Complex Dynamics Behaviors of a Discrete Prey-Predator Model with Beddington-DeAngelis Functional Response

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1 Int. J. Contemp. Math. Sciences, Vol. 7, 2012, no. 45, Complex Dynamics Behaviors of a Discrete Prey-Predator Model with Beddington-DeAngelis Functional Response K. A. Hasan University of Sulaimani, Faculty of Science and Science Educations School of Science-Department of Mathematics, Sulaimani, Iraq kawa79b@gmail.com M. F. Hama University of Sulaimani, Faculty of Science and Science Educations School of Science-Department of Mathematics, Sulaimani, Iraq hamamuzafar@yahoo.com Abstract In this paper the dynamics of a discrete-time prey-predator system is investigated in the closed first quadrant R 2 +. The existence and stability of fixed points are analyzed algebraically.the conditions of existence for flip bifurcation is derived by busing center manifold theorem and bifurcation theory.numerical simulations not only illustrate our results but also exhibit complex dynamical behaviors of the model,such as the periodic-doubling bifurcation in periods 2,4 and 8 and quasi-periodic orbits and chaotic sets Mathematics Subject Classification: 37L10, MSC 37M05, MSC 74H60 Keywords: Discrete model, Beddington-DeAngelis functional response, Stability, Flip bifurcation, center manifold theorem, Numerical simulation 1 Introduction It is well known the Lotka-Voltera prey-predator model is one of the fundamental population models, a predator -prey interaction has been described firstly by two pioneers Lotka(1924)[18] and Voltera (1926) [14] in two independent works. After them,more realistic prey-predator model were introduced by Holling suggesting three types of functional responses for different species to model the phenomena of predation [7]. Predator -prey models have already received much attention from many authors. For example, the stability, and the

2 2180 K. A. Hasan and Muzafar F. Hama existence of periodic solutions of the predator prey models are studied in [12], [15], [16], and [19]. For continuous-time predator-pry models, many authors have chosen delay as the bifurcation parameter to discuss the Hopf bifurcation in [11] and [17]. However, there are few articles discussing the dynamical behaviors of predator-prey models, which include bifurcations and chaos phenomena for the discrete-time models. Liu and Xiao [13], Jing and Yang [9], He and Lai [6], Hu and Zhang [8] obtained the flip bifurcation by using the center manifold theorem and bifurcation theory. But Agiza et al.[1] and Celik et al.[3] only showed the flip bifurcation and Hopf bifurcation by using numerical simulations. A functional response is called of Beddington-DeAngelis type kx if it takes the form P (x, y) =. This type of functional response was (1+by+ax) introduced by Beddington [2] and DeAngelis et al. [4]. The term b y measures the mutual interference between predators. In this paper we consider the following continuous-time prey-predator model with Beddington-DeAngelis type of functional response described by differential equations dx 1 dt dx 2 dt = rx 1 (1 x 1 K ) mx 1 x 2 Bx 2 + hx = c 1 mx 1 x 2 Bx 2 + hx d 1x 2. (1) The model parameters r, K, m, B, h, c 1, and d 1 are assuming only positive values. The prey x 1 grows with intrinsic growth rate r and carrying capacity K in the absence of predation. The predator x 2 consumes the prey x 1 with functional response Beddington-DeAngelis type. The constant c 1 is conversion rates of prey to predator, d 1 is predator death rate for species x 2. By using the following transformation t = rt, x 1 = Kx, x 2 = r y we get the following m system dx dt dy dt = x(1 x) = xy 1 + ax + by cxy 1 + ax + by ey (2) where a = hk, b = Br, c = c 1mK, and e = d 1 m r r. Applying the forward Euler scheme to system (2), we obtain the discrete-time system as follows: [ ] y x x + dx 1 x 1 + ax + by [ cx y y + dy 1 + ax + by e ] (3)

