Qualitative Analysis of SIR Epidemic Models in two Competing Species

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1 ntrnatnal Jurnal f Basc & Appld cncs JBA-J Vl: : 87 Qualtatv Analyss f pdmc Mdls n tw Cmptng pcs Abstract Almst all mathmatcal mdls f nfctus dsass dpnd n subdvdng th ppulatn nt a st f dstnctv classs dpndnt upn xprnc wth rspct t th rlvant dsas. n ur wrk w wll classfy ndvduals as thr a suscptbl ndvdual r an nfctd ndvdual r a rcvrd ndvdual. Tw pdmc mdls n tw cmptng spcs ar frmulatd and analyzd. Th tw spcs ar bth subjct t a dsas. W analyz tw dffrnt typs f ncdnc, standard ncdnc and mass actn ncdnc. Thrshlds ar dntfd whch dtrmn th xstnc f qulbra, whn th ppulatns wll survv and whn th dsas rmans ndmc. Als stablty rsults ar prvd. Usng Hpf bfurcatn thry sm rsults f cmplcatd dynamc bhavr f th mdls ar shwn. Wth th ntrnfctn rat f dsas btwn th tw spcs as a bfurcatn paramtr, t s shwn that th mdl xhbts a Hpf bfurcatn ladng t a famly f prdc slutns. ndx Trm cmpttn mdl, Hpf bfurcatn, uth- Hrwtz crtrn,.. TODUCTO ntally, mdls fr clgcal ntractns and mdls fr nfctus dsass wr dvlpd sparatly. t has bn bsrvd that a strng ntractn may ars btwn ths factrs. Whn mmunty s prmnant, s that rcvrd ndvduals nvr ls thr mmunty, th dsas s calld an dsas. t must b ntd that th bhavr f th mdl bcms much smplr [] f n assums that nfctn ncdnc s prprtnal t th nfctd fractns n ach spcs standard ncdnc and nt t th dnsts f nfctvs mass actn ncdnc. Fr human dsass th cntact rat sms t b nly vry wakly dpndnt n th ppulatn sz, s that th standard ncdnc s a bttr apprxmatn. Mdls fr tw spcs whch shar a dsas wthut cmpttn hav bn dscussd n sm paprs [,,, 7,9]. pdmc mdls n cmptng spcs hav als bn studd prvusly. Andrsn and May [] cnsdrd a hstcmpttr-pathgn mdl whch nvlvs tw drct cmpttrs, n subjct t a pathgn. Thy xamnd th ffct f a pathgn n cnvntnal cmpttn. Vnturn [] analyzd th dynamcs f tw cmptng spcs whn n f thm s subjct t a dsas. n hs mdl wth mass actn ncdnc, h btand lmt cycls. Han t al. [8] studd an pdmc mdl f tw cmpttv spcs wthut dsas-rlatd daths. Thy analyzd th ffct f ntr-nfctn f dsas n th dynamc bhavrs f th mdl. Van dn Drssch and man [] nvstgatd th Crrspndng authr: umayyah Batwa -mal: sbatwa@kau.du.sa arah Al-hkh -mal: salshkh@kau.du.sa. Dgtal Objct dntfr nsrtd by umayyah A. Batwa and arah A. Al-hkh Kng Abdulazz Unvrsty Dpartmnt f Mathmatcs P.O.Bx 45, Jddah 44, KA ntractn f dsas and cmpttn dynamcs n a systm f tw cmptng spcs n whch nly n spcs s suscptbl t dsas. anz and Hthct [] cnsdrd sm mdls f, and typ wth frquncydpndnt ncdnc. Tmpkns, Wht, and Bts usd a varatn f th mdl f Bwrs and Turnr [4] wth dnstyndpndnt dath rats and mass actn ncdnc t study th ffcts f a parapx vrus n cmptng squrrl spcs n th Untd Kngdm. n ths papr, w cnsdr th fllwng cmpttn mdl d d W analyz pdmc mdls n th tw cmptng spcs wth th standard ncdnc and th mass actn ncdnc wr bth spcs can b nfctd. Th papr s rganzd as fllws: n sctn, w frmulat th mdl and xplan sm basc rsults n th cmpttn mdl. n sctn, w analyz th standard ncdnc mdl. n sctn 4, w analyz th mass actn mdl. Fnally, n sctn 5, a cnclusn wll b gvn t summarz ur rsults.. FOMULATO OF TH MODL W cnsdr tw cmptng spcs that survv n th sam habtat n th sam rsurcs [6,P.]