Markovian properties. Physica A 2018;505:688–706. [40] Atangana A, Gómez-Aguilar JF. Decolonisation of fractional calculus rules: breaking commutativity and associativity to capture more natural phenomena. Eur Phys J Plus 2018;133:1–21. [41] Atangana A, Gómez-Aguilar JF. Fractional derivatives with no-index law property: application to chaos and statistics. Chaos Solitons Fractals 2018;114:516–35.
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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Frontiers
Modeling the dynamics of nutrient–phytoplankton–zooplankton system with variable-order fractional derivatives Behzad Ghanbari a, J.F. Gómez-Aguilar b,∗ a b
Department of Engineering Science, Kermanshah University of Technology, Kermanshah, Iran CONACyT-Tecnológico Nacional de México/CENIDET, Interior Internado Palmira S/N, Col. Palmira, C.P. 62490, Cuernavaca Morelos, México
a r t i c l e
i n f o
Article history: Received 31 August 2018 Revised 13 September 2018 Accepted 13 September 2018 Available online 22 September 2018 Keywords: Fractional calculus Variable-order fractional derivatives Nutrient–phytoplankton–zooplankton model Lagrange interpolation
a b s t r a c t We extended the nutrient–phytoplankton–zooplankton model involving variable-order fractional differential operators of Liouville–Caputo, Caputo–Fabrizio and Atangana–Baleanu. Variable-order fractional operators permits model and describe accurately real world problems, for example, diffusion or spread of nutrients or species in different states. Particularly, we model the interaction of nutrient phytoplankton and its predator zooplankton. The variable-order fractional numerical scheme based on the fundamental theorem of fractional calculus and the Lagrange polynomial interpolation was consider. Numerical simulation results are provided for illustrating the effectiveness and applicability of the algorithm to solve variable-order fractional differential equations.
1. Introduction Fractional derivatives become excellent instrument for the description of memory and hereditary properties of various materials and processes. Such effects are in fact neglected in models with classical integer-order. This can be viewed as the main advantage of fractional derivatives. It also plays a crucial role in the description of dynamics between two different points in many other fields [1–11]. Despite of the idea of fractional derivatives and integrals can be considered as a generalization of corresponding standard ones, it is still quite a strange topic, very hard to explain. Because, unlike commonly used differential operators, it is not related to some important geometrical meaning, such as the trend of functions or their convexity. So, sometimes this mathematical tool could be judged “far from reality”. But indeed many physical phenomena have “intrinsic” fractional order description and so fractional order calculus is necessary in order to explain them [1]. There are a several number of definitions of fractional derivatives. For instance, Riemann and Liouville introduced the concept of fractionalorder differentiation with power-law in [2,3]. Caputo and Fabrizio in [4], introduced a new derivative with fractional order based on the exponential-law and Atangana and Baleanu suggested another version of fractional-order derivative which uses the generalized Mittag-Leffler function with strong memory as non-local and nonsingular kernel in [5]. ∗
Corresponding author. E-mail address: [email protected] (J.F. Gómez-Aguilar).
https://doi.org/10.1016/j.chaos.2018.09.026 0960-0779/© 2018 Elsevier Ltd. All rights reserved.
© 2018 Elsevier Ltd. All rights reserved.
