A MATHEMATICAL MODEL OF FOUR SPECIES SYN-ECOSYMBIOSIS COMPRISING OF PREY-PREDATION, MUTUALISM AND COMMENSALISMS-I (FULLY WASHED OUT STATE)

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1 VOL. 6, NO. 4, APRIL 0 ISSN ARPN Jornl of Engineering nd Applied Sciences Asin Reserch Pblishing Nework (ARPN). All righs reserved. A MATHEMATICAL MODEL OF FOUR SPECIES SYN-ECOSYMBIOSIS COMPRISING OF PREY-PREDATION, MUTUALISM AND COMMENSALISMS-I (FULLY WASHED OUT STATE) R. Srilh nd N. Ch. Pbhirmchryl Reserch Scholr, JNTUH, Kkplly, Hyderbd, Indi Professor (Red.) of Mhemics, NIT, Wrngl, Indi E-Mil: bsrilh8@gmil.com ABSTRACT This invesigion dels wih mhemicl model of for species (S, S, S nd S 4 ) Syn-Ecologicl sysem (Flly Wshed o Se). S is predor srviving on he prey S : he prey is commensl o he hos S which iself is in mlism wih he forh species S 4. S nd S 4 re nerl. The mhemicl model eqions chrcerizing he synecosysem consie se of for firs order non-liner copled differenil eqions. There re in ll sixeen eqilibrim poins. Crieri for he sympoic sbiliy of one of he sixeen eqilibrim poins: he flly wshed o se is esblished. The linerised eqions for he perrbions over he eqilibrim poin re nlyzed o esblish he crieri for sbiliy. The sysem is noiced o be loclly sble. Trjecories of he perrbions hve been illsred. Keywords: mhemicl model, species, syn-ecologicl sysem, mlism, commenslism, differenil eqions. INTRODUCTION Mhemicl modeling is n imporn inerdisciplinry civiy which involves he sdy of some specs of diverse disciplines. Biology, Epidemiodology, Physiology, Ecology, Immnology, Bio-economics, Geneics, Phrmocokineics re some of hose disciplines. This mhemicl modeling hs rised o he zenih in recen yers nd spred o ll brnches of life nd drew he enion of every one. Mhemicl modeling of ecosysems ws iniied by Lok [8] nd by Volerr [4]. The generl concep of modeling hs been presened in he reises of Meyer [9], Cshing [], Pl Colinvx [0], Freedmn [], Kpr [5, 6]. The ecologicl inercions cn be brodly clssified s prey-predion, compeiion, mlism nd so on. N.C. Srinivs [] sdied he compeiive eco-sysems of wo species nd hree species wih regrd o limied nd nlimied resorces. Ler, Lkshmi Nryn [7] hs invesiged he wo species prey-predor models. Recenly sbiliy nlysis of compeiive species ws invesiged by Archn Reddy []. Locl sbiliy nlysis for wospecies ecologicl mlism model hs been invesiged by B. Rvindr Reddy e l., [, ]. The presen invesigion is devoed o n nlyicl sdy of for species Syn-Ecologicl sysem. S is predor srviving on he prey S : he prey is commensl o he hos S which iself is in mlism wih he forh species S 4 ; S nd S 4 re nerl. Figre- shows he Schemic Skech of he sysem nder invesigion. The model eqions of he sysem consie se of for firs order non-liner ordinry differenil copled eqions. In ll he sixeen eqilibrim poins of he sysem re idenified nd he sbiliy nlysis is crried o only for he flly wshed o se. The linerized perrbed eqions over he eqilibrim ses re solved nd he rjecories illsred. Figre-. Schemic skech of he Syn Eco- sysem. BASIC EQUATIONS Noion doped N (): The Poplion of he Prey (S ) N (): The Poplion of he Predor (S ) N (): The Poplion of he Hos (S ) of he Prey (S ) nd ml o S 4 N 4 (): The Poplion of S 4 ml o S : Time insn,,, 4 : Nrl growh res of S, S, S, S 4,,, 44 : Self inhibiion coefficiens of S, S, S, S 4, : Inercion (Prey-Predor) coefficiens of S de o S nd S de o S : Coefficien for commensl for S de o he Hos S 4, 4 : Mlly inercion beween S nd S 4,,, 44 : Crrying cpciies of S, S, S, S 4 Frher he vribles N, N, N, N 4 re nonnegive nd he model prmeers,,, 4 ;,,, 44 ;,,, 4 re ssmed o be non-negive consns. 4