. Fr xampl, shp and cws grazng n th sam pastur. Frst, w nd t dfn th fllwng ntatns: ₁t and ₂t ar th dnsts f th tw spcs at tm t, ε s th ntrnsc grwth rats, σ ar th strngth f th ntraspcfc cmpttn, α s th cmpttn cffcnt, ε / σ s th carryng capacty f ach spcs n slatn. W nw assum that bth spcs can b nfctd by a cmmn pathgn, whs cycl fllw an schm,.. fllwng rcvry an ndvdual hav prmannt mmunty. ach spcs wll thn b dvdd n a suscptbl class ₁,₂, nfctd class ₁,₂ and a rcvrd class ₁,₂. W lt β b th ntranfctn rats f dsas n spcs, and β j j b th ntrnfctn rat f dsas btwn th tw spcs, and γ s th rcvry rat. Hr w assum that all th paramtr ar nnngatv. Fr th ttal ppulatn sz, w hav + + =, =, JBA-J Aprl J J

2 ntrnatnal Jurnal f Basc & Appld cncs JBA-J Vl: : JBA-J Aprl J J A. Basc rsult n cmpttn mdl Cnsdr frst th cmpttn mdl : d d. Th fllwng can b fund n []. Thr ar fur qulbrum pnts,,,,, and, whr and Lmma.: ystm. always has th qulbra ₀,₁ and ₂. Assumng that all paramtrs ar pstv, ₀ s always unstabl. As fr th xstnc f an ntrnal qulbrum and th stablty f thm all, thr ar 4 gnrc cass:. f ε₁/σ₁ < ε₂/α₂, ε₂/σ₂ < ε₁/α₁ and σ₁/α₁ > α₂/σ₂, thr xsts als a unqu ntrnal qulbrum ₃. ₃ s glbally asympttcally stabl. n ths cas ₁ and ₂ ar unstabl.. f ε₁/σ₁ > ε₂/α₂, ε₂/σ₂ > ε₁/α₁ and σ₁/α₁ < α₂/σ₂, thr xsts a unqu ntrnal qulbrum ₃, whch s a saddl pnt. Bth ₁ and ₂ ar lcally asympttcally stabl.. f ε₁/σ₁ > ε₂/α₂ and ε₂/σ₂ < ε₁/α₁, thr s n ntrnal qulbrum, ₁ s glbally asympttcally stabl and ₂ s unstabl. 4. f ε₁/σ₁ < ε₂/α₂ and ε₂/σ₂ > ε₁/α₁, thr s n ntrnal qulbrum, ₂ s glbally asympttcally stabl and ₁ s unstabl.. COMPTTO MODL WTH TADAD CDC Cnsdr th fllwng autnmus cmpttn mdl wth standard ncdnc: d d d d d d d d. Lt, and dnt th fractns f th classs, and n th ppulatn, rspctvly =,. ystm. s cnvrtd t d d d d d d d d. ystm. cmprss sx quatns, but nly fur ar ncssary, snc + + =. W chs t us as varabls, and and fr =,, btanng th fllwng 6- dmnsnal systm: d d d d d d. hst-pathgn systm strctng systm. t a sngl hst spcs, n btans th fllwng mdl:.4 whr β s th cntact rat and γ s th rcvry rat. ystm.4 has thr qulbrum pnts,,,,

