Fractional Order Nonlinear Prey Predator Interactions

Transcription

1 International Journal of Computational and Applied Mathemati. ISSN Volume 2, Number 2 (207), pp Reearh India Publiation Frational Order Nonlinear Prey Predator Interation A. George Maria Selvam, R. Janagaraj 2, R. Dhinehbabu 3 and Britto Jaob. S 4,4 Sared Heart College, Tirupattur , India. 2 Kongunadu College of Engineering and Tehnology, Thottiam , India. 3 DMI College of Engineering, Chennai , India. agmh@gmail.om Abtrat Thi paper analye the dynamial behavior of a frational order Prey Predator Model. A diretization proe i applied to obtain it direte verion. The fixed point are obtained and the tability propertie are diued. Time erie and phae portrait are preented and the oillation in the prey predator population i etablihed via limit yle. Keyword: Frational Order, diretization, Lotka - Volterra predator prey ytem, limit yle 200Subjet laifiation: 34A08, 34D20, 92D25. INTRODUCTION Mathematial modeling of interation between peie ha drawn the attention of reearher [, 3, 6]. Population model in eology have been tudied uing ordinary differential equation, differene equation, partial differential equation, frational order differential equation and tohati model. Frational order differential equation (FODE) are uitable to tudy the ytem with memory whih exit in mot biologial ytem. FODE an be ued to model many phenomena whih annot be modeled by integer order differential equation [2, 5]. There are everal definition for frational derivative, ee [4].

2 496 A. George Maria Selvam, R. Janagaraj, R. Dhinehbabu & Britto Jaob.S Stability and dynamial analyi of frational order Lotka Volterra model an be found in [5, 7, 8, 9]. In thi paper, we onider frational order Lotka Volterra predator prey ytem for the tudy of it dynamial behavior. 2. DISCRETIZATION OF FRACTIONAL ORDER MODEL In 926, Volterra ame up with a model to deribe the evolution of predator and prey fih population in the Adriati Sea. They were propoed independently by Alfred J. Lotka in 925 [6, 7]. The equation are x' ax bxy; y' y dxy The dynami i periodi and ytem ha the tendeny to oillate. Alo the model generate neutral tability. In [7], author onidered D x x( y); D y y( x) 0 0 and ompared the dynami of laial model with frational order. Paper [9] diue the tability of the equilibrium point of the frational order ytem D y y ( a by ); D y y ( by ) Introduing the frational order in the laial model, we obtain D x( t) x( t)( a by( t)); D y( t) y( t)( dx( t)) () where t 0and (0,]. Only the poitive and finite value of the parameter have meaningful biologial interpretation. Now, applying the diretization proe for a frational-order ytem deribed in [2, 0], we obtain the direte frational order predator prey ytem a follow: xn xn ( xn ( a byn )); yn yn ( yn( dxn )).(2) ( ) ( ) 3. FIXED POINTS AND STABILITY Fixed point are obtained by olving D x( t) 0 ; D y( t) 0. The fixed point of ytem () are (a) Trivial fixed point E0 (0,0) (Origin); a (b) Interior fixed point of oexitene E, d b (Interior).

3 Frational Order Nonlinear Prey Predator Interation 497 We next tudy the loal tability of the fixed point. The Jaobian matrix J of ytem () evaluated at the fixed point J x The determinant of the Jaobian ( x, y ) i given by (, y ) a by bx dy dx J( x, y ) i Det adx ( a by ) Theorem. The fixed point E0i loally aymptotially table if a,, otherwie untable fixed point. a 0 Proof: The Jaobian matrix at E0 i given by J( E0 ). 0 Hene the eigenvalue of J( E0 ) are a and 2. Thu E0i table when a,. Otherwie E0 i untable fixed point. Theorem 2. The fixed point E i loally aymptotially table if A, where A i a. Otherwie untable fixed point. Proof: The Jaobian matrix at E i given by b 0 d J( E ). ad 0 b The eigenvalue of the matrix J( E ) are,2 a. Hene E i loally aymptotially table when A, and untable A. We will now diu the dynami of the diretized frational order Lotka Volterra predator prey model (2). The dynamial behavior of model (2) i determined by the parameter a, b,, d, and. We will now diu the tability of fixed point of model (2). The Jaobian matrix J of model (2) evaluated at the fixed point ( x, y ) i given by J x ( a by ) bx ( ) ( ) (, y ). * * dy ( dx ) ( ) ( ) (3)

