Applied Mathematics and Computation 218 (2011) 1089–1095

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate/amc

Fractional dynamics of populations

Margarita Rivero a, Juan J. Trujillo c,⇑, Luis Vázquez b, M. Pilar Velasco b

a Departamento de Matemática Fundamental, Universidad de La Laguna, 38271 La Laguna, Tenerife, Spainb Departamento de Matemática Aplicada, Universidad Complutense de Madrid, 28040 Madrid, Spainc Departamento de Análisis Matemático, Universidad de La Laguna, 38271 La Laguna, Tenerife, Spain

a r t i c l e i n f o

Dedicated to Professor H. M. Srivastavaon the Occasion of his Seventieth BirthAnniversary

Keywords:Fractional differential equationsDynamic systemsCritical pointsPhase plane

0096-3003/$ - see front matter � 2011 Elsevier Incdoi:10.1016/j.amc.2011.03.017

⇑ Corresponding author.E-mail addresses: jtrujill@ullmat.es (J.J. Trujillo),

a b s t r a c t

Nature often presents complex dynamics, which cannot be explained by means of ordinarymodels. In this paper, we establish an approach to certain fractional dynamic systems usingonly deterministic arguments. The behavior of the trajectories of fractional non-linearautonomous systems around the corresponding critical points in the phase space is stud-ied. In this work we arrive to several interesting conclusions; for example, we concludethat the order of fractional derivation is an excellent controller of the velocity how thementioned trajectories approach to (or away from) the critical point. Such property couldcontribute to faithfully represent the anomalous reality of the competition among somespecies (in cellular populations as Cancer or HIV). We use classical models, which describedynamics of certain populations in competition, to give a justification of the possible inter-est of the corresponding fractional models in biological areas of research.

� 2011 Elsevier Inc. All rights reserved.

1. Introduction

Fractional calculus emerged from the interest in generalizing the ordinary integrals and derivatives. Different derivativesand integrals of arbitrary (fractional, real or complex) order have been studied. For example, in [8,5] the authors show thatthe fractional calculus constitutes a meeting place of multiple disciplines: stochastic processes, probability, integro-differen-tial equations, integral transforms, special functions, numerical analysis, etc.

Here, we pay attention to fractional differential equations and systems of equations, that is, equations with derivatives ofreal or complex order ([1,5,6]):

F t;XðtÞ; ðDa1 XÞðtÞ; . . . ; ðDan XÞðtÞ� �

¼ 0;

where a1 < a2 < . . . < an and Dai is a fractional derivative of ai order (similarly fractional partial differential equations or sys-tems of equations can be defined).

Such fractional models allow us to consider new complex situations using the behaviors of the trajectories and, therefore,of the solutions of the mentioned models. Moreover, we can study different scales of time and/or space and then problemswith anomalous dynamics. We must remark that the fractional operators are non-local, such characteristic give them capac-ity of memory to the models that involve them.

We consider the dynamic of populations to explain a new deterministic approach to the fractional models with the objec-tive of adding non-locality characteristics to the corresponding classical models. We must remark that in most of the papersdealing with fractional models the authors do not give any justification, although for some subdiffusive models exist a inter-esting approach from a known stochastic point of view.

. All rights reserved.

lvazquez@fdi.ucm.es (L. Vázquez), mvcebrian@mat.ucm.es (M. Pilar Velasco).

1090 M. Rivero et al. / Applied Mathematics and Computation 218 (2011) 1089–1095

The main objective of this paper is to introduce a new approach to the fractional differential systems. The behavior of thetrajectories in the phase space of such systems around the corresponding critical points is studied. In Section 2 the fractionalframework is introduced. The main results are presented in the Section 3.

2. Fractional framework

In this section we introduce some fractional operators, along with properties that will be used in our discussion [7,5].

Definition 2.1 (Riemann–Liouville fractional operator). Let a > 0; n� 1 < a < n; n 2 N; ½a; b� � R and f a suitable realfunction (for example, f 2 L1ða; bÞ):

ðIaaþf ÞðxÞ ¼ 1CðaÞ

Z x

aðx� tÞa�1f ðtÞdt; ðDa

aþf ÞðxÞ ¼ DnðIn�aaþ f ÞðxÞ; ð1Þ

ðIab�f ÞðxÞ ¼ 1CðaÞ

Z b

xðt � xÞa�1f ðtÞdt; ðDa

b�f ÞðxÞ ¼ DnðIn�ab� f ÞðxÞ; ð2Þ

where D is the usual differential operator and x > a in (1), x < b in (2).