3 Discrete prey-predator model 2181 where d is the step size. The main goal of this paper is is to investigate this version as a discrete-time dynamical system in the interior of the first quadrant by using bifurcation theory and center manifold theory. The organization of this paper is as follows. In the second section we discuss the existence and local stability of fixed points in model (3). In the third section we study flip bifurcation for model (3) by choosing d as a bifurcation parameter. In the fourth section we study the numerical simulations,which not only illustrate our results with theoretical analysis but also exhibit complex dynamical behaviors such as the cascade periodic-doubling bifurcation in periods 2,4 and 8 and quasi-periodic orbits and chaotic sets.in section five we give the discussion for our work. 2 The existance and stability of fixed points In this section and through out the paper, we consider the discrete-time model (3)in the closed first quadrant R 2 + on the xy plane. We first discuss the existance of the fixed points for (3), then study the stability of the fixed point by the eigenvalues for the variational matrix of (3) at the fixed point. To determine the fixed points of (3) we have to solve the non- linear system given by x = x + dx (1 x y 1+a x+b y ) y = y + dy ( c x 1+a x+b y e). (4) By composition of the above algebraic system we obtain the following results: Lemma 2.1. System (3): 1. Has two fixed points; for all parameter values, (a) E 0 (0, 0) is origin, (b) E 1 (1, 0) is the axial fixed point. 2. Has the interior fixed point E 2 (x, y ) where y = (c ae)x e be and 0 < e (c ae) < x < 1. Now we study the stability of these fixed points. Note that the local stability of the fixed point (x, y) is determined by the modules of eigenvalues of the charcteristic equation at the fixed point.

4 2182 K. A. Hasan and Muzafar F. Hama where The Jacobian matrix of (3) evaluated at the point (x, y) is given by [ ] j11 j J(x, y) = 12 j 21 j 22 j 11 = 1 + dx ( 1 + ) ( a y + d 1 x (1 + a x + b y) 2 ) y ; 1 + a x + b y j 12 = d x(1 + a x) (1 + a x + b y) 2 ; j 21 = d c y(1 + b y) (1 + a x + b y) 2 ; ( ) ( ) b c d x y c x j 22 = 1 + (1 + a x + b y) 2 (1 + a x + b y) e d and the characteristic equation of the Jacobian matrix J(x, y) can be written as where λ 2 + P (x, y) λ + Q(x, y) = 0 (5) P (x, y) = (j 11 + j 22 ) Q(x, y) = j 11 j 22 j 12 j 21. In order to discuss the stability of the fixed points of (3), we also need the following Lemma. Lemma 2.2. Let F (λ) = λ 2 + P λ + Q. Suppose that F (1) > 0, λ 1, λ 2 are two roots of F (λ) = 0. Then: 1. λ 1 < 1 and λ 2 < 1 if and only if F ( 1) > 0 and Q < 1; 2. λ 1 < 1 and λ 2 > 1 (or λ 1 > 1 and λ 2 < 1) if and only if F ( 1) < 0; 3. λ 1 > 1 and λ 2 > 1 if and only if F ( 1) > 0 and Q > 1; 4. λ 1 = 1 and λ 2 1 if and only if F ( 1) = 0 and P 0, 2; 5. λ 1 and λ 2 are complex and λ 1 = 1 and λ 2 = 1 if and only if P 2 4Q < 0 and Q = 1. Let λ 1 and λ 2 be two roots of (5). We recall some definitions of topological types for a fixed point (x, y).

5 Discrete prey-predator model 2183 Definition 2.3. A fixed point (x, y) is called a sink if λ 1 < 1 and λ 2 < 1, (x, y) is called a source if λ 1 > 1 and λ 2 > 1, (x, y) is called a saddle if λ 1 > 1 and λ 2 < 1 or ( λ 1 < 1 and λ 2 > 1), and (x, y) is called a non-hyperbolic if either λ 1 = 1 or λ 2 = 1. Proposition 2.4. The fixed point E 0 is a source if d > 2 e, E 0 is a saddle if 0 < d < 2 e, and E 0 is a non-hyperbolic if d = 2 e. At the fixed point E 0 the roots of equation (5) are λ 1 = 1+d with d > 0 and λ 2 = 1 de, so λ 1 is not one with model and the second eigenvalue λ 2 = 1 when d = 2. This periodic doubling bifurcation may occure where parameters e vary in the neighborhood d = 2. e The next proposition show the locall dynamics of fixed point E 1 from Lemma(2.2). Proposition 2.5. There are at least four different topological types of E 1 for all permissible values of parameters 1. E 1 is a sink if c 1+a < e and 0 < d < min{2, 2(1+a) e(1+a) c }; 2. E 1 is a source if c 1+a c 3. E 1 is a saddle if d < 2 and ( ) 2 < d < 2(1+a) ; e(1+a) c c > e and 2 < d or ( < e and d > max{2, 2(1+a) }; 1+a e(1+a) c ( ) > e and 0 < 2(1+a) < d < 2 or 1+a e(1+a) c 4. E 1 is non-hyperbolic if either d = 0 or c 1+a = e or d = 2(1+a) e(1+a) c. From the above proposition we obtain, for a fixed point E 1 if (a, b, c, d, e) R where R = {(a, b, c, d, e) d = 2, e c 2(1 + a), d, a, b, c, d, e > 0}, 1 + a e(1 + a) c then one of the eigen values of J(E 1 ) is 1 and the other λ = 1 + ( c e)d is 1+a neither 1 nor 1. Therefor, there may be flib bifurcation of a fixed point E 1, if d varies in the small neighbourhood of d = 2 and (a, b, c, d, e) R. Proposition 2.6. Let E 2 (x, y ) be the positive fixed point of (3). Then 1. E 2 is a sink if one of the following conditions holds (a) P 0 and 0 < d < N P M ; (b) P < 0 and 0 < d < N M ; 2. E 2 is a source if one of the following conditions holds

