Nonlinear Oscillations in a three-dimensional Competition with Inhibition Responses in a Bio-Reactor

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1 Internatonal Journal of Appled Scence Engneerng 7 5 : 9-5 Nonlnear Oscllatons n a three-dmensonal Competton wth Inhbton Responses n a Bo-Reactor Xaonng Xu a Xuncheng Huang ab* a Yangzhou Polytechnc Unversty Yangzhou Jangsu 5 Chna b RDS Research Center Infront Corp Kearny NJ 7 USA Abstract: A three-dmensonal bo-reactor model of explotatve competton of two predator organsms wth nhbton responses for the same renewable organsm wth reproductve propertes s consdered By usng a Lyapunov functon the center manfold theorem the global stablty the exstence of Hopf bfurcaton lmt cycles are proved Ths result s useful n understng the nonlnear oscllaton phenomena n bo-engneerng Keywords: bo-reactor; center manfold theorem; Lyapunov functon; Hopf bfurcaton; lmt cycles; nonlnear oscllaton Introducton The oscllaton s an mportant phenomenon n many natural systems n bology ecology bo-chemstry bo-engneerng A typcal example s the cyclc behavor n bo-reactors In ths paper we study the competton of two predators competes explotatvely for a sngle prey speces n a bo-reactor When the prey speces s non-renewable many results have been reported (see [] for example) But there are not so many results reported for the case when the prey speces s renewable wth reproductve propertes --- a more classc prey Examples of competng for a renewable resource wth some numercal smulatons can be found n Hsu Hubbell Waltman [4] McGehee Armstrong [5] Koch [6] In most of such models t s assumed that the predators consume the nutrent (prey) the consumed nutrent converted to growth s proportonal to consumpton Nutrent uptake (consumpton) s usually taken to be of the Monod (or Mchaels-Menten) form: mxs /( a [4] But n realty other prototypes of functonal responses are also possble [7] In populaton modelng the global stablty of the model s often establshed by constructng a Lyapunov functon Once the Lyapunov functon s obtaned the global stablty follows drectly from the LaSalle s nvarant prncple [9] Then from the global asymptotcal stablty the Hopf bfurcaton follows by the center manfold theorem A lmt cycle of a mathematcal model s related to the nonlnear phenomena of the correspondng system Thus the study of lmt cycles s helpful n understng the oscllaton of the nonlnear system The exstence of lmt cycles s often follows from the Hopf bfurcaton It s known that to establsh the exstence of lmt cycles n n dmensonal system for n s qute dffcult Because the powerful tools n the plane systems lke Poncare-Bendxon theorem cannot be appled * Correspondng author; e-mal: xh@yahoocom Accepted for Publcaton: October 7 7 Chaoyang Unversty of Technology ISSN Int J Appl Sc Eng 7 5 9

2 Xaonng Xu Xuncheng Huang drectly to the ones of n (see counterexamples []) Therefore the method we use n ths paper s also nterestng n mathematcal analyss The Mchaels- Menten type of response s monotonc but the nhbton type s not In the lterature there are not so many results reported for the models wth non-monotonc response functons The study of the three-dmensonal competton model wth the non-monotonc nhbton response certanly has some nterests n bo-mathematcal modelng In ths paper a three-dmensonal bo-reactor model of explotatve competton of two predator organsms wth nhbton responses for the same renewable organsm wth reproductve propertes s consdered We shall use the Lyapunov functon method the center manfold theorem to prove the global stablty the exstence of Hopf bfurcaton lmt cycles We would lke to menton that the Ardto-Rccard type Layapunov functon for the system dscussed n ths paper was used before by Chu Hsu n the three-level food-chan model [78] The model our man results are presented n the next secton the proofs are lsted n the Appendx The Model Man Theorems The competton model of two predators competes explotatvely for a sngle prey speces n chemostat wth nhbton response dfferent death rates takes the form: ds S S x S x γ S( ) dt K ( a ( b δ ( a ( b δ dx S d x dt ( a ( b dx S d x dt ( a ( b S() > x () > x () > () The meanng of the symbols s lsted n the followng table: Table Meanng of the symbols Mathematcal Bologcal Meanng Symbol x two predators S prey speces γ growth rate K carryng capacty of the renewable resource S a b half saturaton constants d death rate of predator x m maxmum predaton rate δ yeld constant for x Notce that n model () the predators: x consumes the prey wth functonal response S of nhbton type ( a ( b For smplcty we wll assume the yeld constant δ n the followng dscusson because f not a varable transformaton can always make the constant yeld as We shall use a corollary to the center manfold theorem to prove the exstence of the 4 Int J Appl Sc Eng 7 5