In most cases, exact solutions of differential equations with integer, non-integer order or variable-order derivatives are very difficult to obtain; this is the principal motivation to develop iterative methods or numerical techniques to solve these equations. For the iterative methods (Adomian decomposition method, the variational iteration method, fractional sub-equation method, homotopy perturbation techniques, among others) [12–17], the principal problem are the stability and the convergence. Traditionally, Adams– Bashforth method has been recognized as a great and powerful numerical method able to provide a numerical solution of fractional differential equations [18–25]. In [26], the authors approximated Liouville–Caputo fractional derivatives by Chebyshev polynomials. Rosenfeld and Dixon in [27] developed a numerical scheme based on scattered data interpolation via reproducing kernel Hilbert spaces to solved Liouville–Caputo fractional order differential equations. In [28], the authors proposed a new three-step fractional Adams–Bashforth scheme for solving linear and nonlinear fractional order differential equations involving the Caputo–Fabrizio operator. Shahbazi and Javidi considered 3/8 Simpsons rule to design a new high order predictor-corrector scheme [29]. Modifications combining the rectangle formula, trapezoid formula, polynomial interpolation or Gauss–Lobatto quadrature can be found in [30,31]. In [32-33], the authors developed a generalized version of Adams–Basforth method to partial differential equations involving Laplace transform, Lagrange polynomial interpolation and the forward-backward scheme. Recently, in [34–36], the authors developed a constant-order and variable-order numerical schemes that combines the fundamental theorem of fractional calculus and the two-step Lagrange polynomial.
B. Ghanbari, J.F. Gómez-Aguilar / Chaos, Solitons and Fractals 116 (2018) 114–120
In the biological models, interaction networks can be visualized as food-chains of species linked by trophic interactions. Phytoplankton provide food for marine life, oxygen for human being and purify the atmosphere by consuming carbon dioxide. Nevertheless, the rapid growth of phytoplankton may reduce the required amount of oxygen needed for the growth of other aquatic plants and animals. In the literature we found mathematical models to consider the interaction nutrient–phytoplankton–zooplankton. A mathematical model that describes three species food chain model consisting of toxin producing phytoplankton, zooplankton and fish population has been developed in [37]. In [38], the authors described a nutrient phytoplankton model by a couple of reactiondiffusion equations with delay. Biological systems presents longrange temporal memory or long-range space interactions, for this reason, the use of fractional derivatives can handle efficiently the dynamics of a disease model and also gives information on each point of the model. In [11] a fractional mathematical model for the interaction of nutrient phytoplankton and its predator zooplankton was investigated numerically. The fractional derivative of Liouville– Caputo type was used to obtain the generalization of the model. For solving the result fractional equations, a new numerical algorithm based on the polynomial interpolation was proposed. In this paper, we consider a variable-order fractional nutrient– phytoplankton–zooplankton system [11] via Liouville–Caputo, Caputo–Fabrizio–Caputo and Atangana–Baleanu–Caputo fractional derivatives. 2. Mathematical model The fractional nutrient–phytoplankton–zooplankton system [11] is generalized by replacing the classical derivative by the operator 0 Dtα (t ) α (t )
0 Dt
α (t )
0 Dt
α (t )
0 Dt
x1 (t ) = α0 − ax1 (t ) − b1 x1 (t )x2 (t ) + c1 x2 (t ) + c2 x3 (t ), (1a) x2 (t ) = b2 x1 (t )x2 (t ) − c3 x2 (t ) − x3 (t ) =
d1 x2 (t )x3 (t ) , e + x2 (t )
d2 x2 (t )x3 (t ) − f x2 (t )x3 (t ) − c4 x3 (t ), e + x2 (t )
(1b)
C α (t ) 0 Dt
f (t )=
x2 ( 0 ) = x2,0 > 0 ,
t
0
(t − τ )−α (t ) f˙ (τ )dτ ,
0 < α (t ) ≤ 1. (2)
CF C α (t ) 0 Dt
f (t ) =
(2 − α (t ))M (α (t )) t 2(1 − α (t )) 0 (t − τ ) ˙ × exp − α (t ) f ( τ )d τ , 1 − α (t )
0 < α (t ) < 1, (3)
where M (α (t )) = 2−α2 (t ) is a normalization function. The variable-order Atangana–Baleanu–Caputo fractional derivative with Mittag-Leffler (ABC) is defined as follows [5,6] ABC α (t ) 0 Dt
f (t ) =
B(α (t )) t (t − τ )α (t ) Eα (t ) − α (t ) 1 − α (t ) 0 1 − α (t ) × f˙ (τ )dτ , 0 < α (t ) ≤ 1,
where B(α (t )) = 1 − α (t ) +
(4)
α (t ) is a normalization function. (α (t ))
Now considering the numerical scheme developed in [36], we obtain numerical simulations for the nutrient–phytoplankton– zooplankton model in Liouville–Caputo; Caputo–Fabrizio–Caputo; and Atangana–Baleanu–Caputo fractional derivatives with variable order α (t). 3. Numerical schemes 3.1. Numerical scheme in Liouville–Caputo sense with variable-order A fractional ordinary differential equation of Liouville–Caputo type with variable-order can be expressed as follows: C α (t ) y 0 Dt
(t ) = f (t , y(t )).
