Hopf-Fold Bifurcation Analysis in a Delayed Predator-prey Models

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1 Vol.37 (SUComS 6), pp Hopf-Fold Bifurcation Analysis in a Delayed Predator-prey Models Shuang Guo, Xiuli Li, Jian Yu, Xiu Ren Department of Mathematics, Teacher Education Institute, Daqing Normal University, Daqing 637, China Shuang Guo: address: guofeixue73@63.com. Abstract. In this paper, we analyzed the distribution of the roots of the associated characteristic equation for the Gause type predator-prey model. the point of bifurcation and a group of conditions of existence of Hopf-Fold bifurcation were obtained at the coexisting equilibrium. There are complex dynamic phenomenons such as periodic motion, quasi--periodic motion and bursting behavior by the numerical simulations. Keywords: predator-prey models; delay; Hopf-Fold bifurcation; bursting behavior Introduction In nature, the population will generally experience some periodic oscillation. It can be regarded as by the impact of delay from a mathematical point. See [-8]. For this model, the purpose of current work is to analysis the effect of delay on the dynamics. We choose x g( x), x p( x), q( y) ry, K px where,, K, p, r are positive parameters. So the model is the following form: dx( t) x dt dy( t) y l yx px e x( t ) x K rz (.)s dt px( t ) dz( t) z ( s mry( t )) dt In this paper, we still let as the bifurcation parameter and consider the delay Gause-type predator-prey model. ISSN: ASTL Copyright 6 SERSC

2 Vol.37 (SUComS 6) Hopf-Fold Bifurcation For the sake of convenience, we non-dimensionalizes the Eq.(.), then the Eq.(.) takes the form: dx( t) x x x ay dt b x dy( t) cx( t ) y l rz dt b x( t ) dz( t) z( s dy( t )) dt ) ( ) (.) which satisfies x, y ) ( ), ) ( ), ( z, ( 3 (,, 3 x ( ), y ( ), z ( ), ( ) (,, ) C([,], ) max ( ) :, Where a, b pk 3 R,and is any norm in pk, 3 R. h e l, c, d mr. p Obviously, the delay can't change the number of equilibria and non-dimensionalizes can't change the properties of system. In the following, we always assume Eq.(.) E x*, y*, z * with has a positive equilibrium exists and denote it by x* ( b) ( b) 4b 4as / d, s y d *, cx * z* r( x * b) The characteristic equation at E is given by 3 D, ) a ( b b ) e, (.) a m 3 dz* where m m ( mn m3n3, aby * ( b x*) bcy * ( b x*) b and mm3n3 * x, n If then have l r b, ax * m, 3 ry* b x * m, m, the model (.) will appear Hopf bifurcation. We let a, b and b m,. We take a derivative with respect to Eq.(.), we 58 Copyright 6 SERSC

3 Vol.37 (SUComS 6) dd(, ) b. d This means is the simple root of= Eq.(.) when Now we substitute i ( imaginary parts gives Simplify (.3), we have m. ) into Eq.(.), separating the real and 3 b cos, sin - Eq.(.4) exists the positive root which satisfied (.4), we have (k ) Denoting ) b (.3) 4 b (.4) b, from equation (.3) and k,, k, ( ) ( ) i( be the root of Eq.(.3) satisfying ( ) ( ), ( x*) denote q* ( m x * Theorem. Suppose q q * and i,, ), then we have the following theorem., then all the roots of Eq.(.3), except and, have negative real parts. Proof On the basis of the foregoing we know that Eq.(.3}) has a single zero root and a simple pair of pure imaginary roots. Suppose that Eq.(.3) has a root with positive real part denoted i when q q * and. Let ( ) i( ) be the root of Eq.(.3) with q q * satisfying ( ) and ( ). Then there exists a positive number,. such that ( ) when ( ) 3 Because (,) b negative real part when. Furthermore, and D has a single zero root and two roots with d( ) d d Re( ) sign d 3 e b b 3 e b sign Re b Copyright 6 SERSC 59

4 Vol.37 (SUComS 6) 3 cos b sign b sign b This contradicts the conclusion of suppose and completes the proof. Then the system (.) can undergo a Hopf -Fold bifurcation at E when q q *. 3 Numerical simulations and discussions We choose a set of parameters: a., b. 3, c. 53, r., s. 5, d. 55, l., Through a simple calculation, we have q *. 578,. 667, we choose four different sets of parameter values ( q, ) (.44,.5), (.985,.75), (,6978,.65) and ( 5.7,.465) which corresponds to the different parameter curves Fig. Fig 4. 6 Copyright 6 SERSC

5 Vol.37 (SUComS 6) Fig.. E is asymptotically stable when q. 44,. 5 Copyright 6 SERSC 6

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7 Vol.37 (SUComS 6) Fig.. The periodic motions near E when q. 985,. 75 Copyright 6 SERSC 63

8 Vol.37 (SUComS 6) Fig.3. The quasi-periodic motion near E when q. 6978, Copyright 6 SERSC

9 Vol.37 (SUComS 6) Fig.4. The bursting behavior near E when q 5. 7, Conclusion In this paper, we investigate the Hopf-Fold bifurcation in a delayed Gause-type predator-prey model by employing the center manifold theory and normal form method. We have derived stability and bifurcation of the zero solution near the critical value. We show that the dynamics of system (.) near the Hopf-Fold point q *, ) ( depends on the time delay. Under the different delays, complex dynamics, such as the Copyright 6 SERSC 65

10 Vol.37 (SUComS 6) quasi-periodic motion, bursting behavior can be observed. Our investigation shows that the oscillating modes in system (.) largely depend on the time delay. Acknowledgments. The work is supported by Heilongjiang Provincial Natural Science Foundation (No. A33) and the PhD Start-up Fund of Daqing Normal University (No. 4ZR9). References. Freedman, H.I., Waltman, P.: Mathematical analysis of some three-species food-chain models [J]. Mathematical Biosciences, 977, 33(3): Hsu, S. B., Hwang, T. W.: Hopf bifurcation analysis for a predator-prey system of Holling and Leslie type [J]. Taiwanese Journal of Mathematics, 999, 3(): Liu, G.R., Yan, W.P., Yan J.R.: Positive periodic solutions for a class of neutral delay Gause-type predator-prey system [J]. Nonlinear Analysis: Theory, Methods and Applications. 9, 7 (): Chen, L.J., Chen, F.D.: Global analysis of a harvested predator-prey model incorporating a onstant prey refuge [J]. International Journal of Biomathematics., 3(): Zhen, B., Xu, J.: Fold-Hopf bifurcation analysis for a coupled fitzhugh-nagumo neural system with time delay [J]. International Journal of Bifurcation and Chaos., (): Kuang, Y., Beretta, E.: Global qualitative analysis of a ratio-dependent predator-prey system, J. Math. Biol. 36(998) Guo, S., Jiang, W.H.: Global stability and Hopf bifurcation for Gause-type predator-prey system [J]. Journal of Applied Mathematics.. ID: 6798, 7 pages. 8. Beretta, E., Kuang Y.: Global analysis in some delayed ratio-dependent predator-prey systems [J]. 9. Nonlinear Analysis, Theory, Methods & Applications. 998, 3(3): Copyright 6 SERSC