3 Nonlnear Oscllatons n a three-dmensonal Competton wth Inhbton Responses n a Bo-Reactor Hopf bfurcaton the lmt cycles of the system Our dscusson s on the set R {( S x x) S x x } wth the followng basc assumptons for the parameters: for each ( B ) : m a b ; ( B ) : m a b ( m a b) 4ab > K ; ( B ) : K > a b; The purpose of these basc assumptons s to guarantee that our dscusson s lmted n the frst quadrant due to bologcal reasons For example because of ( B ) only the usual three equlbrum ponts n R : E : ( K) E: ( λ h( λ )) E:( λ h( λ )) need to be consdered It s easy to verfy that the solutons of system () are bounded postve for all t > St ( ) Kfor t suffcent large [7] For each λ λ are the solutons of the equaton S d ( a ( b Then () ( m a b m a b ab ) ( m a b m a b ab ) λ ( ) 4 λ ( ) 4 () By ( B ) λ > K so there are only three equlbrum ponts n R that we need to consder: E :( K) E:( λ h( λ )) E:( λ h( λ )) where for each h ( S ) s defned as γ S h( ( )( a ( b (4) K It follows that λ represent the break-even concentratons the values of the nutrent where the dervatves of x are zeros And x ( ) h S s the prey soclne when x so s x h ( when x s absent Regardng the prey soclne x h ( as shown n Fgure there exsts an S ( K) such that h( S ) where ( ) S K a b K a b ab Ka Kb / Also there exsts an S ( S K ) such that h( S ) h() γ ab / (5) Fgure The prey soclne x h( when x Int J Appl Sc Eng 7 5 4

4 Xaonng Xu Xuncheng Huang For the stablty of the equlbrum E we have Theorem If λ < λ f S λ then E s globally asymptotcally stable; n other words ( St ( ) x( t) x( t)) ( λ h( λ)) as t The proof can be found n Appendx Now let μ be the bfurcaton parameter rewrte system () n μ as follows: dx f( X μ) (6) dt The stablty of an equlbrum the Hopf bfurcaton are connected by the center manfold theorem Theorem A Let W be an open set n R O () W the analytc functon f s defned as f : W ( μ μ ) R where μ s a small postve number Denote the Jacoban of f at ( X μ ) ( O) as J( f ( O )) Assume () system (6) has ( ) as ts equlbrum pont for any μ ; () the egenvalues of J( f( O)) are ± β ( μ) ± β() α( μ) α() μ μ whch satsfy the cond- β() > α() < tons Then f ( ) s asymptotcally stable at μ unstable on μ > there exsts a suffcently small μ μ > such that system (6) has an asymptotcally stable closed orbt surroundng ( ) The proof of Theorem A can be found n [4] In order to use ths theorem to establsh the exstence of bfurcaton we need the followng theorem Theorem If λ < λ S > λ E s unstable Now assume μ S λ s a bfurcaton parameter We are gong to prove the followng theorem Theorem If λ < λ then system () undergoes a Hopf bfurcaton at μ S λ the perodc soluton created by the Hopf bfurcaton s asymptotcally stable for < S λ << Recently a qute smlar food chan model but wth Monod functonal response s publshed [6] wth some numercal results The lmt cycles n the numercal smulaton of [6] take the followng forms whch gve us some dea about the locatons shapes of lmt cycles n the model () (Fgure ) Fgure Example of lmt cycles of competton n bo-reactor 4 Int J Appl Sc Eng 7 5

