Analysis of a Fractional Order Prey-Predator Model (3-Species)

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1 Global Journal of Computational Science an Mathematics. ISSN Volume 5, Number (06), pp. -9 Research Inia Publications Analysis of a Fractional Orer Prey-Preator Moel (-Species) A. George Maria Selvam, R. Dhineshbabu an D. Abraham Vianny Sacre Heart College, Tirupattur-65 60, S. Inia. DMI College of Engineering, Chennai-600, S. Inia. Knowlege Institute of Technology, Kakapalayam, S. Inia. agmshc@gmail.com Abstract In this paper, we iscuss the ynamical behavior of a fractional orer prey preator moel. The equilibrium points are compute an stability of the equilibrium points are analyze. The phase portraits are obtaine for ifferent sets of parameter values. Numerical simulations are performe an it is shown that the system exhibits rich ynamical behaviors. Keywors an phrases: Fractional orer, stability analysis, Prey-Preator moel. I. INTRODUCTION The concept of fractional-orer calculus was propose early 00 years ago. Fractional orer equations are more suitable than integer orer ones in moeling biological, economic an social systems where memory effects are important. Fractional orer ifferential equations are generalizations of integer orer ifferential equations. There are several efinitions of the fractional erivative / integral. The Riemann-Liouville efinition is [4] n t r f ( τ ) adt f() t = τ Γ n r n ( n r) t ( t ) a τ + for n < r < nan where Γ(). is the Gamma function. The Caputo's efinition of fractional erivatives can be expresse as t ( n) r f ( τ ) adt f() t = τ Γ r n ( n r) ( t τ ) + for ( n < r < n). a

2 A.George Maria Selvam et al Lemma. [, 5] The following linear commensurate fractional-orer autonomous α system D x() t = Ax(), t x(0) = x0 is asymptotically stable if an only if arg λ > α is satisfie for all eigen values ( λ) of matrix A. Also, this system is stable if an only if arg λ > α is satisfie for all eigen values ( λ) of matrix A, an those critical eigen values which satisfy arg λ = α have geometric multiplicity one, where n n n 0 < α <, x R an A R. Lemma. [, 5] Consier the following autonomous system for internal stability α T efinition D x() t = Ax(), t x(0) = x0 with α = [ α, α,..., α n ] an its n-imensional representation: α D x() t = ax() t + ax() t a nxn() t α D x() t = ax() t + ax() t anxn() t... () αn D x () t = a x () t + a x () t a x () t n n n nn n where all α i 's are rational numbers between 0 an. Assume m to be the LCM of the ui enominatorsu i 's ofα i 's, where i, u i, v i Z + α = for i=,,..., n an we set γ =. vi m Define: mα λ a a... a n mα a a... an et λ = 0 () M mα n an an... λ ann The characteristic equation can be transforme to integer-orer polynomial equation if all α i 's are rational number. Then the zero solution of system is globally asymptotically stable if all rootsλ i 's of the characteristic (polynomial) equation satisfy: arg λ > γ, i. II. MODEL DESCRIPTION AND EQUILIBRIUM POINTS Mathematical moeling in population ynamics has gaine a lot of attention uring the last few ecaes. The ynamical relationship between preators an their prey has been an important topic. Many researchers investigate preator-prey population moels with system of first orer ifferential equations. Recently the fractional orer ifferential equations have gaine a lot of attention in many fiel of applie