Definition 2.2 (Caputo fractional derivative). Let a > 0; n� 1 < a < n; n 2 N; ½a; b� � R and f a suitable real function (forexample, f 2 L1ða; bÞ):

CDaaþf

� �ðxÞ ¼ In�a

aþ Dnf� �

ðxÞ ðx > aÞ: ð3Þ

For a suitable function f (for example, f n-times derivable ):

Daaþf

� �ðxÞ ¼ CDa

aþf� �

ðxÞ þXn�1

j¼0

f ðjÞðaÞCð1þ j� aÞ ðx� aÞj�a

: ð4Þ

Thus, we have:

CDaaþ1

� �¼ 0; Da

aþ1� �

¼ ðx� aÞ�a

Cð1� aÞ : ð5Þ

Definition 2.3 (Mittag–Leffler functions). Let be a; b > 0

EaðzÞ ¼X1k¼0

zk

Cðakþ 1Þ ; Ea;bðzÞ ¼X1k¼0

zk

Cðakþ bÞ : ð6Þ

Definition 2.4. A function of type Mittag–Leffler:

ekza :¼ za�1Ea;aðkzaÞ ða > 0; k 2 CÞ: ð7Þ

These functions allow us to generalize the classical exponential function. Such generalized functions have the followingproperties:

Proposition 2.5. Let a > 0 and k 2 C, then:

CDa0þEaðktaÞ ¼ kEaðktaÞ; Da

0þekta ¼ kekt

a : ð8Þ

Proposition 2.6. Eað�kxÞ and e�kxa , with k > 0 and 0 < a < 1, behave like negative exponential functions evolving more slowly

than the exponential functions; however with 1 < a < 2 it has oscillator character (Fig. 1).

3. Species competition

A classic model for competition between two species with population densities x and y is:

dxdt¼ xðk1 � r1x� a1yÞ; ð9Þ

dydt¼ yðk2 � r2y� a2xÞ; ð10Þ

where k1; k2; r1; r2; a1; a2 2 Rþ. This model has two hypotheses [2]:

0 2 4 6 8 1000.10.20.30.40.50.60.70.80.9

10<α<1

x

E α(−x)

0 10 20 30 40 50−1−0.8−0.6−0.4−0.2

00.20.40.60.8

1 1<α<2

x

E α(−x)

0 2 4 6 8 1000.10.20.30.40.50.60.70.80.9

10<α<1

x

e α−x

0 10 20 30 40 50−1−0.8−0.6−0.4−0.2

00.20.40.60.8

11<α<2

x

e α−x

Fig. 1. Graphics of Eað�xÞ and e�xa with a = 0.5(�), 0.7(�), 0.9(:) (pictures 1 and 3 resp.), and with a = 1.5(�), 1.7(�), 1.9(:) (pictures 2 and 4 resp).

M. Rivero et al. / Applied Mathematics and Computation 218 (2011) 1089–1095 1091

1. The population densities x and y grow proportionally to themselves, that is, dxdt ¼ k1xðtÞ; dy

dt ¼ k2yðtÞ, or equivalently,xðtÞ ¼ C1ek1t ; yðtÞ ¼ C2ek2t (classical exponential law). We will modify this hypothesis.

2. When one population grows too much, then the other one decreases, and vice versa. This hypothesis justifies the nonlinearterms of the type xy. In addition, the environment is limited by nature, so that the populations can not grow infinitely. Thishypothesis justifies the nonlinear terms of the type x2 and y2. We will conserve these hypotheses of the nonlinear terms.

Understanding the above first hypothesis such that the growth of populations must follow an exponential function, but notnecessarily the classical exponential function, we can get new degree of freedom. Therefore, for instance, the populations xand y could grow as the generalized exponential functions mentioned above, for instance:

xðtÞ ¼ C1Eaðk1taÞ () CDa0þx

� �ðtÞ ¼ k1xðtÞ;

yðtÞ ¼ C2Ebðk2tbÞ () CDb0þy

� �ðtÞ ¼ k2yðtÞ;

where C1; C2; k1 and k2 are real constants and 0 < a; b 6 1. This option allows us to introduce in the model the parametersa and b that will be used as efficient controllers of the growth rate of the populations, when the medium where the popu-lations live is too complex.