5 Nonlnear Oscllatons n a three-dmensonal Competton wth Inhbton Responses n a Bo-Reactor For the followng dscusson we recall a result from our prevous work [8] Consderng the system dx φ( x)( F( x) π( y)) dt dy ρ( y)( r ψ( x) ξ( y)) dt (7) n whch the bologcal meanngs of the functons parameters n (7) are as n [8] Let E be the equlbrum pont of system (7) Assume the followng assumptons are satsfed: ( ) H φψ π ξ C [ ) F C ( ) F > φ() π() ρ() ξ() φ for all x π ρ ξ for all y > ; Also there exsts an x such that ψ( x ) r for x x ψ ( x) > ;For x k φ( x) s bounded by a lnear functon ( H ) The curve π ( y) F( x) s defned for all x > ψ ( x) ξ ( y) r s for y > r ( H ) There exsts k > x such that F(k) F( k ) < F(x) > for all < x < k ; for any k > k F( k) f F( k ) It s proved n [8] that Theorem B If E s unstable then the system (7) has at least one lmt cycles around E The followng result s for the case when E ( λ h( λ)) s unstable The projectng system of () onto the plane x takes the form: ds S S γ S dt K a S b S dx S d x dt ( a ( b S() > x () > ( ) ( )( ) δ x (8) It follows that the pont E ( λ h( λ)) s the equlbrum of (8) It s easy to see that E s unstable snce S > λ In other words E s on the two dmensonal unstable manfold of E Consder S as x x as y then system (8) s reduced to a specal case of system (7) wth S φ ( ( a ( b π x ( x ) δ r d S r S F( h( ( )( a ( b K ρ( x ) x ξ ( x ) S ψ ( ) ( a ( b r S F( h( ( )( a ( b K ρ ξ ψ S (9) ( x) x ( x) r d ( ( a ( b The basc assumptons ( H H) are satsfed by Theorem C for system (8) there exsts at least a lmt cycle around E Therefore we have Theorem 4 If λ < λ system () has at least one lmt cycle around the equlbrum E Dscusson Competton between speces explotng a common prey speces s probably frequent occurrence n both nature laboratory However not many theoretcal work has been done on such systems [475] Moreover n most of the populaton models the functonal responses are chosen to be some monotonc functons such as Monod (or Mchaels- Menten) type But n real world applcatons t s not always the case The one wth non-monotonc nhbton response s of course worth a further study Int J Appl Sc Eng 7 5 4

6 Xaonng Xu Xuncheng Huang It looks to us the methods used n Secton for the equlbrum E ( λ h ( λ )) s also workng for the equlbrum E( λ h( λ )) For example f we defne h( λ) h( F ( S ( mab) ξ λ ab d ξ λ m ξ It follows that () ( ) S K a b K a b a b Ka Kb / () Theorem A s also vald for S λ Moreover n the proof of Theorem n the Appendx f we use λ λ nstead of λ < λ t seems that the proof s stll workng Therefore an open problem arses what would happen f λ λ the two predator speces havng same break-even concentraton? Both survve? A further study of ths phenomenon must be very nterestng n the bo-mathematcal modelng Some numercal smulaton of lmt cycles for the food chan model wth Monod functonal response can be found n [6] As s well known a lmt cycle n a mathematcal model corresponds to the nonlnear oscllaton phenomena n the bo-reactor system Thus the study of lmt cycles of the model s useful n analyzng the behavor of the reactor Actually the reactng behavor of the food-chan bo-reactor system s very complcated Computer smulaton shows that t ncludes statonary cyclc chaotc coexstence [9] Therefore a further mathematcal analyss of the food-chan s defntely necessary Acknowledgement The authors would lke to thank the revewers for ther very useful comments Appendx Before we prove Theorem we need a useful lemma At frst we defne an auxlary functon F( S ) : ( λ) U ( λ K) R as where S ( mab) ξ λ ab d ξ m ξ λ mab ( S λ ) λ m ς ln S ln λ ( S λ ) for some ς ( S λ ) U ( λ ς Thus t can be verfed that S ( mab) ξ ξ ab d ξ > for S ( K ) S λ m ξ λ () In fact f S λ then ς > λ f S < λ then ς < λ therefore S ( m a b) ξ λ ab d ξ λ mξ mab mζ S λ ζ λ ( )( ) > The followng result s necessary for the nvestgatng of the stablty of the equlbrum E As shown n Fg by the defnton of h ( ) S t s easy to see that there exsts an S ( K) such that h( S ) where ( ) S K a b K a b ab Ka Kb / () Moreover there also exsts S ( S K ) that such 44 Int J Appl Sc Eng 7 5