3 Analysis of a Fractional Orer Prey-Preator Moel (-Species) mathematics incluing population ynamics ue to their ability to provie better escription of ifferent non-linear phenomena [4]. The Lotka-Volterra equations are a pair of first orer ifferential equations use to escribe the ynamics of interactions of two species. In 96, Volterra came up with a moel to escribe the evolution of preator an prey fish populations in the Ariatic Sea. They were propose inepenently by Alfre J.Lotka in 95 [, ]. Several authors formulate fractional orer systems an analyze the ynamical an qualitative behavior of the systems [, 6, 7, 8]. The fractional Lotka-Volterra equations are obtaine from the classical equations by replacing the first orer time erivatives by fractional erivatives []. In this paper, we propose a system of fractional orer prey-preator moel. The stability of equilibrium points is stuie. Numerical solutions an simulations of this moel are provie. We consier the fractional orer moel as follows: α D x() t = ax() t bx () t x() t y() t x() t z() t α D y() t = ( c) y() t + x() t y() t () α D z( t) = ( e) z( t) + fx( t) z( t) + gy( t) z( t) where the parameters abce,,,,, f, gare positive an α, α, α are fractional orers. To evaluate the equilibrium points, let us consier α α α D x() t = 0; D y() t = 0; D z() t = 0. The fractional orer system has five equilibria a E 0 = (0,0,0)(trivial), E =,0,0 b (axial), c a b( c ) E =,,0 (axial), e af b( e ) E =,0, (axial) an f f c ( e ) f ( c ) ( + ag e) + ( f bg)( c ) E 4 =,, (coexistence). g g To accommoate biological meaning, the existence conitions for the equilibria require that they are nonnegative. It obvious E0 an E always exist, E exist when c > an a > b( c ), whilee exist when e > an af > b( e ). The interior equilibrium E 4 ( bg f )( c ) exist when e ( ) > f( c ) an >. ( + ag e) III. STABILITY OF EQUILIBRIA Base on (), to investigate the local stability of each equilibrium point ( *, *, *) * * * provie the Jacobian matrix J( x, y, z ). x y z, we

4 4 A.George Maria Selvam et al * * * * * a bx y z x x * * J = y c+ x 0. (4) * * * * fz gz e + fx + gy For E 0, we have a 0 0 J( E0 ) = 0 c e The eigen values are λ = a, λ = can λ = e. It is clear that E 0 is a sale point, while for E - we have a a a b b a J( E) = 0 c+ 0. b af 0 0 e + b a af The eigen values are λ = a, λ = c+ an λ = e +. Hence E is b b asymptotically stable when a < bc an af < be. Jacobian of E is b c c ( c) J( E ) = a + b( c) 0 0. f g 0 0 e ( c) + [ a + b( c)] which has the following eigen values: λ = f g e ( c) [ a b( c)] + + an b( c) ± b ( c) + 4 ( c)[ a + b( c)] λ, =. Since both of λ an λ are negative, local stability of E is etermine by λ. Hence e + f ( c) b( c)( c) E is stable note when g < an >. a + b( c) a( c) e af b( e ) Local stability of E =,0, is etermine by investigating the eigen f f values of

5 Analysis of a Fractional Orer Prey-Preator Moel (-Species) 5 b e e ( e) f f f J( E ) = 0 c ( e) 0. f g af + b( e) [ af + b( e)] 0 f b( e) ± b ( e) + 4 f ( e)[ af + b( e)] namely λ = c ( e) an λ, =. f f cf b( e)( e) Obviously E is stable when < an f >. Finally the local e a( e) stability of the interior equilibrium point is investigate by consiering the Jacobian matrix A B B J( E4 ) = C 0 0. D E 0 b c e ( ) f( c ) f ( + ag e) + ( c ) f ( f bg) A= ( c), B =, C =, D = g g ( c )( f bg) an E = ( + ag e) +.The characteristic polynomial P( λ) for J( E 4 ) is P( λ) = λ + aλ + aλ + a where a = A; a = B( C+ D); a = BCE. It is obvious that a > 0an a > 0. If aa > a, then Routh Hurwitz criterion implies that all roots of P( λ) have negative real parts, or in other wors, E 4 is a stable point. It can be shown that equation aa a = B[ AC ( + D) + CE] is positive if b( c )[ w + b( c )] f >, where w= ( + ag e) + ( f bg)( c ). This conition is wb ( + ) + b( c ) in contrast to the existence conition of E 4. It means that E4 is unstable. We shall complete this section by summarizing the existence an stability conition of all the equilibrium points in the following table: Equilibrium Point Existence conition Stability Conition E - Always sale 0 E - a < bc an af < be E c > an a > b( c ) e + f ( c) a + b( c) < an g b( c)( c) a( c) >