3 ntrnatnal Jurnal f Basc & Appld cncs JBA-J Vl: : 89,,,,, whr b b s th rprductn numbr f th nfctn. Th tw bundary qulbra and always xst. xst f >. Th fllwng lmma can b asly prvd: Lmma.: ystm.4 always has th qulbra ₀ and ₁. ₀ s always unstabl. As fr th xstnc f an ntrnal qulbrum and th stablty f thm, w hav tw cass:. f <, th dsas-fr qulbrum ₁ s glbally asympttcally stabl n th rgn {,,,, }.. f >, thr xsts a unqu ntrnal qulbrum ₂ whch s glbally asympttcally stabl n th rgn {,,, }. n ths cas ₁ bcms unstabl. qulbrum Pnts Of ystm. Lt,, and 4. ystm. has th fllwng qulbra: ₀=,,,,,, ₁=,,ε₁/σ₁,,,, ₂=,,,,,ε₂/σ₂, ₃=,,,,,, whr,. Fnally, w may fnd an ntrnal qulbrum ₄=,,,,, whr, ar th pstv rt f th fllwng quatns β₁₂₂ - γ₁+ε₁-β₁₁+β₁₁+ γ₁/ ε₁₁+β₁₂ + γ₁/ ε₁₂ ₁= β₂₁₁ - γ₂+ε₂-β₂₂+β₂₂+ γ₂/ ε₂₂+β₂₁+ γ₂/ ε₂₁ ₂ = and, ar = γ₁/ε₁ = γ₂/ε ystm. always has th thr bundary qulbra, ₁ and ₂. xst f ε₁/α₁ > ε₂/σ₂, ε₂/α₂ > ε₁/σ₁ and σ₁/α₁ > α₂/σ₂ r all nqualts ar rvrsd. Lcal stablty f th qulbra W usd th Jacban matrx and uth-hurwtz crtra t gt stablty cndtns. W frst gv th fllwng rsult fr uth-hurwtz Crtra whn k=6 whr th cffcnts a₁,a₂,...,a₆ ar all ral hav ngatv ral parts f and nly f a > fr =..6, a₁a₂a₃ > a₃²+a₁²a₄, a₁a₂-a₃a₃a₄-a₂a₅+a₁a₆>a₁a₄-a₅² and a₁a₅-a₃²a₁a₂-a₃>a₁a₁²a₆-a₃a₁a₄-a₅ Th rsults f ths sctn ar summarzd n th fllwng thrm: Frmulas fr A A 4 and A..A 6 can b fund n th appndx Thrm.: ystm. always has th bundary qulbra ₀,₁ and ₂. ₀ s always unstabl. As fr th xstnc f th bundary qulbrum ₃ and an ntrnal qulbrum and th stablty f thm all, w hav th fllwng cass:. f ε₁/σ₁ < ε₂/α₂, ε₂/σ₂ < ε₁/α₁ and σ₁/α₁ > α₂/σ₂, thr xsts anthr bundary qulbrum ₃. ₃ s lcally asympttcally stabl f and nly f A >, A ₃>, A₄> and A₁A₂A₃>A₃²+A₁²A₄. n ths cas ₁ and ₂ ar unstabl.. f ε₁/σ₁ > ε₂/α₂, ε₂/σ₂ > ε₁/α₁ and σ₁/α₁ < α₂/σ₂, th bundary qulbrum ₃ als xsts, whch s lcally asympttcally stabl f and nly f A₁>, A₃>, A₄> and A₁A₂A₃>A²+A₁²A₄. n ths cas ₁ and ₂ ar lcally asympttcally stabl f and nly f ₁₁<, ₁₂< and ₁₁-₁₂->₁₃₁₄.. f ε₁/σ₁ > ε₂/α₂ and ε₂/σ₂ < ε₁/α₁, th qulbrum ₃ ds nt xst, ₁ s lcally asympttcally stabl f and nly f ₁₁<,₁₂<,₁₁-₁₂->₁₃₁₄. n ths cas ₂ s unstabl. 4. f ε₁/σ₁ < ε₂/α₂ and ε₂/σ₂ > ε₁/α₁, als th qulbrum ₃ ds nt xst, ₂ s lcally asympttcally stabl f and nly f ₁₁<,₁₂<,₁₁-₁₂->₁₃₁₄. n ths cas ₁ s unstabl. 5.Thr may r may nt xsts an ntrnal qulbrum ₄. f ₄ xst, thn t s lcally asympttcally stabl f and nly f A > fr =..6, A₁A ₂A ₃>A ₃²+A ₁²A ₄ A ₁A ₂-A₃A₃A ₄-A₂A₅+A₁A ₆>A ₁A ₄-A ₅² and A₁A ₅-A ₃²A ₁A₂-A ₃>A ₁A ₁²A ₆-A ₃A ₁A ₄-A ₅. Bfurcatn Analyss [5] n ths sctn, w dscuss Hpf bfurcatn thry fr systm.. Th ystm cmprss ght quatns, but nly sx ar ncssary, snc = + +. W chs t us as varabls,, and fr =,, btanng th fllwng 6- dmnsnal systm: Lmm.: uth-hurwtz Crtra whn k=6 Th rts f th fllwng charactrstc quatn f a 6 6 Jacban matrx λ⁶+a₁λ⁵+a₂λ⁴+a₃λ³+a₄λ²+a₅λ+a₆= JBA-J Aprl J J