6 2184 K. A. Hasan and Muzafar F. Hama (a) P 0 and d > N + P M ; (b) P < 0 and d > N M ; 3. E 2 is a saddle if one of the following conditions holds (a) P 0 and 0 < d < N + P M ; (b) P 0 and 0 < N P M < d; 4. E 2 is non hyperbolic if one of the following conditions holds Where (a) p 0 and d = N p M ; (b) p < 0 and d = N M. M = e(1 x )(1 a e c b + 2b x ) N = e(1 x )( a c b) + x P = N 2 4M. From Lemma(2.2), we can see that one of the eigen values of E 2 is 1 and the other is neither 1 nor 1 if the term (4a) in proposition(2.6) holds. Therefor there may be flip bifurcation of E 2 if d varies in the small neighbourhood of B 1 or B 2 where B 1 = {(a, b, c, d, e) d = N p M, p > 0, a > b, b, c, d, e > 0} B 2 = {(a, b, c, d, e) d = N + p M, p > 0, a, b, c, d, e > 0}. Also when the term (4b) in proposition(2.6) holds, we can obtain that the eigenvalues of E 2 are pair of conjugate complex numbers with λ 1 = 1, λ 2 = 1 and hence the conditions in term (4b) of prposition(2.6) can be written as C = {(a, b, c, d, e) d = N M, p < 0, a, b, c, d, e > 0} if the parameter d vary in the small neighbourhood of C, then the Hopf bifurcation will appear. 3 Analysis of bifurcation Based on the analysis in Section 2, we discuss the flip bifurcation of positive equilibriun E 2 (x, y ) in this section. In analysis of the flip bifurcation of E 2 (x, y ) by using center manifold theorem and bifurcation theory in [5], d is an bifurcation paremeter.

7 Discrete prey-predator model 2185 We consider system (3) with (a, b, c, e, d), which is described by y x = x + dx (1 x 1 + a x + b y ) c x y = y + dy ( 1 + a x + b y e). Giving a perturbation d of parameter d. We consider a perturbation of model (6) as follows: x = x + (d + d y )x (1 x 1 + a x + b y ) y = y + (d + d c x )y ( 1 + a x + b y e) where d 1 Let u = x x and v = y y. Then we transform the fixed point E 2 (x, y ) of map (7) into the origin. We have u = u+x +(d+d )(u+x )(1 u x (v + y ) 1 + a(u + x ) + b(v + y ) ) v = v+y +(d+d )(v+y c (u+x ) )( 1+a(u + x ) + b(v + y ) e). Expanding model (8) as taylor series at (u, v, d ) = (0, 0, 0) to second order, then it becoms the following model where u = a 11 u+a 12 v + a 13 u 2 + a 14 uv + a 15 v 2 + b 11 d +b 12 ud +b 13 vd +b 14 u 2 d +b 15 uvd +b 16 v 2 d + O(u, v) 3 v = a 21 u+a 22 v + a 23 u 2 + a 24 uv + a 25 v 2 + b 21 d +b 22 ud +b 23 vd +b 24 u 2 d +b 25 uvd +b 26 v 2 d + O(u, v) 3 a 11 =1+d 2dx +2adx y +bdy 3a 2 dx 2 y 3abdx y 2 b 2 dy 3 a 12 =adx 2 +2bdx y b 2 dx 3 6abdy 2 x a 13 = d+ady 2a 2 dx y 2abdy 2 a 2 y a 14 =a(dx +1) a 2 x (dx +2)+2bdy aby (6d+4y ) a 15 = bdx 2abdx 2 3b 2 dx y b 11 =x x 2 +ax 2 y +bx y 2 a 2 x 3 y 2abx 2 y 2 b 2 y 3 b 12 =1 2x +2ax y 3a 2 x 2 y 4abx y 2 +by 2 b 2 y 3 b 13 =ax 2 +2bx y 2a 2 x 3 6abx y 3 b 14 = 1+ay 3a 2 x y 2aby 2 b 15 =2ax +2by 3a 2 x 2 6aby (4x +6y ) b 16 =bx 3b 2 x y 2abx 2 a 21 =acdx y ( 2+3ax )+bcdy 2 ( 1+4ax +by )+cedy x a 22 =1+acdx 2 ( 1+ax )+bcdx y ( 2+3by )+2abcdx y (1+x ) (6) (7) (8) (9)