(5)
The approximate solution of Eq. (5) is obtained as [36]
(1c)
yn+1 (t ) = y(0 ) +
or ABC Dtα (t ) , called Liouville–Caputo, Caputo–Fabrizio– 0 Caputo or Atangana–Baleanu–Caputo fractional derivatives with variable-order α (t), respectively. The variable-order Liouville–Caputo fractional derivative with power-law (C) is defined as follows [7]:
n
hα (t ) f (tm , ym ) (α (t )) m=0 α (t )(α (t ) + 1 ) 1
((n + 1 − m )α (t ) (n − m + 2 + α (t )) − (n − m )α (t ) hα (t ) f (tm−1 , ym−1 ) ×(n − m + 2 + 2α (t ) )) − α (t )(α (t ) + 1 )
x3 ( 0 ) = x3,0 > 0 ,
where α 0 , a, b1 , b2 , c1 , c2 , c3 , c4 , d1 , d2 , e and f are positive constants. In the above model x1 (t) denote the concentration of nutrient, x2 (t) denotes the biomass of phytoplankton which also produces toxicant harmful to the zooplankton biomass and x3 (t) denote the concentration of zooplankton population, the parameters {a, b1 , b2 , c1 , c2 , c3 , c4 , d1 , d2 , e, f, α 0 } represents the rate of nutrient loss; nutrient uptake rate for the phytoplankton population; nutrient-phytoplankton conversion rate; nutrient recycling rate after the death of phytoplankton; nutrient recycling rate after the death of zooplankton; phytoplankton mortality rate; zooplankton death rate, maximal zooplankton ingestion rate; maximal phytoplankton–zooplankton conversion rate; half saturation constant for a Holling type II functional response; rate of zooplankton decay due to toxin producing phytoplankton and the constant input nutrient concentration, respectively. More details of this model can be found in [11]. The variable-order operator 0 Dtα (t ) can be of type C0 Dtα (t ) ,
CF C D α (t ) t 0
The variable-order Caputo–Fabrizio fractional derivative with exponential-law in Liouville–Caputo sense (CFC) is defined as follows [36]:
with initial conditions
x1 ( 0 ) = x1,0 > 0 ,
1 (1 − α (t ))
115
×((n + 1 − m )α (t )+1 − (n − m )α (t ) (n − m + 1 + α (t ))) . (6) 3.2. Numerical scheme in Caputo–Fabrizio–Caputo sense with variable-order Now, we have the following fractional differential equation with variable-order in Caputo–Fabrizio–Caputo sense CF C α (t ) y 0 Dt
(t ) = f (t , y(t )).
(7)
The numerical solution fo Eq. (7) is obtained by the following expression [36]
(2−α (t ))(1 − α (t )) 3h
α (t )(2−α (t )) f (tn , yn ) (2 − α (t ))(1 − α (t )) h − + α (t )(2 − α (t )) f (tn−1 , yn−1 ).
yn+1 = yn +
2
2
+
4
4
(8)
116
B. Ghanbari, J.F. Gómez-Aguilar / Chaos, Solitons and Fractals 116 (2018) 114–120
3.3. Numerical scheme in Atangana–Baleanu–Caputo sense with variable-order A fractional ordinary differential equation of Atangana–Baleanu–Caputo type with variable-order can be expressed as follows: ABC α (t ) y 0 Dt
(t ) = f (t , y(t )).