4 ntrnatnal Jurnal f Basc & Appld cncs JBA-J Vl: : JBA-J Aprl J J d d d d d d T us th bfurcatn thrm fr ths systm w nd t dscuss Hpf bfurcatn at an ntrnal qulbrum. Hwvr, thr ar n xplct frmula fr an ntrnal qulbrum, and n gnral, nt vn ts xstnc can b prvd. Hnc w wll study th bfurcatn n th vry spcal cas, whr all th analgus paramtrs ar th sam fr spcs and. Althugh t s a vry partcular cas, t dsplays svral ntrstng bhavrs that can shd lght als utsd f ths spcal structur. amly, w lt β₁₁ = β₂₂=β₁,β₁₂=β₂₁=β₂,γ₁=γ₂=γ ε₁ = ε₂=ε,α₁=α₂=α,σ₁=σ₂=σ Thn th systm bcms asymmtrcal wth rspct t th xchang f and systm. d d d d d d.5 Th ntrnal qulbrum pnt s =,,,,, whr,,,. xsts f >. Th Jacban Matrx f ystm.5 at s gvn by J Whr ] [ Th charactrstc quatn crrspndng t J satsfs λ⁶+a₁λ⁵+a₂λ⁴+a₃λ³+a₄λ²+a₅λ+a₆=.6 Whr A₁ = -₁+₇+₁₀ A₂ = ₁²+4 ₁₇+4₁₁₀-₂₆-₄²+₇²+4₇₁₀+₁₀²-₁₁² A₃ = -₁²₇-₁²₁₀+₁₂₆-₁₇²-8₁₇₁₀- ₁₁₀²+₁₁₁²+₂₆₇+ 4₂₆₁₀+₄²₇+₄²₁₀- ₇²₁₀-₇₁₀²+₇₁₁² A₄ = ₁²₇²+ 4₁²₇₁₀+₁²₁₀²-₁²₁₁²-₁₂₆₇- 4₁₂₆₁₀+4₁₇²₁₀+ 4₁₇₁₀²-4₁₇₁₁²+₂²₆²- 4₂₆₇₁₀-₂₆₁₀²+₂₆₁₁²-₄²₇²-4₄²₇₁₀-₄²₁₀²+ ₄²₁₁²+₇²₁₀²-₇²₁₁² A₅ = -₁²₇²g₁₀-₁²₇₁₀²+₁²₇₁₁²+4₁₂₆₇₁₀ +₁₂₆₁₀²-₁₂₆₁₁²- ₁₇²₁₀²+₁₇²₁₁²- ₂²₆²₁₀+ ₂₆₇₁₀²-₂₆₇₁₁²+₄²₇²₁₀ +₄²₇₁₀²-₄²₇₁₁² A₆ = ₁²₇²₁₀²-₁²₇²₁₁²-₁₂₆₇₁₀²+₁₂₆₇₁₁² +₂²₆²₁₀²-₂²₆²₁₁²-₄²₇²₁₀²+₄²₇²₁₁² n th fllwng, w chs th ntrnfctn rat f dsas btwn th tw spcs β₂ as th bfurcatn paramtr, and fx th thr paramtrs. Thrm.: Assum that xsts and A ar all pstv fr =..6, thn thr s a pstv numbr β₂ such that systm.5 xhbts a Hpf bfurcatn ladng t a famly f prdc slutns that bfurcats frm th qulbrum pnt fr sutabl valus f β₂ n a nghbrhd f β₂. Prf: By Lmma. a ncssary and suffcnt cndtns fr th rts f.6 t hav ngatv ral parts ar: A >, A₁A₂A₃>A₃²+A₁²A₄ A₁A₂-A₃A₃A₄-A₂A₅+A₁A₆>A₁A₄-A₅² and

5 ntrnatnal Jurnal f Basc & Appld cncs JBA-J Vl: : 9 A₁A₅-A₃²A₁A₂-A₃>A₁A₁²A₆-A₃A₁A₄-A₅ W wll hav tw pur magnary rts f and nly f A₁A₂A₃=A₃²+A₁²A₄ fr sm valus f β₂, say β₂=β₂.nc at β₂=β thr s an ntrval cntanng β₂ say β₂ -ε,β₂ +ε fr sm ε> fr whch β₂ β₂ -ε,β₂ +ε. Thus fr β₂ β₂ -ε,β₂ +ε, w cannt hav ral pstv rts. fr β₂=β₂, th charactrstc quatn can b factrd nt th frm λ²+θ₁λ+θ₂λ+θ₃λ+θ4λ+θ 5=,θ >,=.5. n partcular, th st f rts s gvn by Pβ₂={,-, -θ₂,-θ₃,-θ4,-θ5}.thn thr ar tw pur magnary rts fr sm valu f β₂ say β₂=β₂. But fr β₂ β₂ -ε,β₂ +ε th rts ar n gnral frm λ₁β₂ = νβ₂+μβ₂ λ₂β₂ = νβ₂-μβ₂ λ₃β₂ = -θ₂ λ₄β₂ = -θ₃ λ 5β₂ = -θ4 λ6β₂ = -θ5 Frst, substtutng λ β₂, =, nt th charactrstc quatn w gt th quatns 5μ 4 ν²-5μ ν 4 -μ 6 -A μ ν + 5A μ 4 ν+a μ⁴-a 6μ²ν²-Aμ²ν- A4 μ²+ v 6 +A v 5 + A ν⁴+ A ν³+a4ν²+a 5ν+A 6 = 6μ 5 ν+6μν 5 -μ³ν +A μ 5 +5A μν 4 -A μ³ν -4A μ³ν- Aμ³+4A μν³+aμν²+a4μν+a5μ = Dffrntatng wth rspct t β₂, w gt Aβ₂ν β₂-bβ₂μ β₂+cβ₂ = Bβ₂ν β₂+aβ₂μ β₂+dβ₂ = nc Aβ₂ Cβ₂ +Bβ₂ Dβ₂, w hav d [ ] d Thrfr, w can apply Hpf bfurcatn thrm [] t prv that systm.5 xhbts a Hpf bfurcatn at ladng t a famly f prdc slutns that bfurcats frm th qulbrum pnt fr sm β₂ β₂ -ε,β₂ +ε. Ths cmplts th prf. V. COMPTTO MODL WTH MA ACTO CDC Cnsdr th fllwng autnmus cmpttn mdl, whch s smlar t th mdl n th prvus sctn, but hr w us th mass actn ncdnc nstad f th standard ncdnc: d d d d d d d d 4. ystm 4. cmprss ght quatns, but nly sx ar ncssary, snc = + +.W chs t us as varabls, and fr =,, btanng th fllwng 