ON THE STABILITY OF DELAY POPULATION DYNAMICS RELATED WITH ALLEE EFFECTS. O. A. Gumus and H. Kose

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1 Mhemicl nd Compuionl Applicions Vol. 7 o. pp O THE STABILITY O DELAY POPULATIO DYAMICS RELATED WITH ALLEE EECTS O. A. Gumus nd H. Kose Deprmen o Mhemics Selcu Universiy 47 Kony Turey ozlem@selcu.edu.r hose@selcu.edu.r Absrc- In recen yers mny scieniss hve ocus on he sudies o he Allee eec in populion dynmics. This pper presens he sbiliy nlysis o equilibrium poins o populion dynmics wih Allee eec which occurs low populion densiy. Key Words- Populion dynmics Allee eec Sbiliy Equilibrium poin. ITRODUCTIO When previous sudies hve been exmined on populion dynmics including dierenil nd dierence equions i is generlly observed h Allee eec cn hve eiher sble or n unsble eec on he sysem [ ]. oneheless discree-ime models re more suible or numericl soluions nd clculions [5]. Allee eec ws irs deined by Allee s negive densiy dependence when he growh re o he populion decreses in low populion densiy. This eec cn consis o socil dysuncion smll populion size inbreeding depression ood exploiion predor voidnce o deence nd diiculies inding in mes. Auhors hve sudied he sbiliy o dieren populion models wihin he rmewor o hese eecs nd developed similr models. Besides sbiliy nlysis is n imporn reserch opic in such sudies. In his presen sudy our purpose is o invesige nd compre he sbiliy o equilibrium poin wih nd wihou Allee eec by considering more generl se o he model sudied in [3]. Le's loo he nonliner generl dely dierence equion where is per cpi growh re which is lwys posiive represens he populion densiy ime nd T is he ime or sexul muriy. Also hs he ollowing orm: where is he uncion describing inercions compeiions mong mure individuls. We ssume h sisies he ollowing condiions: / or [. / is posiive inie number. This pper is orgnized s ollows: In secion irs o ll we give chrcerizion o he sbiliy o he equilibrium poins o Eq.. In secion 3 we wor on he sbiliy nlysis o he equilibrium poins in Eq. wih he Allee eec. In secion 4 we presen numericl simulions h suppor he nlyicl resul. inlly

2 O. A. Gumus nd H. Kose 57 he ls secion o he pper includes conclusions.. STABILITY AALYSIS O Eq. Beore we give he min resuls o his pper we shll remind he ollowing Schur-Cohn crierion see reerences [45]. Theorem. Schur-Cohn Crieri The roos o he chrcerisic polynomil... g lie inside he uni circle i nd only i he ollowing hold: i g ii g iii he mrices B re posiive innerwise. The chrcerisic polynomil which is geing rom linerizion o Eq. round will be 3. g p q r Assume h Eq. hs n equilibrium poins s. Then we ge he ollowing heorem. Theorem. is loclly sble i nd only i he inequliies 3 4 hold. Proo. rom he equilibrium poin deiniion o Eq. we hve. 5 Le s e q p nd r. Given Eq.5. p rom Eq. he vlues o q nd r re

3 On he Sbiliy o Dely Populion Dynmics Reled wih Allee Eecs 58. r q We ge h is loclly sble i nd only i q r p nd r q pr 6 by Theorem. I we wrie he vlues o p q nd r in he irs inequliy o Eq.6 we obin I is esy o see h in his cse nd. Since re negive vlues or [ he ls inequliy is lwys provided. Thereore is conirmed. ow i he vlues o p q nd r re wrien in he second inequliy in 6 we ge I he ls expressions is wrien in he orm o wo inequliies we cn wrie s conirmed.

4 O. A. Gumus nd H. Kose ALLEE EECTS O THE DISCRETE DELAY MODEL In his secion we sudy he locl sbiliy nlysis o he equilibrium poins o Eq. wih he ddiion o Allee eec ime nd. 3.. Allee eec ime - We consider he ollowing non-liner dely dierence equion by he ddiion o Allee eec o discree dely model Eq. 7 where he uncion sisies he properies nd. The conclusion o he biologicl cs requires he ollowing ssumpion on α. 3 i hen ; h is here is no reproducion wihou prners. 4 / or ; h is Allee eec decreses s densiy increses. 5 lim ; h is Allee eec vnishes high densiies. Eq.7 hs he sme posiive equilibrium poins wih since is normlized growh re such h /. Then we ge he ollowing heorem. Theorem 3. is loclly sble i nd only i he inequliies 8 9 hold. Proo. rom he equilibrium poin deiniion or Eq.7 i is cler h. p Liewise i he vlues o q nd r re clculed or Eq.7 we ge

5 On he Sbiliy o Dely Populion Dynmics Reled wih Allee Eecs 6 q. r The irs inequliy in 6 leds he ollowing inequliy or equivlenly nd. or he second inequliy in 6 we rrive rom he ls wo inequliy we cn wrie s required.

6 O. A. Gumus nd H. Kose Allee eec ime - Le us consider he ollowing non-liner dely dierence equion by he ddiion o Allee eec o discree dely model o. equilibrium poin o Eq. is posiive equilibrium poin o Eq.7. Then we hve he ollowing heorem. Theorem 4. is loclly sble i nd only i he inequliies hold. 3 4 Proo. According o Eq. he vlues o p q nd r re s ollows p q r. irsly rom he irs inequliy in 6 we obin. or equivlenly nd

7 On he Sbiliy o Dely Populion Dynmics Reled wih Allee Eecs 6. I we consider he oher inequliy in 6 we ge Ls inequliy leds he ollowing inequliy Allee eec ime We now incorpore n Allee eec ino he discree dely model s ollows: 5 equilibrium poin o Eq.5 is posiive equilibrium poin o Eq.7. Then we cn se he ollowing heorem. Theorem 5. is loclly sble i nd only i he inequliy hold.