(9)
and the numerical solution of Eq. (9) is obtained by the following expression [36]
(α (t ))(1 − α (t )) f (t , y(tn )) (α (t ))(1 − α (t )) + α (t ) n n
1 + hα (t ) f (tm , ym ) ((n + 1 − m )α (t ) (α (t ) + 1 )((1 − α (t ))(α (t )) + α (t ) m=0
yn+1 (t ) = y0 +
(10)
× (n − m + 2 + α (t )) − (n − m )α (t ) (n − m + 2 + 2α (t ))) − hα (t ) f (tm−1 , ym−1 )((n + 1 − m )α (t )+1 − (n − m )α (t ) (n − m + 1 + α (t )))).
4. Numerical results In this section, we present the numerical solutions of the nutrient–phytoplankton–zooplankton model with Liouville–Caputo, Caputo– Fabrizio and Atangana–Baleanu fractional derivatives. Numerical simulations are showed for different values of fractional order.
4.1. Numerical scheme in Liouville–Caputo sense with variable-order Applying the approximate solution given by Eq. (6), we obtain the following equations:
x1,n+1 (t ) = x1,0 +
1
n
(α (t )) m=0
hα (t ) f1 (tn , x1,m , x2,m , x3,m ) ((n + 1 − m )α (t ) (n − m + 2 + α (t )) α (t )(α (t ) + 1 )
− (n − m )α (t ) (n − m + 2 + 2α (t ))) −
hα (t ) f1 (tm−1 , x1,m−1 , x2,m−1 , x3,m−1 ) ( (n + 1 − m )α (t )+1 α (t )(α (t ) + 1 )
− (n − m )α (t ) (n − m + 1 + α (t ))) ,
x2,n+1 (t ) = x2,0 +
1
n
(α (t )) m=0
hα (t ) f2 (tn , x1,m , x2,m , x3,m ) ((n + 1 − m )α (t ) (n − m + 2 + α (t )) α (t )(α (t ) + 1 )
− (n − m )α (t ) (n − m + 2 + 2α (t ))) −
hα (t ) f2 (tm−1 , x1,m−1 , x2,m−1 , x3,m−1 ) ( (n + 1 − m )α (t )+1 α (t )(α (t ) + 1 )
(11)
− (n − m )α (t ) (n − m + 1 + α (t ))) ,
x3,n+1 (t ) = x3,0 +
1
n
(α (t )) m=0
hα (t ) f3 (tn , x1,m , x2,m , x3,m ) ((n + 1 − m )α (t ) (n − m + 2 + α (t )) α (t )(α (t ) + 1 )
− (n − m )α (t ) (n − m + 2 + 2α (t ))) −
hα (t ) f3 (tm−1 , x1,m−1 , x2,m−1 , x3,m−1 ) ( (n + 1 − m )α (t )+1 α (t )(α (t ) + 1 )
− (n − m )α (t ) (n − m + 1 + α (t ))) , where
α0 − ax1 (t ) − bx1 (t )x2 (t ) + c1 x2 (t ) + c2 x3 (t ), d1 x2 (t )x3 (t ) d2 x2 (t )x3 (t ) f2 (t, x1 (t ), x2 (t ), x3 (t )) := b2 x1 (t )x2 (t ) − c3 x2 (t ) − , f3 (t, x1 (t ), x2 (t ), x3 (t )) := − f x2 (t )x3 (t ) − c4 x3 (t ). (12) e + x2 (t ) e + x2 (t ) f1 (t, x1 (t ), x2 (t ), x3 (t )) :=
4.2. Numerical scheme in Caputo–Fabrizio–Caputo sense with variable-order The numerical solution of system (1) in Caputo–Fabrizio–Caputo sense is given by the following equations:
B. Ghanbari, J.F. Gómez-Aguilar / Chaos, Solitons and Fractals 116 (2018) 114–120
x1,n+1 (t ) = x1,n +
−
2
(2 − α (t ))(1 − α (t )) 2
x2,n+1 (t ) = x2,n +
−
2 2