2 VOL. 6, NO. 4, APRIL 0 ISSN ARPN Jornl of Engineering nd Applied Sciences Asin Reserch Pblishing Nework (ARPN). All righs reserved. The model eqions for he growh res of S, S, S, S 4 re d d d N N NN + NN. () N N + NN. () N N + NN. () 4 4 d 4 N N + NN. (4) EQUILIBRIUM STATES The sysem nder invesigion hs sixeen eqilibrim ses defined by i 0, i,,, 4 d re given in he following ble. (5) S. No. Eqilibrim Ses Eqilibrim poin Flly Wshed o se N 0, N 0, N 0, N4 0 Only S 4 srvives 4 N 0, N 0, N 0, N4 44 Only he hos (S )of S srvives N 0, N 0, N, N 0 4 Only he predor S srvives 5 Only he prey S srvives 6 7 Prey (S ) nd predor (S ) wshed o Prey (S ) nd hos (S ) of S wshed o 8 Prey (S ) nd S 4 wshed o 9 Predor (S ) nd Hos (S ) of S wshed o 0 Predor (S ) nd S 4 wshed o Prey (S ) nd predor (S )srvives Only he prey (S ) wshed o Only he predor (S ) wshed o N 0, N, N 0, N 0 N, N 0, N 0, N N 0, N 0, N, N N 0, N, N 0, N 4 44 N 0, N, N, N 0 N, N 0, N 0, N N, N 0, N, N 0 + N, N, N 0, N N 0, N, N, N α + + N, N 0, N, N α where α ( ) + ( 44 44) α ( )

3 VOL. 6, NO. 4, APRIL 0 ISSN ARPN Jornl of Engineering nd Applied Sciences Asin Reserch Pblishing Nework (ARPN). All righs reserved. 4 Only he Hos (S ) of S wshed o + N, N, N 0, N Only S 4 wshed o 6 The co-exisen se (or) Norml sedy se β β N, N, N, N 0 β β where β ( + ) β ( + ) β ( + ) + N N γ + γ γ + γ, N, 4 γ γ + +, N where γ ( + )( ) γ γ ( + )( ) γ ( )( ) The presen pper dels wih he flly wshed o se only. The sbiliy of he oher eqilibrim ses will be presened in he forh coming commnicions. STABILITY OF THE FULLY WASHED OUT EQUILIBRIUM STATE (Sl. No. in he bove Tble) To discss he sbiliy of eqilibrim poin N 0, N 0, N 0, N4 0 Le s consider smll deviions (), (), (), 4 () from he sedy se i.e., N ( ) N + ( ), i,,,4.. (6) i i i Where i () is smll perrbions in he species S i. Sbsiing (6) in (), (), (), (4) nd neglecing prodcs nd higher powers of,,, 4, we ge d d d d d d (7).. (8) (9) d d 4.. (0) 4 4 The chrcerisic eqion of which is ( λ )( λ )( λ )( λ ) 0.. () he roos,,, 4 of which re ll posiive. Hence he Flly Wshed o Se is nsble. The solions of he eqions (7), (8), (9), (0) re 0e () 0e () 0e. (4) 4 4 e. (5) Where 0, 0, 0, re he iniil vles of,,, 4 respecively. There wold rise in ll 576 cses depending pon he ordering of he mgnides of he growh res,,, 4 nd he iniil vles of he perrbions 0 (), 0 (), 0 (), () of he species S, S, S, S 4. Of hese 576 siions some ypicl vriions re illsred hrogh respecive solion crves h wold fcilie o mke some resonble observions. 44