7 Nonlnear Oscllatons n a three-dmensonal Competton wth Inhbton Responses n a Bo-Reactor h( S ) h() γ ab / Lemma If S λ then there exsts a θ > such that max F( θ mn F( (4) S λ λ < S K Proof We dvde the proof nto three dfferent cases: () S < λ S () S λ () λ S We shall fnd a θ such that (4) holds n each case The proof of (): S < λ S Consder the curve: x h ( n the S x plane as t s shown n Fg t can be verfed that there exsts S ˆ [ ) S such that h ( S ˆ ) h ( λ ) By the defnton of F ( S ) we have lm F( F( Sˆ ) S lm F( lm F( S λ S λ > ˆ for S ( U ( λ K] F ( for S ( Sˆ < λ) Suppose max F( > mn F( S λ λ < S K Then there exsts π > such that the equaton F( π has three dstnct roots of whch two are n ( S ˆ ) one n ( λ K] Let r r r be the three roots then < r ˆ < r < S < λ < r K Consder the equaton H ( S ( mab) ξ λ ab h( λ) h( π dξ λ m ξ (5) It has four roots: λ r r r n[ K ] By a smple calculaton 6γ π ( m a b) λ H ( > K ms From Rolle s Theorem there exsts ζ ( K) such that H ( ) ζ Ths contradcton mples the exstence of θ > such that S λ max F ( θ mn F( λ < S K Snce F ˆ ( < for S ( S λ) max F ( S ) < max F ( ) ˆ S S λ S S Obvously ths θ satsfes the hypothess (4) The proof of (): S λ In ths case we have lm F( lm F( lm F( S S λ S λ < for S ( λ ) F ( > for S ( λ K] It s easy to see that any satsfy formula (4) The proof of (): λ S It follows that lm F( S θ ( mn F( ) wll λ < S K γλ ( λ K a b) lm F ( > S λ ( mab) (by L Hosptal law) F( for S ( K] F ( ) S s contnuous n ( K ] Let ηλ η ( K / λ ) η Then γ h( ηλ ) ( K ηλ )( a ηλ )( b ηλ ) K Int J Appl Sc Eng

8 Xaonng Xu Xuncheng Huang F( ηλ) m a b (( ηλ λ ) λ (lnηλ ln λ )) m h( λ ) h( ηλ ) δγλ ( η ) (( η ) λ a b K) ( mab)( η--ln η) η ( K / λ ) η Consder the functon ( ) f ( η) ( η ) ( η ) λ a b K π( m a b)( η ln η) (6) for η ( K / λ ) η where π s a postve constant whch wll be determned later It s easy to see that lm f ( η) f () η f ( η) 6λη ( a b K) π( m a b)/ η Let η We can choose ( λ a b K) π π > mab (7) such that f () In other words when π takes the value π η s an nflecton pont of the curve y f ( η) Snce λ S λ a b K > π > Now by f ( η) 6 λ π( m a b)/ η > f( η ) > f() In other words f ( η) s ncreasng on η ( K / λ ) f ( η ) > f () that s ( η ) (( η ) λ a bk) Thus ( m a b)( ηln η) > π (8) δγλ ( η ) (( η ) λ a bk) F ( ( mab)( ηln η) δγλ δγλ ( λ a bk) > π ( mab) S ( λ K) Moreover f η () then S ( λ ) f( η ) < f() whch mples ( η ) (( η ) λ a bk) Thus ( m a b)( ηln η) < π S ( λ ) (9) δγλ δγλ ( λ a b K) F( < π S ( λ ) ( mab) Therefore we always can choose θ δγλ ( λ a bk) ( mab) such that the hypothess (4) s satsfed We complete the proof of Lemma Proof of Theorem Let for η ( K / λ ) or S ( K) Therefore f ( η) s ncreasng f ( η ) > onη ( K / λ ) Ths mples that f ( η) s ncreasng on η ( K / λ ) S θ ( mab) ξ ξ ab V( S x x) x dξ λ m ξ x θ θ ξ ξ h h λ dξ cx x ( λ) ( ( )) () 46 Int J Appl Sc Eng 7 5