+
3h α (t )(2 − α (t )) f2 (tn , x1,n , x2,n , x3,n ) 4
+
h α (t )(2 − α (t )) f2 (tn−1 , x1n−1 (t ), x2n−1 (t ), x3n−1 (t )), 4
(2 − α (t ))(1 − α (t ))
(2 − α (t ))(1 − α (t ))
3h α (t )(2 − α (t )) f1 (tn , x1,n , x2,n , x3,n ) 4
h α (t )(2 − α (t )) f1 (tn−1 , x1n−1 (t ), x2n−1 (t ), x3n−1 (t )), 4
2 2
+
+
(2 − α (t ))(1 − α (t ))
(2 − α (t ))(1 − α (t ))
x3,n+1 (t ) = x3,n + −
(2 − α (t ))(1 − α (t ))
117
(13)
+
3h α (t )(2 − α (t )) f3 (tn , x1,n , x2,n , x3,n ) 4
+
h α (t )(2 − α (t )) f3 (tn−1 , x1n−1 (t ), x2n−1 (t ), x3n−1 (t ) ), 4
where f1 (t, x1 (t), x2 (t), x3 (t)), f2 (t, x1 (t), x2 (t), x3 (t)) and f3 (t, x1 (t), x2 (t), x3 (t)) are given by Eq. (12).
4.3. Numerical scheme in Atangana–Baleanu–Caputo sense with variable-order In the Atangana–Baleanu–Caputo sense, the numerical representation to the system (1) is obtained as follows:
(α (t ))(1 − α (t )) f (t , x , x , x ) (α (t ))(1 − α (t )) + α (t ) 1 n 1,n 2,n 3,n n
1 + hα (t ) f1 (tn , x1,m , x2,m , x3,m ) (α (t ) + 1 )((1 − α (t ))(α (t )) + α (t ) m=0
x1,n+1 (t ) = x1,0 +
× ((n + 1 − m )α (t ) (n − m + 2 + α (t )) − (n − m )α (t ) (n − m + 2 + 2α (t ))) − hα (t ) f1 (tm−1 , x1,m−1 , x2,m−1 , x3,m−1 ) × ((n + 1 − m )α (t )+1 − (n − m )α (t ) (n − m + 1 + α (t )))),
(α (t ))(1 − α (t )) f (t , x , x , x ) (α (t ))(1 − α (t )) + α (t ) 2 n 1,n 2,n 3,n n
1 + hα (t ) f2 (tn , x1,m , x2,m , x3,m ) (α (t ) + 1 )((1 − α (t ))(α (t )) + α (t ) m=0
x2,n+1 (t ) = x2,0 +
× ((n + 1 − m )α (t ) (n − m + 2 + α (t )) − (n − m )α (t ) (n − m + 2 + 2α (t ))) − hα (t ) f2 (tm−1 , x1,m−1 , x2,m−1 , x3,m−1 ) × ((n + 1 − m )α (t )+1 − (n − m )α (t ) (n − m + 1 + α (t )))),
(α (t ))(1 − α (t )) f (t , x , x , x ) (α (t ))(1 − α (t )) + α (t ) 3 n 1,n 2,n 3,n n
1 + hα (t ) f3 (tn , x1,m , x2,m , x3,m ) (α (t ) + 1 )((1 − α (t ))(α (t )) + α (t ) m=0
x3,n+1 (t ) = x3,0 +
× ((n + 1 − m )α (t ) (n − m + 2 + α (t )) − (n − m )α (t ) (n − m + 2 + 2α (t ))) − hα (t ) f3 (tm−1 , x1,m−1 , x2,m−1 , x3,m−1 )× × ((n + 1 − m )α (t )+1 − (n − m )α (t ) (n − k + 1 + α (t )))),
(14)
118
B. Ghanbari, J.F. Gómez-Aguilar / Chaos, Solitons and Fractals 116 (2018) 114–120
Numerical solutions of different schemes for Example 1 with alpha(t)= 0.85
Fig. 3. Numerical simulations for example 1 in Atangana–Baleanu–Caputo sense. Fig. 1. Numerical simulations of example 1 for Liouville–Caputo, Caputo–Fabrizio– Caputo and Atangana–Baleanu fractional derivatives with α (t ) = 0.85.
