Heteroclinic Bifurcation of a Predator-Prey System with Hassell-Varley Functional Response and Allee Effect

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1 International Journal of ngineering Research And Management (IJRM) ISSN: Volume-05 Issue-0 October 08 Heteroclinic Bifurcation of a Predator-Prey System with Hassell-Varley Functional Resonse and Allee ffect Fuyue Cheng Dongwei Huang Abstract In this aer we consider a Predator and rey system model with Hassell-Varley function and Alle effect. Firstly we discuss the comlex dynamic behaviors including the existence stability of internal equilibrium. Moreover imlementing state feedback controls so that the Heteroclinic bifurcation is existed. Finally choosing as a control arameter we obtain the sufficient conditions for system (.)the existence and uniqueness of order- eriodic solution by using the geometry theory of semi-continuous dynamic systems.based on the method of successor function we rove the existences of order- eriodic solutions. Index Terms Heteroclinic bifurcation; functional resonsel; Allee effect I. INTRODUCTION Hassell-Varley Predators -rey models with Hassell-Varley functional resonse are one of the dominant themes in both ecology and mathematical ecology due to its universal existence and imortance. A general redator-rey model with Hassell-Varley tye functional resonse may discuss in the []. Menwhile Predators -rey models with Alle effect is described by W[ 4]. However the analysis of redators -rey models with Hassell-Varley functional resonse and Alle effect is rare. Furthemore ecological biological are often interfered by human exloiting activities in imulsive atterns. Imulsive model has been used to multile fields widely with threshold such as virus control [56] ecological management and est govern[78]. Chen et al.[9.0] has obtained Geometric theory of the imulsive system. Qualitative roerties such as the uniqueness existence and stability of order- eriodic solutions are robed with the successor function []. The control with threshold has been used widely in the redator-rey model [] in which function resonse with Hassell Varley or Alle effect is absence. Meanwhile the most of works have discussed a the eriodic solution induced by function resonse with Holling II. On the other hand the heteroclinic bifurcation has been widely alied in the continuous dynamic system. However it s alication in imulsive dynamical system was little due to the difficulty caused by discontinuities even though. In this article a redator-rey system model with Hassell-Varley functional and Alle effect is formulated to discuss the heteroclinic bifurcation and the order- eriodic solution induced by state feedback control. The outline is instructed as follows: In Sect. for the redator-rey model without ulse control the existence and stability of the interior equilibrium are discussed. In Sect. heteroclinic cycle is rovided by a stable and ustable mainfold furthermore when the heteroclinic cycle is broken we discuss the order- eriodic solution. II. A PRDATOR-PRY RSPONS MODL P [] resented a general model with Hassell-Varley tye functional interaction. W[] discribed the rey-redator model with Allee effect. In this aer it is assumed that the redator-rey model with Allee effect and Hassell-Varley tye functional interaction which is descibed by /. Then a grazing system for redator (P) -rey (H ) with Hassell-Varley interaction and Allee effect is given: dh H B ah H dt K K dp fh P D dt mp H chp mp H (.) The arameters a c m b f K and D are ositive constants. (.) H and P denote resectively the densities of rey and redator at time T. The arameters K a D denote the constant carrying caacity rey intrinsic growth rate redator death rate resectively. / (.) The redation term c /( mp H ) often reresents a function resonse term. m is half saturation coefficient and c is the maximum consumtion rate er herbivore. Next using t at h H K P b B/ K the model (.) can be rewritten by dh h d d hh b sh h h h F( h ) G( h ) (. ) (. ) Manuscrit received Oct 6 08 Fuyue Cheng School of Science Tianjin Polytechnic University Tianjin 0087 P.R. China Dongwei Huang Key exeriment laboratory of textile Tianjin Polytechnic University Tianjin 0087 P.R. China m K c K s a K m f a D d. f (.) 5

2 Heteroclinic Bifurcation of a Predator-Prey System with Hassell-Varley Functional Resonse and Allee ffect. Qualitative Analysis From the view of biology the dynamics of (.) are centered on the first closed quadrant R. Theorem. We assume a ositive and invariant box ( h R ): 0 h Q0 Q in R t where Q is a ositive constant. such that all solutions of (.) with non-negative initial values aroach as Proof. Obviously h( t) 0 and ( t ) 0 hold for all initial values in R. Again from the result dl dh s L h 0 / on a straight line L : h = 0 we know that the trajectories ass through L from the right to the left. Define V ( h ) h with a undetermined ϑ > 0. When ϑ > k the curvev ( h ) 0 intersects with axis and the line h = at the oints (ϑ0) and F(kϑ ) resectively. Then we have d V F h hh b- d -d. dv Select a large enough ϑ(> k) such that F 0. Then the trajectories of system (.) cross the line F from the uer right into the lower left.define Q = max{ϑ } with ϑ >. Then one gets a ositive and invariant box as a result that all solution with non-negative initial values draw near when t +. The roof is comleted.. Stability analysis In this subsection we will research the qualitative characteristic of system (.) in R. The equilibria satisfy the following equation. dh d h hh b d h h sh h 0 0 (. 4) By simle calculation we can obtain that system (.) has two ositive boundary equilibrium oints (b 0) and ( 0) and a ositive equilibrium ( h ). Next we will describe the stability of equilibria (b 0) ( 0) and ( h ). its Jocobian matrix is J h h (- h ) ( h )( h / 4 The characteristic equation at is b) - sh h / h / / 4 h / h / sh h h / /. (.5) where P Q 0 (.6) 6