8 2186 K. A. Hasan and Muzafar F. Hama a 23 =acdy ( 1+3ax +2by ) a 24 =acdx ( 2+3ax )+2bcdy ( 2+8ax +3by )+cedx a 25 = bcdx ( 1+2ax +3by ) b 21 = bcx y 2 ( 1+b+a)+acx 2 y ( 1+ax ) b 22 = acx y ( 2+3ax )+bcy 2 ( 1+4ax +by )+cey x b 23 =acx 2 ( 1+ax )+bcx y ( 2+3by )+2abcx y (1+x ) b 24 = acy ( 1+3ax +2by ) b 25 = acx ( 2+3ax )+2bcy ( 2+8ax +3by )+cex b 26 = bcx ( 1+2ax +3by ) We construct an invertible matrix a 12 a 12 T = 1 a 11 λ 2 a 11 and use the transformation ( u v ) = T ( x y ). The model (9) becomes into the following form ( ) ( ) ( ) ( x 1 0 x f(u, v, d = + ) y 0 λ 2 y g(u, v, d ) where f(u, v, d ) = 1 det(t ) 2 a 11 )a 13 a 12 a 23 ] u 2 + [(λ 2 a 11 )a 14 a 12 a 24 ] uv + [(λ 2 a 11 )a 15 a 12 a 25 ] v 2 + [(λ 2 a 11 )b 11 a 12 b 21 ] d + [(λ 2 a 11 )b 12 a 12 b 22 ] ud + [(λ 2 a 11 )b 13 a 12 b 23 ] vd + [(λ 2 a 11 )b 14 a 12 b 24 ] u 2 d + [(λ 2 a 11 )b 15 a 12 b 25 ] uvd + [(λ 2 a 11 )b 16 a 12 b 26 ] d v 2 + O( u, v, d )} and g(u, v, d ) = 1 det(t ) 11)a 13 + a 12 a 23 ] u 2 + [(1 + a 11 )a 14 + a 12 a 24 ] uv + [(1 + a 11 )a 15 + a 12 a 25 ] v 2 + [(1 + a 11 )b 11 + a 12 b 21 ] d + [(1 + a 11 )b 12 + a 12 b 22 ] ud + [(1 + a 11 )b 13 + a 12 b 23 ] vd + [(1 + a 11 )b 14 + a 12 b 24 ] u 2 d + [(1 + a 11 )b 15 + a 12 b 25 ] uvd + [(1 + a 11 )b 16 + a 12 b 26 ] d v 2 + O( u, v, d )}. Now, we determine the center manifold W c (0, 0) of model (9) at equilibrium (0, 0) in a small neighborhood of d = 0. By center manifold theorem, we can obtain the approximate representation of the center manifold W c (0, 0) as follows: } W c (0, 0) = {(x, y) : y = a 0 d + a 1 x 2 + a 2 xd + a 3 d 2 + O( x + d ) 3 )