Fig. 4. Numerical simulations of example 2 for Liouville–Caputo, Caputo–Fabrizio– Caputo and Atangana–Baleanu–Caputo fractional derivatives with α (t ) = 0.95.
Fig. 2. Numerical simulations for example 1 in Liouville–Caputo sense.
where f1 (t, x1 (t), x2 (t), x3 (t)), f2 (t, x1 (t), x2 (t), x3 (t)) and f3 (t, x1 (t), x2 (t), x3 (t)) are given by Eq. (12). Example 1. For the following set of parameters: α0 = 0.5, a = 1, b1 = 0.3, b2 = 0.25, c1 = 0.06, c2 = 0.06, c3 = 0.2, c4 = 0.5, d1 = 2.1, d2 = 0.2, e = 1, f = 0.1. Figs. 1–3, show the dynamics of the nutrient–phytoplankton–zooplankton model considering different fractional orders α (t) arbitrarily chosen. The time simulation was t = 60 [s] and step size h = 1 × 10−2 . Example 2. For the following set of parameters: α0 = 1.4, a = 0.7, b1 = 3.2, b2 = 2.4, c1 = 0.06, c2 = 0.06, c3 = 0.68, c4 = 0.21, d1 = 2.1, d2 = 1, e = 1, f = 0.1. Figs. 4–6, show the dynamics of the nutrient–phytoplankton–zooplankton model considering
Fig. 5. Numerical simulations of example 2 for Liouville–Caputo, Caputo–Fabrizio– Caputo and Atangana–Baleanu–Caputo fractional derivatives with α (t ) = 1/(1 + exp(−t ).
B. Ghanbari, J.F. Gómez-Aguilar / Chaos, Solitons and Fractals 116 (2018) 114–120
119
of fractional derivatives with singular power-law, based in this comments, we can conclude the index law is not valid in fractional differentiation. Competing interests The authors declare that there is no conflict of interests regarding the publication of this paper. Authors’ contributions All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript. Acknowledgments
Fig. 6. Numerical simulations of example 2 for Liouville–Caputo, Caputo–Fabrizio– Caputo and Atangana–Baleanu–Caputo fractional derivatives with α (t ) = 0.97 + 0.03 sin(t/10 ).
The authors are grateful to all of the anonymous reviewers for their valuable suggestions. José Francisco Gómez Aguilar acknowledges the support provided by CONACyT: Cátedras CONACyT para jóvenes investigadores 2014 and SNI-CONACyT. References
different fractional orders α (t) arbitrarily chosen. The time simulation was t = 60 [s] and step size h = 1 × 10−2 . The graphical representations show that the model depends notably to the fractional-order and the selected system parameters. In contrast to the constant-order nutrient–phytoplankton– zooplankton model, the memory effect change with time. This means that the memory rate of system is determined by the current time instant. For different time periods we have different memory abilities. In consequence, the constant-order systems permits characterizing the long memory of systems, while the variable-order systems can be used to characterize variable memory effect and capture the areas of transition between dynamic regimes of the physical phenomena. 5. Conclusions Based in the fundamental theorem of fractional calculus and the two-step Lagrange polynomial, we obtain numerical solutions for variable-order fractional nutrient–phytoplankton–zooplankton system. Numerical examples with different variable-orders have been presented to demonstrate the effectiveness of the method. The results obtained with the Liouville–Caputo derivative (based on the singular power-law) not completely describe the memory effects. The Caputo–Fabrizio and Atangana–Baleanu fractional derivatives consider a non-singular kernel based on the exponential-law and the Mittag-Leffler law (which of course the generalization of exponential function). These kernels permits describe the memory of the systems without the inclusion of artificial singularities into the mathematical models, that is to say, the history of the dynamical process are fully described. The generalized Mittag-Leffler function with strong memory involved in the Atangana–Baleanu fractional derivative is a best filter than the