4 VOL. 6, NO. 4, APRIL 0 ISSN ARPN Jornl of Engineering nd Applied Sciences Asin Reserch Pblishing Nework (ARPN). All righs reserved. Cse (i): If 0 < 0 < 0 <, < < < 4 In his cse prey (S ) hs he les nrl birh re nd S 4 domines he prey (S ), predor (S ) nd he hos (S ) of S in nrl growh re s well s in is poplion srengh. Cse (iv): If 0<<0<0, <<<4 In his cse he hos (S ) of S hs he les nrl birh re. Iniilly he hos (S ) of S domines over S 4 nd he predor (S ) ill he imes insn * 4, * respecively. Therefer he dominnce is reversed. ; * * Also he prey (S ) domines over S 4, Predor (S ) ill he ime insn * 4, * respecively nd herefer he dominnce is reversed. ; * * Figre- Cse (ii): If 0 < 0 < 0 <, < < < 4 In his cse predor (S ) hs he les nrl birh re. Iniilly he predor (S ) domines over he prey (S ) ill he ime insn * 0 0 nd here-fer he prey (S ) domined he predor (S ). The * ime my be clled he dominnce ime of he predor (S ) over he prey (S ). Figre-4 Cse (v): If 0 < 0 < 0 <, < < 4 < In his cse he Hos (S ) of S hs he les nrl birh re. Iniilly i is domines over he predor (S ) ill he ime insn * 0 0 nd herefer he dominnce is reversed. Also S 4 domines over he prey (S ) ill he ime insn * 4 nd herefer he dominnce is 4 0 Figre- Cse (iii): If 0 < 0 < < 0, < < < 4 In his cse predor (S ) hs he les nrl birh re. Iniilly he predor (S ) domines over S 4, hos (S ) of S nd prey (S ) ill he ime insn,, respecively nd here fer he dominnce is * * * 4 reversed * * 0 * 0 4,, Figre-5 45

5 VOL. 6, NO. 4, APRIL 0 ISSN ARPN Jornl of Engineering nd Applied Sciences Asin Reserch Pblishing Nework (ARPN). All righs reserved. Cse (vi): If 0 < 0 < < 0, < 4 < < In his cse he prey (S ) hs he les nrl birh re. Iniilly he prey (S ) domines over is hos, S 4 nd Predor (S ) ill he ime insn *, * 4, * respecively nd herefer he dominnce is reversed. Also S 4 domines over he hos (S ) of S, nd he predor (S ) ill he ime insn * 4, * 4 nd herefer he dominnce is reversed. Similrly, he hos (S ) of S domines over he predor (S ) ill he ime insn * nd he dominnce ges reversed herefer. ; ; * 0 * * Cse (viii): If 0 < 0 < 0 <, < < 4 < In his cse he predor (S ) hs he les nrl birh re. Iniilly he predor (S ) domines over he prey (S ), hos (S ) of S ill he ime insn *, * respecively nd herefer he dominnce is reversed. Also he prey (S ) domines over is hos ill he ime insn * nd herefer he dominnce is reversed. Similrly S 4 domines over he hos (S ) of S ill he ime insn * 4 he dominnce ges reversed fer. ; ; * * * Figre-8 Figre-6 Cse (vii): If 0 < 0 < 0 <, < 4 < < In his cse he hos (S ) of S hs he les nrl birh re. Iniilly S 4 domines over boh he prey (S ) nd predor (S ) ill he ime insn * 4, * 4 respecively nd herefer he dominnce is reversed. Also he Prey (S ) domines over he Predor (S ) po he ime insn * nd he dominnce ges reversed fer. nd ; * * * 0 0 ; * 0 * ; * 0 * Cse (ix): If 0 < 0 < < 0, < 4 < < In his cse he prey (S ) hs he les nrl birh re. Iniilly he prey (S ) domines over is hos, predor (S ) nd S 4 ill he ime insn *, *, * 4 respecively nd here fer he dominnce is reversed. Also S 4 domines over he predor (S ) nd he hos (S ) of S ill he imes insn * 4 nd * 4 nd herefer he dominnce is reversed. ; ; * 0 * 0 * nd ; * * Figre-7 46

