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1 Avilble online Pelgi Reserch Librry Advnces in Applied Science Reserch 0 (): 5-65 ISSN: CODEN (USA): AASRFC A Mhemicl Model of For Species Syn-Ecosymbiosis Comprising of Prey-Predion Mlism nd Commenslisms-III (Two of he For Species re wshed o Ses) R. Srilh B. Rvindr Reddy nd N. Ch. Pbhirmchryl JNTUH Kkplly Hyderbd Indi JNTUH College of Engineering Nchplly Krimngr Indi Dep. of Mhemics NIT Wrngl Indi _ ABSTRACT This invesigion dels wih mhemicl model of for species (S S S nd S ) Syn- Ecologicl sysem (Two of he for species re wshed o ses). 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. Frher S nd S re nerl. The model eqions of he sysem consie se of for firs order non-liner ordinry differenil copled eqions. In ll here re sixeen eqilibrim poins. Crieri for he sympoic sbiliy of six of he sixeen eqilibrim poins: Two of he for species re wshed o ses only re esblished in his pper. The linerized eqions for he perrbions over he eqilibrim poins re nlyzed o esblish he crieri for sbiliy nd he rjecories illsred. Key words: Eqilibrim se sbiliy Mlism Commenslisms. _ INTRODUCTION Mhemicl modeling of ecosysems ws iniied by Lok [6] nd by Volerr []. The generl concep of modeling hs been presened in he reises of Meyer [7] Pl Colinvx [8] Freedmn [] Kpr [ ]. The ecologicl inercions cn be brodly clssified s preypredion compeiion mlism nd so on. N.C. Srinivs [] sdied he compeiive ecosysems of wo species nd hree species wih regrd o limied nd nlimied resorces. Ler Lkshmi Nryn [5] hs invesiged he wo species prey-predor models. Recenly sbiliy nlysis of compeiive species ws invesiged by Archn Reddy []. Locl sbiliy nlysis for wo-species ecologicl mlism model hs been invesiged by B. Rvindr Reddy e. l [9 0]. Pelgi Reserch Librry 5
2 R. Srilh e l Adv. Appl. Sci. Res. 0 (): Bsic eqions: Noion Adoped: 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 N () : The Poplion of S ml o S : Time insn : Nrl growh res of S S S S : Self inhibiion coefficiens of S S S S : Inercion (Prey-Predor) coefficiens of S de o S nd S de o S : Coefficien for commensl for S de o he Hos S : Mlly inercion beween S nd S : Crrying cpciies of S S S S Frher he vribles N N N N re non-negive nd he model prmeers ; ; re ssmed o be non-negive consns. The model eqions for he growh res of S S S S re dn dn dn dn N N N N + N N. (.) N N + N N. (.) N N + N N. (.) N N + N N. (.). Eqilibrim Ses: The sysem nder invesigion hs sixeen eqilibrim ses re given by dn i 0 i (.) I. Flly wshed o se: N 0 N 0 N 0 N 0 () II. Ses in which hree of he for species re wshed o nd forh is srviving () N 0 N 0 N 0 N () N 0 N 0 N N 0 () N 0 N N 0 N 0 (5) N N 0 N 0 N 0 III. Ses in which wo of he for species re wshed o while he oher wo re srviving Pelgi Reserch Librry 5