exponential and powerlaw functions. Furthermore, the Atangana–Baleanu fractional order derivative is at the same time Liouville–Caputo and Caputo– Fabrizio thus possesses Markovian and non-Markovian properties. These operators describe different waiting times distribution as such is observed in the ecological interactions. The Caputo–Fabrizio and Atangana–Baleanu derivatives not obeying the index law imposed in fractional calculus [39–41]. Due to this apparent limitation, these non-singular operators present crossover behavior and permits describe more appropriate real world problems than that
[1] Caponetto R, Dongola G, Fortuna L, Petrás I. World scientific series on nonlinear science series A. Fractional order systems modeling and control applications; 2010. [2] Samko SG, Kilbas AA, Marichev OI. Fractional integrals and derivatives: theory and applications. Yverdon: Gordon and Breach; 1993. [3] Podlubny I. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Fractional differential equations. San Diego, Calif, USA: Academic Press; 1999. [4] Caputo M, Fabrizio M. A new definition of fractional derivative without singular kernel. Progr Fract Differ Appl 2015;1:73–85. [5] Atangana A, Baleanu D. New fractional derivatives with nonlocal and non-singular kernel. Theory Appl Heat Transf Model Therm Sci 2016;20(2):763–9. [6] Yang XJ. Fractional derivatives of constant and variable orders applied to anomalous relaxation models in heat-transfer problems. Therm Sci 2017;21(3):1161–71. [7] Moghaddam BP, Yaghoobi S, Machado JT. An extended predictor-corrector algorithm for variable-order fractional delay differential equations. J Comput Nonlinear Dyn 2016;1:1–11. [8] Moghaddam BP, Yaghoobi S, Machado JT. An extended predictor-corrector algorithm for variable-order fractional delay differential equations. J Comput Nonlinear Dyn 2016;1:1–11. [9] Khare S, Misra O, Dhar J. Role of toxin producing phytoplankton on a plankton ecosystem. Nonlinear Anal Hybrid Syst 2010;4(3):496–502. [10] Chatterjee A, Pal S, Chatterjee S. Bottom up and top down effect on toxin producing phytoplankton and its consequence on the formation of plankton bloom. Appl Math Comput 2011;218(7):3387–98. [11] Asl MS, Javidi M. Novel algorithms to estimate nonlinear FDEs: applied to fractional order nutrient–phytoplankton–zooplankton system. J Comput Appl Math 2018;339:193–207. [12] Duan JS, Rach R, Wazwaz AM. Steady-state concentrations of carbon dioxide absorbed into phenyl glycidyl ether solutions by the adomian decomposition method. J Math Chem 2015;53(4):1054–67. [13] Lu L, Duan J, Fan L. Solution of the magnetohydrodynamics Jeffery–Hamel flow equations by the modified adomian decomposition method. Adv Appl Math Mech 2015;7(05):675–86. [14] Ma HC, Yao DD, Peng XF. Exact solutions of non-linear fractional partial differential equations by fractional sub-equation method. Thermal Sci 2015;19(4):1239–44. [15] Guo S, Mei L. Exact solutions of space-time fractional variant Boussinesq equations. Adv Sci Lett 2012;10(1):700–2. [16] Mohyud-Din ST, Nawaz T, Azhar E, Akbar MA. Fractional sub-equation method to space-time fractional Calogero–Degasperis and potential Kadomtsev–Petviashvili equations. J Taibah Univ Sci 2017;11(2):258–63. [17] Jafari H, Ghorbani M, Ghasempour S. A note on exact solutions for nonlinear integral equations by a modified homotopy perturbation method. New Trends Math Sci 2013;1(2):22–6. [18] Diethelm K, Ford NJ, Freed AD. A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn 2002;29(1–4):3–22. [19] Diethelm K, Ford NJ, Freed AD. Detailed error analysis for a fractional adams method. Numer Algorithms 2004;36(1):31–52. [20] Galeone L, Garrappa R. Fractional Adams–Moulton methods. Math Comput Simul 2008;79(4):1358–67. [21] Li C, Tao C. On the fractional adams method. Comput Math Appl 2009;58(8):1573–88.