6 VOL. 6, NO. 4, APRIL 0 ISSN ARPN Jornl of Engineering nd Applied Sciences Asin Reserch Pblishing Nework (ARPN). All righs reserved. ; * * ; * 0 * nd * Figre-9 Cse (x): If < 0 < 0 < 0, < < 4 < In his cse he prey (S ) hs he les nrl birh re. Iniilly he prey (S ) domines over S 4 ill he ime insn * 4 nd herefer he dominnce is reversed. Also he predor (S ) domines over S nd S 4 ill he ime insns *, * 4 respecively nd herefer he dominnce is reversed. * * 4 ; 4 nd * 0 0 Figre- Cse (xii): If < 0 < 0 < 0, 4 < < < In his cse S 4 hs he les nrl birh re. Iniilly he prey (S ) domines over is hos nd predor (S ) ill he ime insn *, * respecively nd herefer he dominnce is reversed. * 0 * 0 ; 0 0 Figre-0 Cse (xi): If < 0 < 0 < 0, < < < 4 In his cse he predor (S ) hs he les nrl birh re. Iniilly he predor (S ) domines over S 4 nd prey (S ) ill he ime insn * 4, * respecively nd herefer he dominnce is reversed. Also he hos (S ) of S domines over he prey (S ) nd S 4 ill he ime insn *, * 4 respecively nd herefer he dominnce is reversed. Similrly he prey (S ) domines over S 4 ill he ime insn * 4 nd he dominnce ges reversed fer. Figre- TRAJECTORIES OF PERTURBATIONS The rjecories in he - plne given by ( ) ( ) nd re shown in Figre

7 VOL. 6, NO. 4, APRIL 0 ISSN ARPN Jornl of Engineering nd Applied Sciences Asin Reserch Pblishing Nework (ARPN). All righs reserved. [4] George F Simmons: Differenil Eqions wih pplicions nd hisoricl noes. T McGrw- Hill, New Delhi. [5] Kpr J. N Mhemicl Modeling. Wiley - Esern. [6] Kpr J.N Mhemicl Models in Biology nd Medicine Affilied Es - Wes. Figre- Also he rjecories in he plne given by ( ) ( ) nd re shown in Figre Similrly he rjecories in he - 4, -, - 4, plnes re ( ) ( ),( ) ( ), ( ) ( ),( ) ( ) respecively. 0 0 [7] Lkshmi Nryn K A Mhemicl sdy of Prey-Predor Ecologicl Models wih pril covers for he prey nd lernive food for he predor. PhD Thesis. J. N. T. Universiy. [8] Lok A. j. 95. Elemens of Physicl biology. Willims nd Wilkins, Blimore. [9] Meyer W.J Conceps of Mhemicl Modeling. McGrw - Hill. [0] Pl Colinvx Ecology. John Wiley nd Sons Inc., New York. [] Rvindr Reddy B., Lkshmi Nryn, K. nd Pbhirmchryl N.Ch A model of wo mlly inercing species wih limied resorces for boh he species. Inernionl J. of Engg. Reserch nd Ind. Appls. (II): 8-9. REFERENCES Figre-4 [] Rvindr Reddy B., Lkshmi Nryn K. nd Pbhirmchryl N.Ch. 00. A model of wo mlly inercing species wih limied resorces nd hrvesing of boh he species consn re. Inernionl J. of Mh. Sci nd Engg. Appls. (IJMSEA). 4(III): [] Srinivs N.C. 99. Some Mhemicl specs of modeling in Bio Medicl Sciences. PhD Thesis. Kkiy Universiy. [4] Volerr V. 9. Leconssen l heorie mhemiqe de l leie po lvie. Ghier - Villrs, Pris. [] Archn Reddy R On he sbiliy of some mhemicl models in biosciences- inercing species. PhD Thesis. JNTU. [] Cshing J. M Inegro - differenil eqions nd Dely Models in Poplion Dynmics. Lecre Noes in Biomhemics. Springer- Verlg, Heidelberg. Vol. 0. [] Freedmn H. I Deerminisic Mhemicl Models in Poplion Ecology. Mrcel - Decker, New York. 48