9 Nonlnear Oscllatons n a three-dmensonal Competton wth Inhbton Responses n a Bo-Reactor θ c ( ) wll be determned later It s easy to see that V( S x x) C ( R R) R {( S x x) S x x } V( λ h( λ )) V( S x x) > for ( Sxy ) R /{ E} The dervatve of V along the trajectory of system () s V& ( S x x ) θ ( mab) S S ab S S S γ ms K ( a ( b ( a ( b x S x x ( ) ( ) S m a b ab S θ θ θ ξ ξ x h λ x θx dξ d x λ mξ ( a ( b θ S θ S cθ x d x cx d x ( a ( b ( a ( b Denote where V& ( S x x ) V V V ( m a b) SS ab γ V x K S a S b S h θ ( )( )( ) ( λ) ( a ( b K S ( mab) SS ab θ d ξ λ mξ θ λ V cx x d ( a λ)( b λ) θ ( mab) S S ab S V x x ms ( a ( b S S λ cθ d c ( a ( b ( a ( b ( a λ)( b λ) By Theorem there exsts θ > such that (4) holds Notce that λ s as defned n () ( m a b) SS ab ( Sλ )( S λ ) () If S < λ ( m a b) SS ab ( a ( b S λ ξ ξ mξ < h( λ ) h ( < θ ( m a b) ab d ξ Therefore S ( mab) ξ ξ ab h( h( λ) > θ dξ λ m ξ thus V Smlarly f S λ we also have V Also snce λ < λ V for any c We now just need to show that there exsts a c constant or a functon of S such that V In the case of c s a functon of S as we wll see c ( λ ψ( λ) a negatve term θ θ of cxx ψ( λ) xx wll be added nto V ( S x x ) whch s stll negatve We shall fnd the c n two cases: () a a or b b () a < a b < b In the frst case of a a or b b snce λ < λ by Markus theorem ([]) we can follow the same argument of Theorem 4 n [] for any number c the soluton of system () satsfes that ( St ( ) xt ( ) yt ( )) ( λ h( λ )) as t Ths means that E( λ h( λ )) s globally asymptotcally stable We just need to fnd a c for the second case of a < a b < b Let ( mab) SS ab Δ ( ( a ( b m cθ d ( mab) S S ab ( a ( b S λ c ( a ( b ( a λ)( b λ) It follows that λ λ () ( Sλ)( λ Δ ( ( Sλ)( λ cθd m( a ( b ( a ( b ( Sλ )( ab Sλ ) c ; ( a S )( b S )( a )( b ) Int J Appl Sc Eng

10 Xaonng Xu Xuncheng Huang or Sλ ( λ ( a ( b Δ ( ( a ( a ( b ( b m Defne c ( ab Sλ)( a ( b cθd( λ ( a ( b ( a λ)( b λ) a S b S m ( a λ)( b λ) () ( )( ) Ψ ( ( ab Sλ )( a ( b θd ( λ ( a ( b That s (4) Ψ S a a b b m ( a b Sλ ) θλ ( ( ) Snce a S b S ( a λ)( b λ) Θ( Ψ ( a a b b ( ab Sλ) θλ ( a S b S ( a λ)( b λ) where a S ( b ( a λ)( b λ) (5) a a b b a a b b Θ ( θ θ( λ a S b S ( a b S a θλ ( a b b λ < (6) ( ab Sλ )( a ( b Δ ( θd ( λ ( a ( b ( a λ)( b λ) ( λ( Sλ) ( ( λ ) ( ) Ψ Ψ ( a ( a ( b ( b (snce Ψ( ncreases) (7) Note that Δ( s always negatve f S λ Therefore V& ( S x y) V V V By the LaSalle s nvarant prncple all trajectores tend to the largest nvarant set n Λ {( Sxy ) V } Ths requres S λ y To make { S S λ} nvarant under the condton y t follows λ λ S γλ x K ( a λ)( b λ) In other words γ λ x ( a λ )( b λ ) h( λ ) K (8) Therefore { E } s the only nvarant set n Λ We thus complete the proof of Theorem Proof of Theorem Consder the Jocoban of system () at E J( E ) ( a j ) j It follows that ts characterstc equaton s ( r a )( r a r a a ) (9) It follows that Ψ ( > snce a a > b b > By ( B ) λ > K then ab λλ > λ K or ab λk> whch mples λ λ> λ> ab S ab K ab K Snce ab λ > we can choose c ( λ Ψ ( λ ) > It follows that ( ab λ ) a ( / K) h( ) a γ λ λ ( a λ) ( b λ) λ ( a λ )( b λ ) ( ab λ ) λ a h( ) a d λ ( a λ) ( b λ) ( a λ)( b λ) Assume the three roots of (9) as r r r It follows that r r a rr aa > (snce ab > λ ) r a < (snce λ < λ ) 48 Int J Appl Sc Eng 7 5