9 Discrete prey-predator model 2187 where O( x + d ) 3 is a function of (x, d ), at least of the third order, and a 0 = (1 + a 11)b 11 + a 12 b 21 λ 2 det(t ) a 1 = a 2 = a 3 = 1 det(t )(λ 2 2) {((1 + a 11)a 13 + a 12 a 23 ) a ((1 + a 11 )a 14 + a 12 a 24 ) + ((1 + a 11 )a 15 + a 12 a 25 )} 1 det(t )λ 2 (λ 2 1) {2a 1 ((λ 2 a 11 )b 11 a 12 b 21 ) a 12 ((1 + a 11 )b 12 + a 12 b 22 ) + ((1 + a 11 )b 13 + a 12 b 23 ) (1 + a 11 ) λ 2 ((1 + a 11 )b 11 + a 12 b 21 )} 1 det(t )λ 2 { a 2 12a 2 0 ((1 + a 11 )a 13 + a 12 a 23 ) 2 ((λ 2 a 11 )b 11 a 12 b 21 ) +a 0 (λ 2 a 11 ) ((1 + a 11 )b 13 + a 12 b 23 ) + a 12 a 0 ((1 + a 11 )b 12 + a 12 b 22 ) +a 12 a 2 0 (λ 2 a 11 ) ((1 + a 11 )a 14 + a 12 a 24 )}. We write now u, v in term of x, y, d as u = a 12 x + a 12 a 0 d + a 12 a 1 x 2 + a 12 a 2 xd + a 12 a 3 d 2 ) v = (1 + a 11 )x + (λ 2 a 11 ) (a 0 d + a 1 x 2 + a 2 xd + a 3 d 2 Therefore, the map G which is restricted to the center manifold W c (0, 0) is: where (10) G (x) = x + f(u, v, d ) = x + h 0 d + h 1 x 2 + h 2 xd + h 3 d 2 + h 4 x 2 d + h 5 xd 2 +h 6 x 3 + h 7 d 3 + O( x + d ) 3 h 0 = (λ 2 a 11 )b 11 a 12 b 21 det(t ) h 1 = 1 det(t ) {a2 12 ((λ 2 a 11 )a 13 a 12 a 23 ) + (1 + a 11 ) 2 ((λ 2 a 11 )a 15 a 12 a 25 ) a 12 (1 + a 11 ) ((λ 2 a 11 )a 14 a 12 a 24 )} h 2 = 1 det(t ) {2a2 12a 0 ((λ 2 a 11 )a 13 a 12 a 23 ) + a 12 ((λ 2 a 11 )b 12 a 12 b 22 ) +a 0 a 12 (λ 2 1 2a 11 ) ((λ 2 a 11 )a 14 a 12 a 24 ) b 13 (1 + a 11 )(λ 2 a 11 ) +a 12 b 23 (1 + a 11 ) 2a 0 (λ 2 a 11 )(1 + a 11 ) ((λ 2 a 11 )a 15 a 12 a 25 )} h 3 = 1 det(t ) {a2 12a 0 (λ 2 a 11 ) ((λ 2 a 11 )a 14 a 12 a 24 ) + a 2 12a 2 0a 13 (λ 2 a 11 ) a 3 12 a2 0 a 23 + a 2 0 (λ 2 a 11 ) 2 ((λ 2 a 11 )a 15 a 12 a 25 ) +a 0 a 12 ((λ 2 a 11 )b 12 a 12 b 22 ) + a 0 (λ 2 a 11 ) ((λ 2 a 11 )b 13 a 12 b 23 )}