3 International Journal of ngineering Research And Management (IJRM) ISSN: Volume-05 Issue-0 October 08 And P h (- h ) ( h )( h b) - h sh / 4 / ( h h ) (.7) Q / 4 / 4 / sh h sh h h (- h ) ( h )( h b) - / / / 4 h h h /. (.8) Through calculations we get Q ( ) 0 Q ( ) 0 Thus (b 0) and ( 0) are saddle. For system (.) the stability of is existed if and only if P 0 and 0 Q (see Fig ). Fig. The stability of interior equilibrium d=/ r=/ =5s=5b=/0.. PRIODICITY DRIVN BY HTROCLINIC CYCL AND ORDR- PRIODIC SOLUTION According to the above analysiswe know that the system (.) has two saddles and a stable ositive oint ( h ). It may be undesirable from biological or economical view. So in this section we will take state feedback control strategy to reduce the quantities of rey and redator by h and resectively when the rey reaches the level Thus we consider the following model:. 7

4 Heteroclinic Bifurcation of a Predator-Prey System with Hassell-Varley Functional Resonse and Allee ffect dh sh h hh b h d h d h h( t ) h( t) h ( t ) ( t). h (.). Where h t ) and t ) denote the level of redator and rey after a imulsive control is emloyed at time t. ( ( Theorem. Suose P 0 and 0. Thenfor any 0 system (.) has an heteroclinic cycle if. M: and N: ( ) resectively. For at oint A and the stable manifold Proof. For system (4.) Imulse set and hase set can be denoted by convenience we denote the unstable manifold L of intersects imulsive set L of intersects hase set N at oint B. By the trajectory roerty of system (.) we have that L and L are above the dh vertical isocline 0. The imulse function f ( h) ( ) h is monotonically increasing about h and monotonically decreasing about and so there must exist a fixed value 0 f ( h ) h A h. That mans there must exist a fixed value 0 such that oint A is maed on to the oint B after imulsive effect; that is B such that oint A is maed on to the oint B after imulsive effect. So the the closed curve AB A A constitutes a cycle which asses through the saddles and (see Fig. ). Therefore the system (.) has an order- heteroclinic cycle. Fig.. The existence of order- heteroclinic cycle. d=/ r=/ =5s=5b=/0. 8

5 International Journal of ngineering Research And Management (IJRM) ISSN: Volume-05 Issue-0 October 08 If 0 oint A is maed on to oint A and ( ) h ( ) h M A M A. That means the heteroclinic cycle is broken. Denote the intersection oint of h nullcline dh / 0 and the hase set N as the right oint G and the left oint resectively. First there exists a trajectory assing oint G and which intersects imulsive set M at the oint N and F( G) h N h 0. Second there exists a M G. Then the imulsive function mas M G to oint G trajectory assing oint and which intersects imulsive set M at the oint oint N and F( ) h N h 0. Q between and G in hase set such that F ( Q) 0 M. Then the imulsive function mas M to By the continuous character of successor function system (.) has a oint. It means system (.) has an order- eriodic solution. G G Fig.. The existence of order- eriodic solution. d=/ r=/ =5s=5b=/0. DISCUSS In this aer for a redator-rey model with with Hassell-Varley functional interaction and Alle effect. Firstly we discuss the comlex dynamic behaviors including the existence stability of internal equilibrium. Moreover imlementing state feedback controls so that the Heteroclinic bifurcation is existed. Finally choosing as a control arameter we obtain the sufficient conditions for system (.)the existence and uniqueness of order- eriodic solution if the Heteroclinic cycle is broken. Imulsive control is an effective scheme but the control cost also is non-negligible we will secially calculate cost in the future work. RFRNC [] Zhang Y. Gao S. Fan K. Wang Q.: Asymtotic behavior of a non-autonomous redator-rey model with hassell-varley tye functional resonse and random erturbation. Journal of Alied Mathematics and Comuting 49(-) (05). [] Berec L. Angulo. Courcham F.: Multile Allee effects and oulation management. Trends col. vol (007) [] CourchamF.Clutton-BrockT.GrenfellB.:Inversedensit y deendence and the Allee effect. Trends col. vol (999) [4] Guo H.J. Chen L.S. Song X.Y.: Mathematical models of restoration and control of a single secies with Allee effect. Al. Math. Model (00) [5] Georgescu P. Hsieh Y.H.: Global dynamics of a redator-rey model with stage structure for the redator. Siam Journal on Alied Mathematics 67(5) 79-95(007). [6] Tang S. Xiao Y. Chen L. Cheke R.A.: Integrated est management models and their dynamical behaviour. Bulletin of Mathematical Biology 67() 5-5 (005). [7] Zhang M. Song G. Chen L.: A state feedback imulse model for comuter worm control. Nonlinear Dynamics 85() -9 (06). [8] Fu J. Wang Y.: The mathematical study of est management strategy. Discrete Dynamics in Nature and Society 0(4) (0). [9] Chen L.: Pest control and geometric theory of semi-continuous dynamical system. J. Beihua Univ. () 9-(0).. [0] Chen L.: Theory and alication of semi-continuous dynamical system. J. Yulin Norm. Univ. 4() -0 (0). [] Huang M. Duan G. Song X.: A redator-rey system with imulsive state feedback control. Math. Al. 5() (0). 9

6 Heteroclinic Bifurcation of a Predator-Prey System with Hassell-Varley Functional Resonse and Allee ffect [] Fang D. Pei Y. Lv Y. Chen L.: Periodicity induced by state feedback controls and driven by disarate dynamics of a herbivore lankton model with cannibalism. Nonlinear Dynamics (5) -6 (07). [] Jiang G. Lu Q. Qian L.: Comlex dynamics of a holling tye II reyredator system with state feedback control. Chaos Solitons Fractals () (007). 0