Then, we arrive directly to the following generalized fractional model:

CDa

0þx ¼ xðk1 � r1x� a1yÞ; ð11ÞCDb

0þy ¼ yðk2 � r2y� a2xÞ; ð12Þ

where k1; k2; r1; r2; a1 and a2 are positive real constants.

By example, the carcinogenic cells have a uncontrolled growth and, on the contrary, in unfavorable environmental situ-ations (wars, epidemics. . .) the human population grows slowly. These anomalous situations demand a different expositionof this problem in many cases, for example fractional differential systems.

Let us consider the following specific example:

CDa

0þx ¼ xð1� x� yÞ; ð13ÞCDb

0þy ¼ y12� 1

4y� 3

4x

� �; ð14Þ

and let us suppose that b ¼ ra, with r 2 N. Then, this system has four isolated critical points that we will study and comparewith the ordinary case:

3.1. Case ðx0; y0Þ=(0,0)

For this point, the original system is a almost linear system and we can consider the associated linear case:

CDa0þx ¼ x; ð15Þ

CDb0þy ¼ 1

2y: ð16Þ

Keeping b ¼ ra in mind, we can rewrite the system:

CDa0þ

x

y1

..

.

yr

0BBBB@

1CCCCA ¼

1 0 0 . . . 00 0... ..

.Ir�1

0 00 1=2 0 . . . 0

0BBBBBB@

1CCCCCCA

x

y1

..

.

yr

0BBBB@

1CCCCA; ð17Þ

where y1 ¼ y and yiþ1 ¼ CDa0þyi; i ¼ 1; . . . ; r � 1.

Using Diethelm’s numerical algorithm ([3,4]) to solve this system, we obtain different phase planes, Fig. 2. The point (0,0)is an unstable improper node. For the fractional case a 6 b < 1 the trajectories of the solutions move away of (0,0) more

1092 M. Rivero et al. / Applied Mathematics and Computation 218 (2011) 1089–1095

quickly in the X axis and more slowly in the Y axis that in the ordinary case. For a < 1 < b the solutions evolve more quicklyin both X and Y axis, and for 1 < a 6 b the trajectories move more slowly in the X axis and more quickly in the Y axis incomparison to the ordinary case.

3.2. Case ðx0; y00=(1,0)

Making the change of variable x ¼ 1þ u and y ¼ 0þ v , we transfer the critical point to the origin and we obtain a almostlinear system whose associated linear case is:

Fig. 2.(–), a ¼

CDa0þu ¼ �u� v ; ð18Þ

CDb0þv ¼ �

14

v: ð19Þ

Keeping b ¼ ra in mind, as in the first case we can write the corresponding system with r equations of order a. Thereforesolving such system for different values of a, we obtain the phase planes (Fig. 3). The point (1,0) is a stable improper nodebecause the trajectories approach to the critical point. Moreover, in the fractional case the solutions move more slowly thanin the ordinary case for a 6 b < 1, whereas for higher values of a the velocity of trajectories increases ða < 1 < bÞ, turninginto strange trajectories for 0 < 1 < a 6 b.

3.3. Case ðx0; y0Þ=(0,2)

Making the change of variable x ¼ 0þ u and y ¼ 2þ v , we transfer the critical point to the origin and we obtain a almostlinear system whose associated linear case is:

CDa0þu ¼ �u; ð20Þ

CDb0þv ¼ �

12

v � 32

u: ð21Þ

Keeping b ¼ ra in mind, as in the first case we can write the corresponding system with r equations of order a. Thereforesolving such system for different a; b, we can represent the phase planes (Fig. 4). Similarly to the previous case, the point(0,2) is a stable improper node and the trajectories approach to the critical point more slowly for a 6 b < 1 and more quicklyfor a < 1 < b in the fractional case than in the ordinary one. However, for 1 < a 6 b the trajectories of the solutions have achaotic behavior.