8 O. A. Gumus nd H. Kose 63 Proo. I he vlues o p q nd r re wrien in Eq.5 we hve r q p Le's consider sbiliy condiions in 6. Thus we ge. The ls inequliy cn be wrien s ollows nd. ow when he process is reguled or he second inequliy in 6 we obin Consequenly 6 7 nd 8 re conirmed. Corllry 6. Allee eecs ime nd decreses he sbiliy o Eq. Proo. Le s e y x nd. z Eq. is sble i nd only i

9 64 On he Sbiliy o Dely Populion Dynmics Reled wih Allee Eecs x y y y x y y x I he vlues o x y nd z re wrien in he sbiliy condiions o Eq.7 Eq. nd Eq.5 we obin x z y x y z z z y y x z y y yz z y x yz z y x y z x x y z y y z x nd x z y x y z y yz y x y yz y x respecively. I is cler h or ech vlue o x nd y which provides inequliy 9 les one o he condiions nd is no sisied or ech z>. In oher words sble equilibrium poin o Eq. is no sble or equions UMERICAL SIMILATIOS In his secion we numericlly presen our he nlyicl resul obined in he ormer secions by using MATLAB progrmming. We grph he D rjecories o he populion dynmics model wih nd wihou Allee eec ime nd in ig. ig. nd ig. 3 respecively. In his igures we e he uncion see or insnce [7] nd he Allee uncion i i / i i nd where is posiive consn. I is obvious rom he grph h he comprisons o he populion densiy digrms lso veriy he sbilizing impc o he Allee eecs. In hese compuions he iniil condiions re en s..3.4 nd. 9 h yield he corresponding equilibrium poin s In ddiion he prmeer vlue is en s.3. ormlized growh re is.4 such h. 9

10 O. A. Gumus nd H. Kose 65 ig.. Densiy-ime grphs o he models nd wih.9 /.3. ig.. Densiy-ime grphs o he models nd wih.9 /.3.

11 66 On he Sbiliy o Dely Populion Dynmics Reled wih Allee Eecs ig. 3. Densiy-ime grphs o he models nd wih.9 /.3 These numericl simulion re consisen wih he nlyicl resul obin in he ormer secions nd suppors he mhemicl nlysis. 5. COCLUSIO ormer sudies indice h Allee eec hs dieren eecs on dieren populions. Mhemicl ormulions o he populion will provide inormion o us bou he cors eecing populion nd he developmen o h group o living beings in he uure. This siuion is imporn in h i conribues o he esblishmen o equilibrium in lie cycle which is siued in Biology. In his pper we sudied on hird degree dely dierence model under compeiive eec. irsly we obined he sbiliy condiions o he equilibrium poin o his model. Then we invesiged he sbiliy o he equilibrium poin o he model ogeher wih Allee eec. We compred he sbiliy o hese models wih nd wihou Allee eec. In conclusion we observed h Allee eec reduced sbiliy in he model clss h we sudied. 6. REERECES. W. C. Allee Animl Aggreions: A Sudy in Generl Sociology. Universiy o Chicgo Press Chicgo 93.. L. J. S. Allen An Inroducion o Mhemicl Biology Person ew Jersey C. Çeli H. Merdn O. Dumn Ö. Aın Allee eecs on populion dynmics wih dely Chos Solions & rcls S. Elydi An Inroducion o Dierence Equions. Springer ew Yor S. Elydi R. J. Scer Populion models wih Allee eec: A new model Journl o

12 O. A. Gumus nd H. Kose 67 Biologicl Dynmics M. S. owler GD. Ruxon Populions dynmics consequences o Allee eecs J. Theor Biol K. Goplsmy Globl sbiliy in he dely-logisic equion wih discree-delys Huson J. Mh D. Hdjivgousi S. Ichiroglou Exisence o sble loclized srucures in populion dynmics hrough he Allee eec Chos Soluions & rcls S. R. J. Jng Allee eecs in discree-ime hos prsioid model wih sge srucure in he hos Discree Conin Dyn Sys Ser B Y. Kung Dely Dierenil Equions wih Applicions in Populion Dynmics. Acdemic Press ew Yor M. A. McCrhy The Allee eec inding mes nd heoreicl models Ecologicl Modeling H. Merdn O. Dumn Ö. Aın C. Çeli Allee eecs on populion dynmics in coninuous overlpping cse Chos Soluions & rcls H. Merdn O. Dumn On he sbiliy nlysis o generl discree-ime populion model involving predion nd Allee eecs Chos Solions & rcls H. Merdn O. A. Gumus Sbiliy nlysis o generl discree-ime populion model involving dely nd Allee eecs Mhemicl nd Compuionl Applicions in press 5. J. D. Murry Mhemicl Biology Springer-Verlg ew Yor O. Dumn H. Merdn Sbiliy nlysis o coninuous populion model involving predion nd Allee eecs Chos Solions & rcls I. Scheuring Allee eec increses he dynmicl sbiliy o populions J. Theor. Biol PA. Sephens WJ. Suherlnd Consequences o he Allee eec or behvior ecology nd conservion Trends Ecol. Evol SR. Zhou Y. Liu G. Wng The sbiliy o predor-prey sysems subjec o he Allee eecs Theor. Popul. Biol