3 R. Srilh e l Adv. Appl. Sci. Res. 0 (): (6) N 0 N 0 N N This se cn exis only when > 0. (7) N 0 N N 0 N (8) N 0 0 N N N (9) N N 0 N 0 N + (0) N N 0 N N 0 + () N N N 0 N This se cn exis only when > 0 IV. Ses in which one of he for species is wshed o while he oher hree re srviving + + N N N N () 0 α + + N N N N () 0 α Where α ( + ) + ( ) α ( ) + N N N N () (5) N β β N N N 0 β β Where β ( + ) β ( + ) β ( + ) + V. The co-exisen se (or) Norml sedy se (6) γ + γ γ + γ N N N + N + γ γ Where γ ( + )( ) γ + γ ( + )( ) γ ( )( ) The presen pper dels wih wo of he for species re wshed o ses only. The sbiliy of he oher eqilibrim ses will be presened in he forh coming commnicions. Pelgi Reserch Librry 5
4 R. Srilh e l Adv. Appl. Sci. Res. 0 (): Sbiliy of wo of he for species wshed o eqilibrim ses: (Sl. Nos in he bove Eqilibrim ses). Sbiliy of he Eqilibrim Se N 0 N 0 N N from he sedy se Le s consider smll deviions ( ) ( ) ( ) ( ) i.e. N ( ) N ( ) i (..) i i i Sbsiing (..) in (.) (.) (.) (.) nd neglecing prodcs nd higher powers of we ge d d l (..)... (..) d d N + N (..) N N... (..5) l + N... (..6) Here The chrcerisic eqion of which is ( λ l)( λ ) λ ( N N) λ ( ) NN The chrcerisic roos of (..7) re ( N + N ) ± ( N N ) + N N λ l λ λ... (..7) Two roos of he eqion (..7) re posiive nd he oher wo roos re negive. Hence he eqilibrim se is nsble. The solions of he eqions (..) (..) (..) (..5) re l 0e (..8) 0e (..9) ( ) ( ) λ + N + N λ + N + N e + e λ λ λ λ 0 ( λ + N ) + 0 N 0 ( λ + N ) + 0 N λ λ e + e λ λ λ λ where re he iniil vles of respecively. 0 0 λ 0 0 λ... (..0)... (.. There wold rise in ll 576 cses depending pon he ordering of he mgnides of he growh res nd he iniil vles of he perrbions 0 () 0 () 0 () 0 () of he species S S S S. Of hese 576 siions some ypicl vriions re illsred hrogh respecive solion crves h wold fcilie o mke some resonble observions. The solions re illsred in figres &. Pelgi Reserch Librry 5
5 R. Srilh e l Adv. Appl. Sci. Res. 0 (): 5-65 Cse (i) : If 0 < 0 < 0 < 0 < l < < In his cse iniilly he predor (S ) domines over he prey (S ) ill he ime insn nd here fer he dominnce is reversed. I is eviden boh he species prey nd Predor re going wy from he eqilibrim poin while he oher wo species converge o he eqilibrim poin. Hence he eqilibrim se is nsble. Fig. Cse (ii) : If 0 < 0 < 0 < 0 < < < l In his cse iniilly he hos (S ) of S domines over he prey (S ) S nd he predor (S ) ill he ime insn respecively nd here fer he dominnce is reversed. Also S domined over by he predor (S ) ill he ime insn reversed. Fig. nd here fer he dominnce is. Sbiliy of he Eqilibrim Se 7 N 0 N N 0 N Sbsiing (..) in (.) (.) (.) (.) nd neglecing prodcs nd higher powers of we ge d d r (..) + (..) d d l (..) +... (..) Pelgi Reserch Librry 55