3.4. Case ðx0; y0Þ ¼ 12 ;

12

� �

Making the change of variable x ¼ 12þ u and y ¼ 1

2þ v , we transfer the critical point to the origin and we obtain a almostlinear system whose associated linear case is:

CDa0þu ¼ �1

2u� 1

2v ; ð22Þ

CDb0þv ¼ �

38

u� 18

v; ð23Þ

with eigenvalues k1 ¼ �5þffiffiffiffi57p

16 ; k2 ¼ �5�ffiffiffiffi57p

16 .Keeping b ¼ ra in mind, as in the first case we can write the corresponding system with r equations of order a. Therefore

solving such system, we obtain different phase planes in according to the values of the parameters a; b (Fig. 5). The critical

0 1 2 3 4 5 6 7 8x 104

0

100

200

300

400

500

600

700

800

x

y

0 1 2 3 4 5 6 7 8x 104

0

50

100

150

200

250

x

y

Phase plane of (0, 0) for ordinary case (-) and fractional case: a ¼ 0:3; b ¼ 0:6 (–), a ¼ 0:6; b ¼ 1:2 (:), a ¼ 1:2; b ¼ 2:4 (- � -) in picture 1; a ¼ b ¼ 0:3b ¼ 0:6 (:), a ¼ b ¼ 1:2 (- � -) in picture 2.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

x

y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

x

y

Fig. 3. Phase plane of (1,0) for ordinary case (-) and fractional case: a ¼ 0:3; b ¼ 0:6 (–), a ¼ 0:6; b ¼ 1:2 (:), a ¼ 1:2; b ¼ 2:4 (- � -) in picture 1; a ¼ b ¼ 0:3(–), a ¼ b ¼ 0:6 (:), a ¼ b ¼ 1:2 (- � -) in picture 2.

0 0.2 0.4 0.6 0.8 1 1.20

1

2

3

4

5

6

7

8

x

y

0 0.2 0.4 0.6 0.8 1 1.20

1

2

3

4

5

6

7

8

x

y

Fig. 4. Phase plane of (2,0) for ordinary case (-) and fractional case: a ¼ 0:3; b ¼ 0:6 (–), a ¼ 0:6; b ¼ 1:2 (:), a ¼ 1:2; b ¼ 2:4 (- � -) in picture 1; a ¼ b ¼ 0:3(–), a ¼ b ¼ 0:6 (:), a ¼ b ¼ 1:2 (- � -) in picture 2.

M. Rivero et al. / Applied Mathematics and Computation 218 (2011) 1089–1095 1093

point 12 ;

12

� �is an unstable saddle point and the solutions move more slowly in the fractional case a 6 b < 1 than in the or-

dinary case. Increasing the values of a and b in the fractional case, we observe that the trajectories have a higher velocity thatin the ordinary case.

4. Predator–Prey

Other important model is the predator–prey case or Lotka–Volterra model [2]:

Fig. 5.a ¼ b ¼

dxdt¼ xðk1 � a1yÞ; ð24Þ

dydt¼ yð�k2 þ a2xÞ: ð25Þ

Here, we consider this model under anomalous situations in the growth of these populations. For this reason, we introducefractional operators which allow us to control the velocity of the trajectories of the system.

0 0.1 0.2 0.3 0.4 0.5 0.62

2.05

2.1

2.15

2.2

2.25

2.3

2.35

2.4

2.45

2.5

x

y

0 0.1 0.2 0.3 0.4 0.5 0.62

2.05

2.1

2.15

2.2

2.25

2.3

2.35

2.4

2.45

2.5

x

y

Phase plane of (1/2,1/2) for ordinary case (-) and fractional case: a ¼ 0:3; b ¼ 0:6 (–), a ¼ 0:6; b ¼ 1:2 (:), a ¼ 1:2; b ¼ 2:4 (- � -) in picture 1;0:3 (–), a ¼ b ¼ 0:6 (:), a ¼ b ¼ 1:2 (- � -) in picture 2.

1094 M. Rivero et al. / Applied Mathematics and Computation 218 (2011) 1089–1095

So, let us study the following differential system:

Fig. 6.(–), a ¼

CDa0þx ¼ xð1� yÞ; ð26Þ

CDb0þy ¼ yð�1þ xÞ; ð27Þ

where b ¼ ra, with r a natural number. Its critical points are (0,0) and (1,1), that we will study separately.