10 2188 K. A. Hasan and Muzafar F. Hama h 4 = 1 det(t ) {a2 12 (2a 2 + 2a 0 a 1 + 1) ((λ 2 a 11 )a 13 a 12 a 23 ) +a 12 ((λ 2 a 11 )a 14 a 12 a 24 ) (a 2 (λ 2 1 2a 11 ) + 2a 0 a 1 (λ 2 a 11 ) (1+a 11 )) + ((λ 2 a 11 )a 15 a 12 a 25 ) ( 2a 2 (λ 2 a 11 )(1 + a 11 ) + 2a 0 a 1 (λ 2 a 11 ) 2) +a 1 a 12 (1+a 11 ) 2 ((λ 2 a 11 )b 12 a 12 b 22 )+a 1 b 13 (λ 2 a 11 ) 2 a 1 a 12 b 23 (λ 2 a 11 )} h 5 = 1 det(t ) {a2 12 (2a 3 + 2a 0 a 3 + 2a 0 ) ((λ 2 a 11 )a 13 a 12 a 23 ) +a 12 ((λ 2 a 11 )a 14 a 12 a 24 ) (( a 0 a 3 )(λ 2 1 2a 11 )+2a 0 a 2 (λ 2 a 11 )) + ((λ 2 a 11 )a 15 a 12 a 25 ) ( 2(a 2 +a 0 )(λ 2 a 11 )(1+a 11 )+2a 0 a 1 (λ 2 a 11 ) 2) +a 2 a 12 (1+a 11 ) 2 ((λ 2 a 11 )b 12 a 12 b 22 )+a 2 b 13 (λ 2 a 11 ) 2 a 2 a 12 b 23 (λ 2 a 11 )} h 6 = 1 det(t ) {2a2 12a 1 ((λ 2 a 11 )a 13 a 12 a 23 ) +a 1 a 12 (λ 2 1 2a 11 ) ((λ 2 a 11 )a 14 a 12 a 24 ) 2a 0 (λ 2 a 11 )(1 + a 11 ) ((λ 2 a 11 )a 15 a 12 a 25 )} h 7 = 1 ( ) det(t ) {a2 12 2a0 a 3 + a 2 0 ((λ2 a 11 )a 13 a 12 a 23 ) +a 12 (λ 2 a 11 )(2a 0 a 3 +2a 2 0) ((λ 2 a 11 )a 14 a 12 a 24 ) +(λ 2 a 11 ) 2 (2a 0 a 12 + a 2 0) ((λ 2 a 11 )a 15 a 12 a 25 )}. In order to undergo a flip bifrucation for map (10), we require that two discriminatory quantities β 1 and β 2 are not zero, where ( 2 G β 1 = x d + 1 ) G 2 d 2 G x 2 (0,0) = h 0 h 1 + h2 0 and ( ( ) ) G 1 β 2 = 6 x + G 2 2 x 2 (0,0) = h 6 + h 2 1. Thus, from the above analysis and the theorem [5], wo obtain the following result. Theorem 3.1. If β 1 0, then model (6) undergos a flip bifurcation at equilibrium E 2 (x, y ) when the parameter d varies in the small neighborhood of the origin. Moreover, if β 1 > 0( resp., β 1 < 0), the the period 2 points that bifurcation from E 2 (x, y ) are stable (resp., unstable). 4 Numerical simulations In this section, we give the bifurcation diagrams, phase portraits of model (3) to confirm the above theoretical analysis and show the comples dynamical behaviors by using numerical simulations. The bifurcation parameters are considered in the following three cases: Case 1 : Choosing a = 0.3, b = 0.5, c = 0.1, and e = 0.2, initial value

11 Discrete prey-predator model 2189 (x 0, y 0 ) = (0.5, 0.6) and d [0, 2.96]. We see that model (3) has a fixed point E 1 (1, 0) and (a, b, c, d) R. Figure1 show the correctness of proposition (2.5). From Figure 1 we see that the fixed point E 1 (1, 0) is stable for d < 2, and loses its stability when d = 2. Further, when d > 2 there is the period- doubling bifurcation. Moreover a chaotic set is emerged with increasing of d. Figure 1: The flip bifurcation of x(n) with a=0.3, b=0.5, c=0.1, and e=0.2, d [0, 2.96] and initial (x 0, y 0 ) = (0.5, 0.6). Case 2 : Choosing a = 1.3, b = 3, c = 2, and e = 0.4, initial value (x 0, y 0 ) = (0.4, 0.6) and d [1.5, 3.5] and we only vary parameter d to see the variation of dynamics behaviors about model (3), on the basis of proposition (2.6), we know that the map (3) has only one positive fixed point. After calculation for Figure 2: The bifurcation diagram of model (3) in the (d, x)-plane for a = 1.3, b=3, c=2, and, e = 0.4, d [1.5, 3.5] and initial (x 0, y 0 ) = (0.4, 0.6).