6 R. Srilh e l Adv. Appl. Sci. Res. 0 (): 5-65 Here r l +... (..5) The chrcerisic eqion of which is ( λ r )( λ + )( λ l )( λ + ) 0... (..6) Cse (A): When r < 0 (i.e. when < ) The roos r re negive nd l is posiive. Hence he eqilibrim se is nsble. The solions of he eqions (..) (..) (..) (..) re r e (..7) 0 [ 0 ] e + e ( r + ) ( r + ) 0 0 r (..8) l 0e... (..9) [ ] 0 0 l 0 e + e ( l + ) ( l + )... (..0) The solion crves re s shown in figres &. Cse (i): If 0 < 0 < 0 < 0 nd < l < < r In his cse iniilly S domines over he Hos (S ) of S ill he ime insn nd here fer he dominnce is reversed. Also he commensl species is observed o be going wy from he eqilibrim poin while he oher hree species converge o he eqilibrim poin. Hence he eqilibrim se is nsble. Cse (ii): If 0 < 0 < 0 < 0 ndl < < r <. In his cse iniilly he Prey (S ) domines over S he hos (S ) of S nd he Predor (S ) ill he ime insn respecively nd here fer he dominnce is reversed. Also S domines over he Hos (S ) of S ill he ime insn reversed. nd here fer he dominnce is Pelgi Reserch Librry 56
7 R. Srilh e l Adv. Appl. Sci. Res. 0 (): 5-65 Cse (B): When r > 0 (i.e. when > ) The roos re negive nd r l re posiive. Hence he eqilibrim se is nsble. In his cse he solions re sme s in cse (A) nd he solions re illsred in figres 5 & 6. Cse (i): If 0 < 0 < 0 < 0 nd r < l < <. In his cse iniilly S domines over he Prey (S ) nd he Hos (S ) of S ill he ime insn respecively nd here fer he dominnce is reversed. Also he Prey (S ) domines he Hos (S ) of S ill he ime insn nd here fer he dominnce is reversed. Fig.5 Cse (ii): If 0 < 0 < 0 < 0 nd < r < l <. In his cse iniilly he Predor (S ) domines over he Hos (S ) of S nd S ill he ime insn respecively nd he dominnce ges reversed here fer. Also he Prey (S ) domines he Hos (S ) of S ill he ime insn nd here fer he dominnce is reversed. Fig.6 Pelgi Reserch Librry 57
8 R. Srilh e l Adv. Appl. Sci. Res. 0 (): Sbiliy of he Eqilibrim Se 8 N 0 N N N 0 Sbsiing (..) in (.) (.) (.) (.) nd neglecing prodcs nd higher powers of we ge d d s (..) d + (..) d +... (..) n... (..) s n +... (..6) Here The chrcerisic eqion of which is ( λ s )( λ + )( λ + )( λ n ) 0... (..7) Cse (A): When s < 0 (i.e. when + < ) The roos s re negive nd n is posiive. Hence he eqilibrim se is nsble. The solions of he eqions (..) (..) (..) (..) re s e (..8) 0 [ 0 ] e + e ( s + ) ( s + ) 0 0 [ 0 ] e + e ( n + ) ( n + ) 0 0 s n (..9)... (..0) n 0e... (..) The solion crves re exhibied in figres 7 & 8. Cse (i): If 0 < 0 < 0 < 0 nd < < s < n In his cse iniilly he Hos (S ) of S domines over S ill he ime insn nd here fer he dominnce is reversed. Also he Predor (S ) domines over he S ill he ime insn nd here fer he dominnce is reversed. Fig.7 Pelgi Reserch Librry 58
9 R. Srilh e l Adv. Appl. Sci. Res. 0 (): 5-65 Cse (ii): If 0 < 0 < 0 < 0 nd < n < < s In his cse iniilly he Prey (S ) domines over S ill he ime insn nd here fer he dominnce is reversed. Also he Predor (S ) domines over he S ill he ime insn nd here fer he dominnce is reversed. Similrly he Hos (S ) of S domines over S ill he ime insn nd he dominnce is ges reversed here fer. Fig.8 Cse (B): When s > 0 (i.e. when + > ) The roos re negive nd s n re posiive. Hence he eqilibrim se is nsble. In his cse he solions re sme s in cse (A) nd he solions re illsred in figres 9 & 0. Cse (i): If 0 < 0 < 0 < 0 nd s < < < n In his cse iniilly he Hos (S ) of S domines he Predor (S ) nd Prey (S ) ill he ime insn respecively nd here fer he dominnce is reversed. Also he Predor (S ) domines over he Prey (S ) ill he ime insn nd he dominnce ges reversed here fer. Similrly S domines over he Predor (S ) nd he Prey (S ) ill he ime insn respecively nd here fer he dominnce is reversed. Cse (ii): If 0 < 0 < 0 < 0 nd < n < s < Pelgi Reserch Librry 59