4.1. Case ðx0; y0Þ=(0,0)

For this point, the original system is a almost linear system and we can consider the associated linear case:

CDa0þx ¼ x; ð28Þ

CDb0þy ¼ �y: ð29Þ

Keeping b ¼ ra in mind, we can rewrite the system:

CDa0þ

xy1

..

.

yr

0BBBB@

1CCCCA ¼

1 0 0 . . . 00 0... ..

.Ir�1

0 00 �1 0 . . . 0

0BBBBBB@

1CCCCCCA

xy1

..

.

yr

0BBBB@

1CCCCA; ð30Þ

where y1 ¼ y and yiþ1 ¼ CDa0þyi; i ¼ 1; . . . ; r � 1.

Studying numerically (Diethelm’s numerical algorithm [3,4]) the trajectories of the solutions for different values of a; b,we can represent the phase planes (Fig. 6). The critical point (0,0) is an unstable saddle point. Moreover the solutions movemore quickly in the ordinary case than in the fractional case a 6 b < 1 and we observe the opposite behavior (slower in theordinary case that in the fractional one) for a < 1 < b and 1 < a 6 b, with strange trajectories in this last case.

4.2. Case ðx0; y0Þ=(1,1)

With x ¼ 1þ u; y ¼ 1þ v , we transfer the critical point to the origin and obtain a almost linear system whose associatedlinear case:

CDa0þu ¼ �v ; ð31Þ

CDa0þv ¼ u: ð32Þ

This system has two pure imaginary eigenvalues �i. Being b ¼ ra, we rewrite the system:

CDa0þ

u

v1

..

.

v r

0BBBB@

1CCCCA ¼

0 �1 0 . . . 00 0... ..

.Ir�1

0 01 0 0 . . . 0

0BBBBBB@

1CCCCCCA

u

v1

..

.

v r

0BBBB@

1CCCCA; ð33Þ

where v1 ¼ v and v iþ1 ¼ CDa0þv i; i ¼ 1; . . . ; r � 1.

0 20 40 60 80 1000

0.05

0.1

0.15

0.2

0.25

x

y

0 20 40 60 80 1000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

x

y

Phase plane of (0,0) for ordinary case (-) and fractional case: a ¼ 0:3; b ¼ 0:6 (–), a ¼ 0:6; b ¼ 1:2 (:), a ¼ 1:2; b ¼ 2:4 (- � -) in picture 1; a ¼ b ¼ 0:3b ¼ 0:6 (:), a ¼ b ¼ 1:2 (- � -) in picture 2.

0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

3

3.5

4

x

y

0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

x

y

Fig. 7. Phase plane of (1,1) for ordinary case (-) and fractional case: a ¼ 0:3; b ¼ 0:6 (–), a ¼ 0:6; b ¼ 1:2 (:), a ¼ 1:2; b ¼ 2:4 (- � -) in picture 1; a ¼ b ¼ 0:3(–), a ¼ b ¼ 0:6 (:), a ¼ b ¼ 1:2 (- � -) in picture 2.

M. Rivero et al. / Applied Mathematics and Computation 218 (2011) 1089–1095 1095

Using the mentioned numerical algorithm to solve this system, we obtain the phase planes (Fig. 7). In the ordinary case,we obtain periodic orbits and the critical point is a center. However in the fractional case the solutions lose the periodicityand form spirals that stay inside the periodic orbit of the ordinary case for a 6 b < 1, that go into and go out of the periodicorbit of the ordinary case for a < 1 < b, and that stay outside the periodic orbit of the ordinary case for 1 < a 6 b.

Acknowledgments

This paper has been partially supported by MICINN of Spain (project MTM2010-16499, AYA2009-14212-C05-05, and FPUAP2007-00864) to which the authors are very thankful.

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Sydney-Toronto, 1977.[3] K. Diethelm, N.J. Ford, A.D. Freed, Detailed error analysis for a fractional Adams method, Numer. algor. 26 (2004) 31–52.[4] K. Diethelm, N.J. Ford, A.D. Freed, Yu. Luchko, Algorithms for the fractional calculus: a selection of numerical methods, Comput. Meth. Appl. Mech. Eng.

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