12 2190 K. A. Hasan and Muzafar F. Hama (a) d = 2.7 (b) d = 3.11 (c) d = 3.23 (d) d = 3.25 (e) d = (f) d = 3.26 (g) d = 3.27 Figure 3: Phase portraits for various values of d corresponding to Figure 2

13 Discrete prey-predator model 2191 the positive fixed point of map (3), the flip bifurcation emerges from the fixed point ( 5, 25) at d = and (a, b, c, d, e) = (1.3, 3, 2, , 0.4) 6 36 B 1. It show the correctness of proposition (2.6). From Figure 2 we see that the fixed point is stable for d < , and loses its stability when d = We also observe that threr is a cascade of period-doubling bifurcation. The phase portraits which are associate with Figure 2 are disposed in Figure 3 for d (2.6855, 3.253) there are periodic- 2, 4, 6, 8, 16, 32 when d = 3.26, 3.27 we can see the chaotic sets. Case 3 : Choosing a = 1, b = 0.5, c = 4, and e = 9, initial value (x 10 0, y 0 ) = Figure 4: The bifurcation diagram of map (1) with d [1, 2.2] and (x 0, y 0 ) = (0.4, 0.9). (0.4, 0.9) and d [1, 2.2] according prposition (2.6), we know that the map (3) has only one positive fixed point. After calculation of the positive fixed Figure 5: Local amplification corresponding to Fig.4

14 2192 K. A. Hasan and Muzafar F. Hama (a) d = 1.59 (b) d = (c) d = (d) d = (e) d = (f) d = Figure 6: Phase portraits for various values of d corresponding to Figure 4 point of map (3), the flip bifurcation emerges from the fixed point ( 9, 11) at d = and (a, b, c, d, e) = (1, 0.5, 4, 1.598, 0.9) C. It show the correctness of proposition (2.6). From Figure 4 we observe that the fixed point ( 9, 11 ) is stable for d<1.598, that it loses its stability when d = and that an invariant circle appears when the parameter d exceeds Figure 5 is local amplification for d [1.82, 2.1]. The phase portraits which are associate with Figure 4 is displayed in Figure 6 and Figure 7, which clearly depicts how a smooth invariant circle bifurcates from the stable fixed point ( 9, 11 ). When d exceed there appears a circular curve enclosing the fixed point ( 9, 11 ), and its radius becomes larger for d increases from to 1.82 als if d increases there are

15 Discrete prey-predator model 2193 (a) d = (b) d = (c) d = (d) d = (e) d = (f) d = (g) (h) d = 2.09 Figure 7: Phase portraits for various values of d corresponding to Figure 4

16 2194 K. A. Hasan and Muzafar F. Hama period-6 orbits, period 36 orbits, period 64 orbits, quasi- periodic orbits and attracting chaotic sets. 5 Conclusion In this paper we conclude that the discrete-time pry-predator model (3) has complex dynamics and we show that the unique positive fixed point of model (3) can undergo flip bifurcation and Hopf bifurcation where d varies in the small neighborhood of B 1 or B 2 and C. More over system (3) displays much interesting dynamical behaviors,including period -6,36,64 orbits invariant cycle, cascade of periodic -doubling, quasi- periodic orbits and the chaotic sets. These result reveal far richer dynamics of the discrete model compared to the continuous model. References [1] H.N. Agiza, E.M. Elabbasy, H. El-Metwally, A.A. Elsadany, Chaotic dynamics of a discrete prey-predator model with holling type II,Nonliear Anal. 10 (2009) [2] J.R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency,j. Animal Ecol. 44 (1975) [3] C. Celik, O. Duman, Allee effect in a discrete-time predator-prey system,chaos Solitons Fractals 40 (2009) [4] D.L. DeAngelis, R.A. Goldstein, R.V. O Neill, A model for trophic interaction,ecology, 56 (1975) [5] J. Guckenheimer, P. Holmes, Nonlinear Oscillations, dynamical systems and bifurcation of vector fields, Springer- Verlag, New york, (1983). [6] Z.He,X.Lai, Bifurcation and chaos behavior of a discrete-time predatorprey system,nonliear Anal. 12 (2011) [7] C.S. Holling, The functional response of predator to prey density and its role in mimicry and population regulation,mem. Ent. Soc. Canada, 45 (1965) [8] Z. Hu, Z.Teng, L. Zhang, Stability and bifurcation analysis of a discrete predator-prey model with nonmonotonic functional response,nonliear Anal. 12 (2011)

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