10 R. Srilh e l Adv. Appl. Sci. Res. 0 (): 5-65 In his cse iniilly he Hos (S ) of S domines he Predor (S ) ill he ime insn nd here fer he dominnce is reversed. Also S domines over he Predor (S ) ill he ime insn nd here fer he dominnce is reversed. Fig. 0. Sbiliy of he Eqilibrim Se 9 N N 0 N 0 N Sbsiing (..) in (.) (.) (.) (.) nd neglecing prodcs nd higher powers of we ge d + (..) d q (..) d d l... (..) +... (..) Here q + l + (..5) The chrcerisic eqion of which is ( λ + )( λ q )( λ l )( λ + ) 0... (..6) The roos q l re posiive nd re negive. Hence he eqilibrim se is nsble. The solions of he eqions (..) (..) (..) (..) re l 0 q [ 0 + ] e + e e ( l + ) ( q + ) ( l + ) ( q + ) (..7) q l e (..8) e... (..9) l [ 0 ] e + e ( l + ) ( l + ) 0... (..0) The solion crves re s shown in figres &. Pelgi Reserch Librry 60
11 R. Srilh e l Adv. Appl. Sci. Res. 0 (): 5-65 Cse (i): If 0 < 0 < 0 < 0 nd < < l < q In his cse iniilly he Prey (S ) domines over he Hos (S ) of S nd he Predor (S ) ill he ime insn respecively nd here fer he dominnce is reversed. Also S domines over he Hos (S ) of S nd he Predor (S ) ill he ime insn respecively nd here fer he dominnce is reversed. Similrly he hos (S ) of S domines over he Predor (S ) ill he ime insn nd here fer he dominnce is reversed. Cse (ii): If 0 < 0 < 0 < 0 nd q < l < <. In his cse iniilly he Prey (S ) domines over S nd he hos (S ) of S ill he ime insn respecively nd here fer he dominnce is reversed. Also S domines over he Hos (S ) of S ill he ime insn nd here fer he dominnce is reversed. Fig..5 Sbiliy of he Eqilibrim Se 0 + N N 0 N N 0 Pelgi Reserch Librry 6
12 R. Srilh e l Adv. Appl. Sci. Res. 0 (): 5-65 Sbsiing (..) in (.) (.) (.) (.) nd neglecing prodcs nd higher powers of we ge d M N + N (.5.) d r (.5.) d d +... (.5.) n... (.5.) Here M... (.5.5) r + N n + N... (.5.6) The chrcerisic eqion of which is ( λ + M )( λ r )( λ + )( λ n ) 0... (.5.7) The roos r n re posiive nd M re negive. Hence he eqilibrim se is nsble. The solions of he eqions (.5.) (.5.) (.5.) (.5.) re N ( + M ) N ( r + M ) 0 0 M 0 + e ( r + M)( + M) n r N ( 0 η7) e + η7e ( r + M) N0e ( + M) + ( r + M)( + M) (.5.8) r e (.5.9) 0 [ 0 ] e + e ( n + ) ( n + ) 0 0 n... (.5.0) n 0e... (.5.) 0 Where η 7 ( n + ) The solion crves re exhibied in figres &. Cse (i): If 0 < 0 < 0 < 0 nd < M < n < r In his cse iniilly he Hos (S ) of S domines over he Predor (S ) ill he ime insn nd here fer he dominnce is reversed. Pelgi Reserch Librry 6