6- dmnsnal systm: d d d d d d 4. hst-pathgn systm strctng systm 4. t a sngl hst spcs, n btans th fllwng mdl: d 4. d d whr β s th cntact rat and γ s th rcvry rat. ystm 4. has thr qulbrum pnts,,,,,,,,, whr s th rprductn numbr f th nfctn. ystm 4. always has th tw bundary qulbra and. xst f >. Lmma 4.: ystm 4. always has th qulbra ₀ and ₁. ₀ s always unstabl. As fr th xstnc f an ntrnal qulbrum and th stablty f thm, w hav tw cass: JBA-J Aprl J J

6 ntrnatnal Jurnal f Basc & Appld cncs JBA-J Vl: : 9. f ₂₀<, th dsas-fr qulbrum ₁ s glbally asympttcally stabl n th rgn {,,, }.. f ₂₀>, thr xsts a unqu ntrnal qulbrum ₂ whch s glbally asympttcally stabl n th rgn {,,,.}. n ths cas ₁ bcms unstabl. qulbrum Pnts Of ystm 4. Lt,. ystm 4. has th fllwng qulbra,,,,,,,,,,,,,,,,,,,,, 4,,,,,, 5=,,,,,, whr,. Fnally, w may fnd an ntrnal qulbrum 6=,,,,, whr, ar th pstv rt f th fllwng quatns β₁₂₂ + β₁₁ - γ₁-ε₁-β₁₁+γ₁/ε₁-β₁₂+γ₁/ε₂₁ = β₂₁₁ + β₂₂ - γ₂-ε₂-β₂₂+ γ/ε ₂-β₂₁+ γ/ε ₁₂ = ystm 4. always has th thr bundary qulbra, ₁ and. xst f >, 4 xsts f > and 5 xsts f ε₁/α₁>ε₂/σ₂, ε₂/α₂> ε₁/σ₁ and σ₁/α₁ > α₂/σ₂ r all nqualts ar rvrsd. Lcal stablty f th qulbra As w dd n sctn w usd uth-hurwtz crtra t gt th fllwng rsult: Frmulas fr A A 4 and A 4..A 46 can b fund n th appndx Thrm 4.: ystm 4. always has th bundary qulbra ₀,₁ and ₃. ₀ s always unstabl. As fr th xstnc f th thr qulbra and th stablty f thm all, w hav th fllwng cass:.f ₂₁>, th qulbrum ₂ xsts. n ths cas ₁ s unstabl..f ₂₂>, th qulbrum ₄ xsts. n ths cas ₃ s unstabl..f ε₁/σ₁<ε₂/α₂, ε₂/σ₂<ε₁/α₁ and σ₁/α₁>α₂/σ₂, thr xsts anthr bundary qulbrum ₅. ₅ s lcally asympttcally stabl f and nly f A >,A>,A ₄> and A ₁A ₂A ₃>A ₃²+A ₁²A ₄. n ths cas ₁ and ₃ ar unstabl. 4.f ε₁/σ₁>ε₂/α₂, ε₂/σ₂>ε₁/α₁ and σ₁/α₁ < α₂/σ₂, th bundary qulbrum ₅ als xsts, whch s lcally asympttcally stabl f and nly f A₁>,A ₃>,A ₄> and A ₁A₂A ₃>A ₃²+A ₁²A ₄. n ths cas ₁ s lcally asympttcally stabl f and nly f ₂₁< and ₃ s lcally asympttcally stabl f and nly f ₂₂<. 5.f ε₁/σ₁>ε₂/α₂ and ε₂/σ₂<ε₁/α₁, th qulbrum ₅ ds nt xst, ₁ s lcally asympttcally stabl f and nly f ₂₁<. n ths cas ₂ s unstabl. 6.f ε₁/σ₁<ε₂/α₂ and ε₂/σ₂>ε₁/α₁, als th qulbrum ₅ ds nt xst, ₃ s lcally asympttcally stabl f and nly f ₂₂<. n ths cas ₁ s unstabl. 7.Thr may r may nt xsts an ntrnal qulbrum ₆. f ₆ xst, thn t s lcally asympttcally stabl f and nly f A 4> fr =..6, A4₁A 4₂A 4₃>A 4₃²+A 4₁²A 4₄ A 4₁A 4₂-A4₃A4₃A 4₄-A4₂A4₅+A4₁A 4₆>A 4₁A 4₄-A 4₅² and A4₁A 4₅-A 4₃²A 4₁A4₂-A 4₃>A 4₁A 4₁²A 4₆-A 4₃A 4₁A 4₄-A 4₅ Bfurcatn Analyss T us th bfurcatn thrm fr systm 4. w nd t dscuss Hpf bfurcatn at an ntrnal qulbrum. Agan w wll study th bfurcatn n th vry spcal cas whr β₁₁ = β₂₂=β₁,β₁₂=β₂₁=β₂,γ₁=γ₂=γ ε₁ = ε₂=ε,α₁=α₂=α,σ₁=σ₂=σ Th ntrnal qulbrum pnt s =,,,,, whr,. xsts f. Th Jacban Matrx Of ystm 4. at s gvn by h h h h4 h5 h6 h7 h8 h9 Whr h h J h4 h5 h h h h 9 h6 h7 h8 h h h h h h4 h h h h8 h9 h h h Th charactrstc quatn crrspndng t J satsfs Whr λ⁶+b₁λ⁵+b₂λ⁴+b₃λ³+b₄λ²+b₅λ+b₆= B₁ = -₁+h₇+h₁₀ JBA-J Aprl J J