4 498 A. George Maria Selvam, R. Janagaraj, R. Dhinehbabu & Britto Jaob.S The harateriti equation of the Jaobian matrix i 2 Tr Det 0. (4) where Tr i the trae and Det i the determinant of the Jaobian matrix they are J( x, y ) and Tr a dx by ( ) 2 ( ) 2 ( ) ( ) Det a dx by adx by a ( ) ( ) (5) To tudy the tability propertie of the model (2), we preent the following reult that an be eaily proved by uing the relation between root and oeffiient of the harateriti equation (4). Let and 2 be the two root of Eq. (4), are the eigen value of the Jaobian matrix f f x y J g g x y evaluated at the fixed point of ytem (). Then we have Lemma 3. (i) A fixed point loally aymptotially table. (ii) A fixed point untable. (iii) A fixed point ). (iv) A fixed point Theorem 4. If ( x, y ) i alled a ink if and 2, o the ink i ( x, y ) i alled oure if and 2, o the oure i loally ( x, y ) i alled a addle if and 2 (or and 2 ( x, y ) i alled non-hyperboli if either or 2. 2 ( ) 0 then the fixed point E0 i a addle point. If 2 ( ), then E0 i a oure and if 2 ( ), then E0 i non-hyperboli.

5 Frational Order Nonlinear Prey Predator Interation 499 Proof: The Jaobian matrix J at E0 i given by a 0 ( ) J( E0). 0 ( ) Hene, the eigenvalue are a ( ) and 2 ( ). Sine a 0, then. Thu the fixed point E0 i a addle point if 2 ( ) and non-hyperboli if 2 ( ). 2 ( ) 0, oure if a Theorem 5.The poitive fixed point E, of the ytem (2) i loally d b aymptotially table if 0, where A ( ) uh that A a. Proof: The Jaobian matrix evaluated at the fixed point E ha the form b ( ) d J( E ). ad ( ) b The trae and determinant of the Jaobian matrix J( E ) are given by Tr( J( E )) 2, Det ( a). ( ) 2 Hene the eigenvalue are,2 ( a). ( ) Thu E i loally aymptotially table when 0. The next lemma i an immediate onequene from Theorem (5). a Lemma 6. The poitive fixed point E, d b where A ( ) uh that A a hold. of the ytem (2) i untable if 0,

6 500 A. George Maria Selvam, R. Janagaraj, R. Dhinehbabu & Britto Jaob.S 4. PERIODIC SOLUTIONS AND LIMIT CYCLES In thi etion, example are preented to illutrate the periodi oillation in the prey predator population. All of the trajetorie form loed orbit. Mainly, we preent the orbit of the olution x and y with phae plane diagram for the frational order predator-prey ytem (2). Eologit applied the laial model to the data on the Canadian lynx now hoe hare interation whih exhibited periodi oillation reulting in limit yle. A loed trajetory implie yle periodi oillation. Example. For 0.99, a 0.; b 0.09; 0.45; d.and 0.0,0.03,0.05 and 0.08, the time plot and phae portrait are preented. Thee value reult in olution with different amplitude. Oillatory behavior of the population i etablihed by the exitene of limit yle, See Figure. Example 2. For the initial value x0.5, y 0.6 and 0.99; 0.05, a ; b ; and d 0.999, the Eigen value are, i0.008, o that, Hene the interior fixed point E i untable. The orbit and the phae diagram illutrate the reult, See Figure 2.

7 Frational Order Nonlinear Prey Predator Interation 50 Figure 2. Time Serie and Phae Portrait for interior fixed point E. REFERENCES [] Berrymann, A.A: The origin and evolution of predator-prey theory, Eology 73, (992). [2] H. A. A. El-Saka, The Frational-Order SIR and SIRS Epidemi Model with Variable Population Size, Math. Si. Lett. 2, No. 3, (203). [3] A. A. Eladany A. E. Matouk, Dynamial behavior of frational - order Lotka - Volterra predator - prey model and it diretization, J. Appl. Math. Comput. DOI 0.007/ [4] Ivo Petra, Frational order Nonlinear Sytem - Modeling, Analyi and Simulation, Higher Eduation Pre, Springer International Edition, April 200. [5] A.George Maria Selvam, R.Janagaraj and D. Abraham Vianny, Analyi of Frational Order Prey - Predator Interation, Amerian Journal of Engineering Reearh (AJER), e-issn: p-issn : , Volume-4, Iue-9, pp [6] Leah Edeltein-Kehet, Mathematial Model in Biology, SIAM, Random Houe, New York, [7] Letiia Adriana Ramírez Hernández, Mayra Guadalupe Garía Reyna, Juan Martínez Ortiz, Population dynami with frational equation (Predator- Prey), Ata Univeritaria, vol. 23, núm. 2, noviembre-, 203, pp. 9-. [8] Margarita Rivero, Juan J. Trujillo, Lui Vázquez, M. Pilar Velao, Frational dynami of population, Applied Mathemati and Computation 28 (20) [9] F. Merrikh-Bayat, More Detail on Analyi of Frational-Order Lotka- Volterra Equation, arxiv:40.003v [math.ds].

8 502 A. George Maria Selvam, R. Janagaraj, R. Dhinehbabu & Britto Jaob.S [0] Ravi P Agarwal, Ahmed MA El-Sayed and Sanaa M Salman, Frationalorder Chua ytem: diretization, bifuration and hao, Advane in Differene Equation 203, 203:320.