13 R. Srilh e l Adv. Appl. Sci. Res. 0 (): 5-65 Cse (ii): If 0 < 0 < 0 < 0 nd < M < r < n In his cse iniilly he Hos (S ) of S domines over he Predor (S ) S nd he Prey (S ) ill he ime insn respecively nd here fer he dominnce is reversed. Also he Predor (S ) domines over he Prey (S ) ill he ime insn nd here fer he dominnce is reversed. And he Predor (S ) domines over S ill he ime insn nd he dominnce is ges reversed here fer. Similrly S domines he Prey (S ) ill he ime insn nd here fer he dominnce is reversed..6 Sbiliy of he Eqilibrim Se + N N N 0 N Sbsiing (..) in (.) (.) (.) (.) nd neglecing prodcs nd higher powers of we ge d d d d N N + N (.6.) N N (.6.) (.6.) (.6.) The chrcerisic eqion of which is λ + (N + N ) λ + NN ( λ )( λ ) (.6.5) The chrcerisic roos of (.6.5) re ( N + N) ± ( N+ N) NN λ λ λ --- (.6.6) Two roos of he eqion (.6.5) re posiive nd he oher wo roos re negive. Hence he eqilibrim se is nsble. The rjecories re given by Pelgi Reserch Librry 6
14 R. Srilh e l Adv. Appl. Sci. Res. 0 (): 5-65 ( ) N + N φ ( λ ) e λ λ ( ) + + φ ( 0 φ)( λ λ) N N 0 + φ( λ ) e λ e λ λ ( ) λ λ ( ) λ + ξ e + φ λ λ λ N + N φ ( λ ) ξ e ( φ )( λ λ ) N + + N + φ ( λ ) λ e (.6.7) (.6.8) 0e ---- (.6.9) 0e --- (.6.0) Here φ β N φ φ ( + P ) P N 0 + ψ + β N ( ) β ( + )N N β N + N 0 ( λ + P ) ( λ + P ) ψ N + N ξ ξ N N The solions re illsred in figres 5 & 6. Cse (i) : If 0 < 0 < 0 < 0 nd < < < In his cse iniilly he hos (S ) of S domines over S he prey (S ) nd he predor (S ) in nrl growh re s well s in is iniil poplion srengh. I is eviden h ll he for species going wy from he eqilibrim poin. Hence he eqilibrim se is nsble s shown in figre. Fig. 5 Cse (ii): If 0 < 0 < 0 < 0 < < < In his cse iniilly he prey (S ) domines over S nd he hos (S ) of S ill he ime insn respecively nd here fer he dominnce is reversed. Also he predor (S ) domines Pelgi Reserch Librry 6
15 R. Srilh e l Adv. Appl. Sci. Res. 0 (): 5-65 over S nd he hos (S ) of S ill he ime insn dominnce is reversed. Fig. 6 respecively nd here fer he REFERENCES [] Archn Reddy R on he sbiliy of some mhemicl models in biosciencesinercing species Ph.D hesis 009 JNTU. [] Freedmn HI Deerminisic Mhemicl Models in Poplion Ecology Mrcel Decker New York 980. [] Kpr JN Mhemicl Modeling Wiley Esern 988. [] Kpr JN Mhemicl Models in Biology nd Medicine Affilied Es Wes 985. [5] Lkshmi Nryn K A Mhemicl sdy of Prey-Predor Ecologicl Models wih pril covers for he prey nd lernive food for he predor Ph.D hesis 00 J.N.T.Universiy. [6] Lok AJ Elemens of Physicl biology Willims nd Wilkins Blimore 95. [7] Meyer WJ Conceps of Mhemicl Modeling Mc Grw Hill 985. [8] Pl Colinvx Ecology John Wiley nd Sons Inc. New York 986. [9] Rvindr Reddy B Lkshminryn K nd Pbhirmchryl NCh Advnces in Theoreicl nd Applied Mhemics Vol.5 No. (00) -. [0] Rvindr Reddy B Lkshminryn K nd Pbhirmchryl NCh Inernionl J. of Mh. Sci & Engg. Appls. (IJMSEA) Vol. No. III (Ags 00) [] Trinov NC Some Mhemicl specs of modeling in Bio Medicl Sciences Ph.D hesis 99 Kkiy Universiy. [] Volerr V Leconssen l heorie mhemiqe de l leie po lvie Ghier Villrs Pris 9. Pelgi Reserch Librry 65
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