7 ntrnatnal Jurnal f Basc & Appld cncs JBA-J Vl: : 9 B₂= h₁²+4 h₁h₇+4h₁h₁₀-h₂h₆-h₄²+h₇²+4h₇h₁₀+h₁₀²-h₁₁² B₃=-h₁²h₇-h₁²h₁₀+h₁h₂h₆-h₁h₇²-8h₁h₇h₁₀-h₁h₁₀²+ h₁h₁₁²+h₂h₆h₇+4h₂h₆h₁₀+h₄²h₇+h₄²h₁₀-h₇²h₁₀- h₇h₁₀²+h₇h₁₁² B₄=h₁²h₇²+4h₁²h₇h₁₀+h₁²h₁₀²-h₁²h₁₁²-h₁h₂h₆h₇-h₁h₂h₆h₁₀+ 4h₁h₇²h₁₀+4h₁h₇h₁₀²-4h₁h₇h₁₁²+h₂²h₆²-4h₂h₆h₇h₁₀-h₂h₆h₁₀²+ h₂h₆h₁₁²- h₄²h₇²-4h₄²h₇h₁₀-h₄²h₁₀²+ h₄²h₁₁²+h₇²h₁₀²-h₇²h₁₁² B₅=-h₁²h₇²h₁₀-h₁²h₇h₁₀²+h₁²h₇h₁₁²+4h₁h₂h₆h₇h₁₀+ h₁h₂h₆h₁₀²-h₁h₂h₆h₁₁²-h₁h₇²h₁₀²+h₁h₇²h₁₁²- h₂²h₆²h₁₀+ h₂h₆h₇h₁₀² -h₂h₆h₇h₁₁²+h₄²h₇²h₁₀+h₄²h₇h₁₀²- h₄²h₇h₁₁² B₆=h₁²h₇²h₁₀²-h₁²h₇²h₁₁²-h₁h₂h₆h₇h₁₀²+h₁h₂h₆h₇h₁₁²+ h₂²h₆²h₁₀²-h₂²h₆²h₁₁²-h₄² h₇²h₁₀²+h₄²h₇²h₁₁² n th fllwng, w chs β₂ as th bfurcatn paramtr, and fx th thr paramtrs. Th prf f th fllwng thrm s smlar t thrm.. Thrm 4.: Assum that xsts and B ar all pstv fr =.6, thn thr s a pstv numbr β₂ such that systm 4. xhbts a Hpf bfurcatn ladng t a famly f prdc slutns that bfurcats frm th qulbrum pnt fr sutabl valus f β₂ n a nghbrhd f β₂. V. COCLUO W summarz th rsult f ths papr n th fllwng tabls, QUL- BUM Cmpttn mdl wth standard ncdnc TC? TABL? s s ₁₁<, ₁₂< and ₁₁-₁₂->₁₃₁₄ and ε₂/α₂ < ε₁/σ₁ s ₁₁<, ₁₂< and ₁₁-₁₂->₁₃₁₄ and ε₁/α₁ < ε₂/σ₂ ε₁/α₁ > ε₂/σ₂, ε₂/α₂ > ε₁/σ₁ and σ₁/α₁ > α₂/σ₂ r all nqualts ar rvrsd A>, A₃>, A₄> and A₁A₂A₃>A₃²+A₁²A₄ 4 May xst A> fr =..6, A₁A₂A₃>A₃²+A₁²A₄ A₁A₂-A₃A₃A₄- A₂A₅+A₁A₆>A₁A₄- A₅² and A₁A₅-A₃²A₁A₂-A₃> A₁A₁²A₆-A₃A₁A₄- A₅ Cmpttn mdl wth mass actn ncdnc QUL- TC? TABL? BUM s s ₂₁< and ε₂/α₂ < ε₁/σ₁ ₂₁> rsult s ₂₂< and ε₁/α₁ < ε₂/σ₂ 4 ₂₂> rsult 5 ε₁/α₁ > ε₂/σ₂, ε₂/α₂ > ε₁/σ₁ and σ₁/α₁ > α₂/σ₂ r all nqualts ar rvrsd A>,A>,A₄> and A₁A₂A₃>A₃²+A₁²A₄ 6 May xst A4> fr =..6, A4₁A4₂A4₃>A4₃²+A4₁²A4₄ A4₁A4₂-A4₃A4₃A4₄- A4₂A4₅+A4₁A4₆>A4₁A4₄- A4₅² and A4₁A4₅-A4₃²A4₁A4₂- A4₃>A4₁A4₁²A4₆- A4₃A4₁A4₄-A4₅ FOMULA FO A----A4 APPD A A₂=γ₁γ₂+γ₁ε₂+γ₂ε₁+ε₁ε₂-γ₂β₁₁-γ₁β₂₂-ε₂β₁₁-ε₁β₂₂+β₁₁ β₂₂ -β₁₂β₂₁+σ₂γ₁ +σ₂γ₂ +σ₂ε₁ +σ₂ε₂ -σ₂β₁₁ - σ₂β₂₂ +σ₁γ₁ + σ₁γ₂ +σ₁ε₁ +σ₁ε₂ -σ₁β₁₁ -σ₁β₂₂ -α₁α₂ + σ₁σ₂ A₃=σ₁γ₁γ₂ +σ₁γ₁ε₂ +σ₁γ₂ε₁ +σ₁ε₁ε₂ - σ₁γ₂β₁₁ - σ₁γ₁β₂₂ -σ₁ε₂β₁₁ -σ₁ε₁β₂₂ +σ₁ β₁₁β₂₂ -σ₁β₁₂β₂₁ +σ₂γ₁γ₂ +σ₂γ₁ε₂ + σ₂γ₂ε₁ +σ₂ε₁ε₂ -σ₂γ₂β₁₁ - σ₂γ₁β₂₂ -σ₂ε₂β₁₁ -σ₂ε₁β₂₂ + σ₂β₁₁β₂₂ -σ₂β₁₂β₂₁ -α₁α₂γ₁ +σ₁σ₂γ₁ -α₁α₂γ₂ + σ₁σ₂γ₂ -α₁α₂ε₁ +σ₁σ₂ε₁ -α₁α₂ε₂ +σ₁σ₂ε₂ + α₁α₂β₁₁ -σ₁σ₂β₁₁ +α₁α₂ β₂₂ -σ₁σ₂β₂₂ A₄=-α₁α₂γ₁γ₂ +σ₁σ₂γ₁γ₂ -α₁α₂γ₁ε₂ - α₁α₂γ₂ε₁ +σ₁σ₂γ₁ε₂ +σ₁σ₂ γ₂ε₁ -α₁α₂ε₁ε₂ +σ₁σ₂ε₁ε₂ +α₁α₂γ₂β₁₁ -σ₁σ₂γ₂β₁₁ + α₁α₂γ₁β₂₂ -σ₁σ₂γ₁β₂₂ +α₁α₂ ε₂β₁₁ -σ₁σ₂ε₂β₁₁ +α₁α₂ε₁β₂₂ -σ₁σ₂ε₁β₂₂ -α₁α₂β₁₁β₂₂ + α₁α₂β₁₂β₂₁ +σ₁σ₂β₁₁β₂₂ -σ₁σ₂ β₁₂β₂₁ FOMULA FO A----A6 c₁=β₁₁-γ₁-ε₁-β₁₁ -β₁₁ -β₁₂ c₂=-β₁₁ -β₁₂ c₃=β₁₂- - c₄=γ₁ c₅=-ε₁ c₆=-σ₁ c₇=-α₁ c₈=β₂₁ JBA-J Aprl J J