6 6 A.George Maria Selvam et al E e > an af > b( e ) e ( ) < cfan b( e)( e) a( e) >. f E e ( ) > f( c ) an 4 b( c )[ w + b( c )] f > ( c )( bg f) wb ( + ) + b( c ) > + ag e IV. DYNAMIC BEHAVIOR WITH NUMERICAL SOLUTIONS Numerical solution of the fractional-orer Prey-Preator system is given as follows []: k α ( α) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) xt = axt bx t xt yt xt zt h c xt k k k k k k k j k j j= v k α ( α) ( ) ( ) ( ) ( ) ( ) ( ) ( ) yt = cyt + xt yt h c yt k k k k j k j j= v k α ( α) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) zt = ezt + fxt zt + gyt zt h c zt k k k k k k j k j j= v where Tsim is the simulation time, k,,..., N is the start point (initial conitions). = for N [ T / h] =, an( x(0), y(0), z(0) ) Example. Let us consier the parameters with values a= ; b= 0.5; c = 4; = 5; e= 4; f = 5; g = 0.4 an the erivative orer α = α = α = For these parameters, the corresponing eigen values are λ = 0 an λ, = ±i for E 4, which satisfy arg λ > α. It means the system () is stable, see fig-. Also the characteristic equation of the linearize system () at the equilibrium point E4 is λ + 0.λ + 8λ = 0. sim

7 Analysis of a Fractional Orer Prey-Preator Moel (-Species) 7 Figure :Time Series an Phase iagram of Fixe Point E4 with Stability. Example. Let us consier the parameters values a= 8; b= 0.05; c = 4; = ; e= 7; f = 9; g = 4 an the erivative orerα = α = α = For these parameters, the corresponing eigen values are λ =.66 an λ, = 0.88 ± i.579for E 4, which satisfy arg λ > α. It means the system () is unstable, see fig-. Also the characteristic equation of the linearize system () at the equilibrium point E4 is λ + 0.5λ λ = 0.

8 8 A.George Maria Selvam et al Figure :Time Series an Phase iagram of Fixe Point E4 with Unstability. V. REFERENCES [] Ab-Elalim A. Elsaany, H. A. EL-Metwally, E. M. Elabbasy, H. N. Agiza, Chaos an bifurcation of a nonlinear iscrete prey-preator system, Computational Ecology an Software, 0, (): [] Hala A. El-Saka, Fractional-Orer Partial Differential Equation for Preator- Prey, Journal of Fractional Calculus an Applications Vol. 5() July 04, pp [] Ivo Petras, Fractional orer Nonlinear Systems-Moeling, Analysis an Simulation, Higher EucationPress, Springer International Eition, April 00. [4] Leticia Ariana Ramrez Hernnez, Mayra Guaalupe Garca Reyna an Juan Martnez Ortiz, Population ynamics with fractional equations (Preator-Prey), Acta Universitaria, vol., núm., noviembre-, 0, pp. 9-. [5] Mehi Dalir an Maji Bashour, Applications of Fractional Calculus, Applie Mathematical Sciences,Vol. 4, 00, no., 0-0. [6] Ping Zhou an Rui Ding, Control an Synchronization of the Fractional-Orer Lorenz Chaotic System via Fractional-Orer Derivative, Mathematical Problems in Engineering, Hinawi Publishing, Volume 0 : -4. [7] S. Z. Ria, M. Khalil, Hany A. Hosham, Soheir Gaellah, Preator-Prey

9 Analysis of a Fractional Orer Prey-Preator Moel (-Species) 9 Fractional-Orer Dynamical System with both the Populations Affecte by Diseases, Journal of Fractional Calculus an Applications, Vol. 5(S) No., pp. -. [8] Yingjia Guo, The Stability of Solutions for a Fractional Preator-Prey System, Abstract an Applie Analysis, Volume 04, Article ID 445, 7 pages.

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