11 Nonlnear Oscllatons n a three-dmensonal Competton wth Inhbton Responses n a Bo-Reactor Therefore f a > r r have postve real part E s unstable; < r r have negatve real part E s stable Notce that ( ab λ ) γ λ a K a b γ ( λ / ) ( λ )( λ ) ( a λ) ( b λ) K γλ ( λ K a b λ K a b ab ) Ka ( λ)( b λ) ( ) ( ) Snce S s the only pont such that h( S ) on ( K ] t follows that f S < λ a < E s stable; f S > λ a > E s unstable but wth a one-dmensonal stable manfold The proof of Theorem s complete Proof of Theorem Snce λ μ S ( a λ)( b λ) m system () can be λ wrtten n μ as follows: S x x ϕ ( S x x μ) ( μ) ( μ) ϕ S x x ϕ S x x Use the varable changes: S S λ x x h ( λ ) x x system () n varables S x x s dx f( X μ) () dt whose Jacoban s denoted as J( S x x ) Consder system () ts Jacoban at μ ( S x x ) () J( f( O )) J ( S x x ) ( S x x ) () μ J( S x x ) Sx x λ h λ ( ) ( ( )) S λ Its characterstc equaton has the egenvalues: ± β () α () where β () λ( ab λ ) > ( a λ) ( b λ) α λ λ λ () d < (snce < ) ( a λ)( b λ) By Theorem A when λ < λ S λ the equlbrum E s globally asymptotcally stable by Theorem when S > λ t s unstable Therefore the hypotheses of Theorem B are satsfed Actually we have () The equlbrum of system (): O () n the S x x coordnate or E ( λ h( λ )) n Sx x s globally asymptotcally stable f μ ; () t s unstable f μ > Therefore system (8) undergoes a Hopf bfurcaton at μ so does system () at S λ From Theorem B t follows that for a suffcent small μ μ > system () has an asymptotcally stable closed orbt surroundng () In other words for < S λ << system () has an asymptotcally stable closed orbt surroundng E( λ h( λ )) The proof of Theorem s complete References [ ] Smth H L Waltman P 995 The Theory of the Chemostat Cambrdge Cambrdge Unv Press [ ] Wolkowcz G Global Lu Z 99 dynamcs of a mathematcal model of competton n the chemostat: general Int J Appl Sc Eng

12 Xaonng Xu Xuncheng Huang response functon dfferent death rates SIAM Journal of Appled Math 5(): - [ ] Hsu S B Hubbell S P Waltman P A 978 contrbuton to the theory of competng predators Ecologcal Monograph 8: 7-49 [ 4] Hsu S B Hubbell S P Waltman P 978 Competng predators Sam Journal of Appled Mathematcs 5(4): [ 5] McGehee R Armstrong R A 977 Some mathematcal problems concernng the ecologcal prncple of compettve excluson Journal of Dfferental Equatons : 5-7 [ 6] Koch A L 974 Compettve coexstence of two predators utlzng the same prey under constant envronment condtons Journal of Theoretcal Bology 44: 7-86 [ 7] Yang R D Humphrey A E 975 Dynamcs steady state studes of phenol bodegradaton n pure mxed cultures Botechnology Boengneerng 7: -5 [ 8] Huang X C Zhu L M 5 Lmt cycles n a general Kolmogorov model Nonlnear Analyss: Theory Method Applcatons 6(8): 9-44 [ 9] Hsu S B 5 A survey of constructng Lyapunov functons for mathematcal models n populaton bology Tawanese Journal of Mathematcal 9(): 5-7 [] Huang X C Zhu L M Wang Y M A note on competton n the boreactor wth toxn Journal of Mathematcal Chemstry to appear n 7 [] D Heedene R N 96 A thrd order autonomous dfferental equaton wth almost perodc solutons Journal of Mathematcal Analyss Applcatons : 44-5 [] Schwetzer P A 974 Counterexample to the Serfert conjecture openng closed leaves of folatons Amercan Journal of Mathematcs :: 86-4 [] Zhang J 987 The Geometrc Theory Bfurcaton Problem of Ordnary Dfferental Equatons Pekng Unversty Press Bejng [4] Huang X C Zhu L M Wang Y M A note on competton n the boreactor wth toxn Journal of Mathematcal Chemstry to appear n 7 [5] Huang X C 988 A Mathematcal Analyss on Populaton Models PhD Dssertaton Mlwaukee Marquette Unversty [6] Wang Y Q Jng Z J 6 Global qualtatve analyss of a food chan model Acta Mathematca Scenta 6A: 4-4 [7] Chu C H Hsu S B 998 Extncton of top-predator n a three-level food-chan model Journal of Mathematcal Bology 7: 7-8 [8] Chu C H 999 Lyapunov functons for the global stablty of competng predators Journal of Mathematcal Analyss Applcatons : -4 [9] Deng B Food chan chaos due to juncton-fold pont Chaos (): [] Deng B 4 Food chan chaos wth canard exploson Chaos 4(4): Int J Appl Sc Eng 7 5