8 ntrnatnal Jurnal f Basc & Appld cncs JBA-J Vl: : 94 c₉=β₂₂-γ₂-ε₂-β₂₂ -β₂₂ -β₂₁ c₁₀=-β₂₂ -β₂₁ c₁₁=γ₂ c₁₂=-ε₂ c₁₃=-α₂ c₁₄=-σ₂ A₁=-c₁-c₅-c₆-c₉-c₁₂- c₁₄ A ₂=c₁c₅-c₂c₄+c₁c₆+c₁c₉-c₃c₈+c₅c₆+c₅c₉+c₆c₉+c₁c₁₂+ c₁c₁₄+c₅c₁₂+c₆c₁₂+c₅c₁₄+c₆c₁₄-c₇c₁₃+c₉c₁₂+c₉c₁₄-c₁₀c₁₁+ c₁₂c₁₄ A₃=-c₁c₅c₆+c₂c₄c₆-c₁ c₅c₉+c₂c₄c₉-c₁c₆c₉+c₃c₅c₈+ c₃c₆c₈c₅c₆c₉-c₁c₅c₁₂+c₂c₄c₁₂-c₁c₆c₁₂-c₁c₅c₁₄+c₂c₄c₁₄-c₁c₆c₁₄+ c₁c₇c₁₃ -c₁c₉c₁₂+c₃ c₈c₁₂-c₅c₆c₁₂-c₁c₉c₁₄+c₃c₈c₁₄- c₅c₆c₁₄+ c₅c₇c₁₃-c₅c₉c₁₂-c₆c₉c₁₂-c₅c₉c₁₄-c₆c₉c₁₄+c₇c₉c₁₃+ c₁c₁₀c₁₁+ c₅c₁₀c₁₁-c₁c₁₂c₁₄+c₆c₁₀c₁₁-c₅c₁₂c₁₄-c₆c₁₂c₁₄+c₇c₁₂c₁₃c₉c₁₂c₁₄+c₁₀c₁₁c₁₄ A₄=c₁c₅c₆c₉-c₂c₄c₆c₉-c₃c₅c₆c₈+c₁c₅c₆c₁₂-c₂c₄c₆c₁₂+ c₁c₅c₆c₁₄-c₁c₅c₇c₁₃-c₂c₄c₆c₁₄+c₂c₄c₇c₁₃+c₁c₅c₉c₁₂c₂c₄c₉c₁₂+c₁c₆c₉c₁₂-c₃c₅c₈c₁₂+c₁c₅c₉c₁₄-c₂c₄c₉c₁₄c₃c₆c₈c₁₂+c₁c₆c₉c₁₄-c₁c₇c₉c₁₃-c₃c₅c₈c₁₄-c₃c₆c₈c₁₄+ c₃c₇c₈c₁₃+c₅c₆c₉c₁₂+ c₅c₆c₉c₁₄-c₅c₇c₉c₁₃-c₁c₅c₁₀c₁₁+ c₂c₄c₁₀c₁₁-c₁c₆c₁₀c₁₁+c₁c₅c₁₂c₁₄- c₂c₄c₁₂c₁₄ -c₅c₆c₁₀c₁₁+ c₁c₆c₁₂c₁₄- c₁c₇c₁₂c₁₃+c₁c₉c₁₂c₁₄-c₃c₈c₁₂c₁₄ +c₅c₆c₁₂c₁₄c₅c₇c₁₂c₁₃+c₅c₉c₁₂c₁₄+c₆c₉c₁₂c₁₄-c₇c₉c₁₂c₁₃ -c₁c₁₀c₁₁c₁₄c₅c₁₀c₁₁c₁₄-c₆c₁₀c₁₁c₁₄+c₇c₁₀c₁₁c₁₃ A ₅=-c₁c₅c₆c₉c₁₂+c₂c₄c₆c₉c₁₂+c₃c₅c₆c₈c₁₂-c₁c₅c₆c₉c₁₄+ c₁c₅c₇c₉c₁₃+c₂c₄c₆c₉c₁₄-c₂c₄c₇c₉c₁₃+c₃c₅c₆c₈c₁₄c₃c₅c₇c₈c₁₃+c₁c₅c₆c₁₀c₁₁-c₂c₄c₆ c₁₀c₁₁-c₁c₅c₆c₁₂c₁₄+ c₁c₅c₇c₁₂c₁₃+c₂ c₄c₆c₁₂c₁₄-c₂c₄c₇c₁₂c₁₃-c₁c₅c₉c₁₂c₁₄+ c₂c₄c₉c₁₂c₁₄-c₁c₆c₉c₁₂c₁₄+c₁c₇c₉ c₁₂c₁₃+c₃c₅c₈c₁₂c₁₄ +c₃c₆c₈c₁₂c₁₄-c₃ c₇c₈c₁₂c₁₃-c₅c₆c₉c₁₂c₁₄+c₅c₇c₉c₁₂c₁₃+ c₁c₅c₁₀c₁₁c₁₄-c₂c₄c₁₀c₁₁c₁₄+c₁c₆c₁₀ c₁₁c₁₄-c₁c₇c₁₀c₁₁c₁₃+ c₅c₆c₁₀c₁₁c₁₄-c₅ c₇c₁₀c₁₁c₁₃ A ₆=c₁c₅c₆c₉c₁₂c₁₄-c₁c₅c₇c₉c₁₂ c₁₃-c₂c₄c₆c₉c₁₂c₁₄+ ₂c₄c₇c₉c₁₂c₁₃-c₃ c₅c₆c₈c₁₂c₁₄+c₃c₅c₇c₈c₁₂c₁₃c₁c₅c₆c₁₀ c₁₁c₁₄+c₁c₅c₇c₁₀c₁₁c₁₃ +c₂c₄c₆c₁₀c₁₁c₁₄c₂c₄c₇c₁₀c₁₁c₁₃ FOMULA FO A----A4 A₁=γ₁+γ₂+ε₁+ε₂+σ₁ -β₁₁ +σ₂ -β₂₂ A₂=γ₁γ₂+γ₁ε₂+γ₂ε₁+ε₁ε₂+σ₂γ₁ +σ₂γ₂ +σ₂ε₁ + σ₂ε₂ -γ₁β₂₂ -ε₁β₂₂ -σ₁β₁₁ ²-σ₂β₂₂ ²+σ₁γ₁ + σ₁γ₂ +σ₁ε₁ +σ₁ε₂ -γ₂β₁₁ -ε₂β₁₁ -α₁α₂ +σ₁σ₂ -σ₂β₁₁ -σ₁β₂₂ +β₁₁β₂₂ -β₁₂β₂₁ A₃=-σ₂γ₁β₂₂ ²-σ₂ε₁β₂₂ ²+σ₁γ₁γ₂ +σ₁γ₁ε₂ + σ₁γ₂ε₁ +σ₁ε₁ε₂ +σ₂γ₁γ₂ +σ₂γ₁ε₂ +σ₂γ₂ ε₁ + σ₂ε₁ε₂ -σ₁γ₂β₁₁ ²-σ₁ ε₂β₁₁ ²-α₁α₂γ₁ + σ₁σ₂γ₁ -α₁α₂γ₂ +σ₁σ₂γ₂ -α₁α₂ ε₁ + σ₁σ₂ε₁ -α₁α₂ε₂ +σ₁σ₂ε₂ -σ₁γ₁β₂₂ σ₂γ₂β₁₁ -σ₁ε₁β₂₂ -σ₂ε₂β₁₁ + α₁α₂β₁₁ ² - σ₁σ₂β₁₁ ² +α₁α₂β₂₂ ²σ₁σ₂β₂₂ ²+ σ₁β₁₁β₂₂ ² -σ₁β₁₂β₂₁ ² + σ₂β₁₁β₂₂ ²-σ₂β₁₂β₂₁ ² A₄=-α₁α₂β₁₁β₂₂ ² ²+α₁α₂β₁₂β₂₁ ² ² +σ₁σ₂β₁₁β₂₂ ² ²-σ₁σ₂β₁₂β₂₁ ² ²+α₁α₂γ₂β₁₁ ² σ₁σ₂γ₂β₁₁ ² +α₁ α₂γ₁β₂₂ ²-σ₁σ₂γ₁β₂₂ ²+α₁α₂ ε₂β₁₁ ² -σ₁σ₂ε₂β₁₁ ² +α₁α₂ε₁ β₂₂ ²σ₁σ₂ε₁β₂₂ ²- α₁α₂γ₁γ₂ +σ₁σ₂γ₁γ₂ α₁α₂ γ₁ε₂ -α₁α₂γ₂ε₁ +σ₁σ₂γ₁ε₂ + σ₁σ₂γ₂ε₁ -α₁α₂ε₁ε₂ + σ₁σ₂ε₁ε₂ FOMULA FO A4----A46 b₁=β₁₁ -β₁₁ -γ₁-ε₁-β₁₂ b₂=β₁₁ -β₁₂ b₃=β₁ - σ₁ +β₁₂ b₄=-β₂ - b₅=- α₁ b₆=- γ₁ b₇=- σ - α₁ b₈=- σ b₉=-α₁ b₁₀=-σ₁ b₁₁=-α₁ b₁₂=β₂₁ - - b₁₃=-α₂ b₁₄=β₂₂ -β₂₂ -β₂₂ -γ₂-ε₂-β₂₁ b₁₅=-β₂₂ -β₂₁ b₁₆=β₂₂ -σ₂ +β₂₁ b₁₇=-α₂ b₁₈=γ₂ b₁₉=-σ₂ -α₂ b₂₀=-σ₂ b₂₁=-α₂ b₂₂=-σ₂ A 4₁=-d₁+d₇+d₁₀+d₂₂+d₁₄+d₁₉ A 4₂=d₁d₇-d₂d₆+d₁d₁₀+d₁d₂₂+d₁d₁₄-d₄d₁₂+d₇d₁₀+d₁d₁₉+ d₇ d₂₂+d₇d₁₄+d₇d₁₉+d₁₀d₂₂-d₁₁d₂₁+ d₁₀d₁₄+d₂₂d₁₄+ d₁₀d₁₉+d₂₂d₁₉ +d₁₄ d₁₉-d₁₅d₁₈ A4₃=-d₁d₇d₁₀+d₂d₆d₁₀-d₁ d₇d₂₂+d₂d₆d₂₂-d₁d₇d₁₄+d₂d₆d₁₄ +d₄d₇d₁₂-d₁d₇d₁₉+d₂d₆d₁₉-d₁ d₁₀d₂₂+d₁d₁₁d₂₁-d₁d₁₀d₁₄ +d₄d₁₀d₁₂- d₁d₂₂d₁₄+d₄d₁₂d₂₂-d₁d₁₀d₁₉-d₇ d₁₀d₂₂+ d₇d₁₁d₂₁-d₇d₁₀d₁₄-d₁d₂₂d₁₉-d₁d₁₄d₁₉+d₁d₁₅d₁₈-d₇d₂₂d₁₄+d₄ d₁₂d₁₉ -d₇d₁₀d₁₉-d₇d₂₂d₁₉-d₇d₁₄d₁₉+d₇d₁₅d₁₈-d₁₀d₂₂d₁₄+ d₁₁d₂₁d₁₄-d₁₀ d₂₂d₁₉+d₁₁d₂₁d₁₉-d₁₀d₁₄d₁₉+d₁₀d₁₅d₁₈d₂₂d₁₄d₁₉+d₂₂d₁₅d₁₈ A 4₄=d₁d₇d₁₀d₂₂-d₁d₇d₁₁d₂₁-d₂d₆d₁₀d₂₂+d₂d₆d₁₁d₂₁+ d₁d₇d₁₀d₁₄ -d₂d₆d₁₀d₁₄-d₄d₇d₁₀d₁₂+ d₁d₇d₂₂d₁₄d₂d₆d₂₂d₁₄-d₄d₇d₁₂d₂₂+ d₁d₇d₁₀d₁₉-d₂d₆d₁₀d₁₉+d₁d₇d₂₂d₁₉d₂d₆d₂₂d₁₉+d₁d₇d₁₄d₁₉ -d₁d₇d₁₅d₁₈-d₂d₆d₁₄d₁₉+d₂d₆d₁₅d₁₈ JBA-J Aprl J J