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B. Ghanbari, J.F. Gómez-Aguilar / Chaos, Solitons and Fractals 116 (2018) 114–120
[22] Daftardar-Gejji V, Jafari H. Analysis of a system of non autonomous fractional differential equations involving Caputo derivatives. J Math Anal Appl 2007;328:1026–33. [23] Daftardar-Gejji V, Sukale Y, Bhalekar S. A new predictor-corrector method for fractional differential equations. Appl Math Comput 2014;244:158–82. [24] Odibat ZM, Momani S. Application of variational iteration method to nonlinear differential equations of fractional order. Int J Nonlinear Sci Numer Simul 2006;7(1):27–34. [25] Lin R, Liu F, Anh V, Turner I. Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation. Appl Math Comput 2009;212(2):435–45. [26] Sugiura H, Hasegawa T. Quadrature rule for Abel’s equations: uniformly approximating fractional derivatives uniformly approximating fractional derivatives. J Comput Appl Math 2009;223:459–68. [27] Rosenfeld JA, Dixon WE. Approximating the Caputo fractional derivative through the Mittag–Leffler reproducing kernel hilbert space and the kernelized Adams–Bashforth–Moulton method. SIAM J Numer Anal 2017;55(3):1201–17. [28] Owolabi KM, Atangana A. Analysis and application of new fractional Adams–Bashforth scheme with Caputo–Fabrizio derivative. Chaos Solitons Fractal 2017;105:111–19. [29] Asl MS, Javidi M. An improved PC scheme for nonlinear fractional differential equations: error and stability analysis. J Comput Appl Math 2017;324:101–17. [30] Li C, Yi Q, Chen A. Finite difference methods with non-uniform meshes for nonlinear fractional differential equations. J Comput Phys 2016;316:614–31. [31] Zhao L, Deng W. Jacobian-predictor-corrector approach for fractional differential equations. Adv Comput Math 2014;40(1):137–65.
[32] Gnitchogna R, Atangana A. New two step laplace Adam–Bashforth method for integer a noninteger order partial differential equations. Numer Methods Partial Differ Equ 2017;34(5):1739–58. [33] Alkahtani BST. Atangana–Batogna numerical scheme applied on a linear and non-linear fractional differential equation. Eur Phys J Plus 2018;133(3):1–11. [34] Atangana A, Owolabi KM. New numerical approach for fractional differential equations. Math Model Nat Phenom 2018;13(1):1–17. [35] Toufik M, Atangana A. New numerical approximation of fractional derivative with non-local and non-singular kernel: application to chaotic models. Eur Phys J Plus 2017;132(10):1–16. [36] Solís-Pérez JE, Gómez-Aguilar JF, Atangana A. Novel numerical method for solving variable-order fractional differential equations with power, exponential and Mittag–Leffler laws. Chaos Solitons Fractals 2018;115:175–85. [37] Upadhyay RK, Chattopadhyay J. Chaos to order: role of toxin producing phytoplankton in aquatic systems. J Nonlinear Anal Model Control 2005;10(2):383–96. [38] Dai C, Zhao M, Yu H. Dynamics induced by delay in a nutrient-phytoplankton model with diffusion. Ecol Complex 2016;26:29–36. [39] Atangana A. Non validity of index law in fractional calculus: a fractional differential operator with Markovian and non-Markovian properties. Physica A 2018;505:688–706. [40] Atangana A, Gómez-Aguilar JF. Decolonisation of fractional calculus rules: breaking commutativity and associativity to capture more natural phenomena. Eur Phys J Plus 2018;133:1–21. [41] Atangana A, Gómez-Aguilar JF. Fractional derivatives with no-index law property: application to chaos and statistics. Chaos Solitons Fractals 2018;114:516–35.