9 ntrnatnal Jurnal f Basc & Appld cncs JBA-J Vl: : 95 d₄d₇d₁₂d₁₉+ d₁d₁₀d₂₂d₁₄-d₁d₁₁d₂₁d₁₄-d₄d₁₀d₁₂d₂₂+ d₄d₁₁d₁₂d₂₁+d₁d₁₀d₂₂d₁₉-d₁d₁₁d₂₁d₁₉ + d₁d₁₀d₁₄d₁₉d₁d₁₀d₁₅d₁₈+d₇d₁₀d₂₂d₁₄- d₇d₁₁d₂₁d₁₄-d₄d₁₀d₁₂d₁₉ +d₁d₂₂d₁₄d₁₉-d₁d₂₂d₁₅d₁₈-d₄d₁₂d₂₂d₁₉+d₇d₁₀d₂₂d₁₉d₇d₁₁d₂₁d₁₉+d₇d₁₀d₁₄d₁₉-d₇d₁₀d₁₅d₁₈+d₇d₂₂d₁₄d₁₉d₇d₂₂d₁₅d₁₈+d₁₀d₂₂d₁₄d₁₉ -d₁₀d₂₂d₁₅d₁₈-d₁₁d₂₁d₁₄d₁₉ +d₁₁d₂₁d₁₅d₁₈ A 4₅=- d₁d₇d₁₀d₂₂d₁₄+d₁d₇d₁₁d₂₁d₁₄+d₂d₆d₁₀ d₂₂d₁₄d₂d₆d₁₁d₂₁d₁₄+d₄d₇d₁₀d₁₂d₂₂-d₄d₇d₁₁d₁₂d₂₁-₁d₇d₁₀d₂₂d₁₉+ d₁d₇d₁₁d₂₁d₁₉+ d₂d₆d₁₀d₂₂d₁₉-d₂d₆d₁₁d₂₁d₁₉-d₁d₇d₁₀ d₁₄d₁₉+d₁d₇d₁₀d₁₅d₁₈ +d₂d₆d₁₀d₁₄d₁₉-d₂ d₆d₁₀d₁₅d₁₈+ d₄d₇d₁₀d₁₂d₁₉-d₁d₇d₂₂d₁₄d₁₉+ d₁d₇d₂₂d₁₅d₁₈+d₂d₆d₂₂d₁₄d₁₉ -d₂d₆d₂₂ d₁₅d₁₈+d₄d₇d₁₂d₂₂d₁₉-d₁d₁₀d₂₂d₁₄d₁₉+ d₁d₁₀d₂₂d₁₅d₁₈+d₁d₁₁d₂₁d₁₄d₁₉-d₁d₁₁d₂₁d₁₅d₁₈ + d₄d₁₀d₁₂d₂₂d₁₉-d₄d₁₁d₁₂d₂₁d₁₉-d₁₀d₂₂d₁₄d₁₉+ d₇d₁₀d₂₂d₁₅d₁₈+d₇d₁₁d₂₁d₁₄d₁₉-d₇ d₁₁d₂₁d₁₅d₁₈ A 4₆=d₁d₇d₁₀d₂₂d₁₄d₁₉-d₁d₇d₁₀d₂₂d₁₅d₁₈-d₁d₇d₁₁d₂₁d₁₄d₁₉ +d₁d₇d₁₁d₂₁d₁₅d₁₈-d₂ d₆d₁₀d₂₂d₁₄d₁₉+d₂d₆d₁₀d₂₂d₁₅d₁₈ +d₂d₆d₁₁d₂₁ d₁₄d₁₉-d₂d₆d₁₁d₂₁d₁₅d₁₈-d₄d₇d₁₀d₁₂d₂₂d₁₉ + d₄d₇d₁₁d₁₂d₂₁d₁₉ FC [] Andrsn,. M., & May,. M. "Th nvasn, prsstnc, and sprad f nfctus dsass wthn anmal and plant cmmunts", Phls. Trans.. c. Lnd. B 4, 5-57, 986 [] Bgn, M., & Bwrs,. G., Bynd hst-pathgn dynamcs, n: B. T. Grnfll, A. P. Dbsnds., clgy f dsas n natural ppulatns, Cambrdg Unvrsty Prss, ,995 [] Bgn, M., Bwrs,. G., Kadanaks,., & Hdgkn, D., Dsas and cmmunty structur: Th mprtanc f hst slf- rgulatn n a hsthst-pathgn mdl, Am. at. 9, -599 [4] Bwrs,. G., & Turnr, J. "Cmmunty structur and th ntrplay btwn ntrspcfc nfctn and cmpttn", J. Thrt. Bl. 87, 95-9, 997 [5] l-shkh, M. M., & Mahruf,. A., tablty and bfurcatn f a smpl fd chan n a chmstat wth rmval rats, Chas, lutns and Fractals, ,5 [6] Farkas, M. "Dynamcal Mdls n Blgy", Acadmc Prss nc, [7] Grnman, J. V., & Hudsn, P. J., nfctd cxstnc nstablty wth and wthut dnsty-dpndnt rgulatn, J. Thrt. Bl. 85, 45-56,997 [8] Han, L., Ma,., & h, T. "An pdmc mdl f tw cmpttv spcs", Math. Cmput. Mdllng 7, [9] Hthct, H. W., Wang, W., & L,., pcs cxstnc and prdcty n hst-hst-pathgn mdls, J. Math. Bl. 5, 69-66,5. [] Hlt,. D., & Pckrng, J., nfctus dsas and spcs cxstnc: A mdl f Ltka-Vltrra frm, Am. at. 6, 96-,985 [] anz,. A., & Hthct, H. W., Cmptng spcs mdls wth an nfctus dsas, Math. Bsc. ng., 9-5, 6 [] P. van dn Drssch, M.L.man, "Dsasd nducd scllatns btwn tw cmptng spcs", AM J. Appl.Dyn.yst., 6-69, 4 [] Vnturn,. "Th ffcts f dsass n cmptng spcs", Math. Bsc. 74, JBA-J Aprl J J