Int. J. Opn Problms Compt. Math., Vol., o., Jun 008 A Pry-Prdator Modl with an Altrnativ Food for th Prdator, Harvsting of Both th Spcis and with A Gstation Priod for Intraction K. L. arayan and. CH. P. Ramacharyulu Dpartmnt of Humanitis & Scincs, Hydrabad-500070, India. Dpartmnt of Mathmatics and Humanitis, Warangal - 506 004, India -mail: narayankundru@yahoo.com Abstract In th classical Lotka - Voltrra Pry - Prdator modl, thr is no protction for Pry from th Prdator and Prdator sustains on th Pry alon. Whn th Pry population falls blow a crtain lvl, th prdator would migrat to anothr rgion in sarch of food and rturn only whn th Pry-Population riss to th rquird lvl. A population modl with tim dlay was proposd by Kapur, (c.f. Dly diffrntial and intgro-diffrntial quations in population dynamics, J.Math.Phy.Sci., 4, 07-9, 980) and this modl motivatd th prsnt invstigation. In th prsnt invstigation w studid a pry - prdator modl incorporating i) th prdator is providd with an altrnativ food in addition to th pry, ii) both th pry and th prdator ar harvstd proportional to thir population sizs and iii) a gstation priod for intraction. Th modl is charactrizd by a coupl of first ordr intgro-diffrntial quations. All th four quilibrium points of th modl ar idntifid and stability critria ar discussd. Som thrshold rsults ar illustratd. Kys words: Equilibrium points, ormal Study Stat, ormalizd Krnls, Pry, Prdator, Stability, Thrshold Diagrams, and Thrshold Rsults., Introduction Som of th pry-prdator modls wr discussd by Michal Olinnck [4], May [5], Varma [6] Colinvaux [7], Frdman [8], arayan [9]. A population modl with tim dlay was proposd by Kapur []. Voltrra formulatd a distributd tim dlay modl for pry - prdator cological modls. Kapur, [] discussd
A Pry-Prdator Modl with an Altrnativ Food 7 th solution in th closd form for that modl. Inspird from that, w discussd a mor gnral modl by taking an altrativ food for th prdator and harvsting of both th spcis. Th modl is charactrizd by a coupl of first ordr ordinary dlay-diffrntial quations. All th four quilibrium points of th modl ar idntifid and stability critria ar discussd. In consonanc with th principl of comptitiv xclusion (Gaus [3]) som thrshold rsults ar illustratd. Basic Equations Th modl quations for a two spcis Pry-Prdator systm is givn by th following systm of first ordr dlay - diffrntial quations mploying th following notation: and ar th populations of th pry and prdator with th natural growth rats a and a rspctivly, is rat of dcras of th pry du to insufficint food, is rat of dcras of th pry du to inhibition by th prdator, is rat of incras of th prdator du to succssful attacks on th pry, is rat of dcras of th prdator du to insufficint food othr than th pry, k and k ar rat of dcras of th pry and prdator du to harvsting, k ( t s), k ( t s) ar 3 4 wight factors to giv th influnc at tim t of, of tim s( t) i.. k ( t s), k ( t s) ar rat of changs of 3 4, aftr a tim intrval ( t s). Furthr both th variabls and ar non-ngativ and th modl paramtrs a, a,,,,, k, k ar assumd to b non-ngativ constants. d T = a( k) k4( ts) ( s) ds (.) dt d T = a( k) + k3( ts) ( s) ds (.) dt put t s = z, i.. s = t z (.3) k3( z), k4( z) 0, ar tim dlayd so that k3( z) dz = k4( z) dz = (.4) 0 0 ar normalizd krnls. ow w rwrit th basic quations as d = a( k) k4( z) ( tz) dz dt (.5) 0
73 K. L. arayan and. CH. P. Ramacharyulu d dt = a( k) + k3( z) ( tz) dz (.6) 0 3 Equilibrium Points Th systm undr invstigation has four quilibrium stats:. Th fully washd out stat with th quilibrium points = 0; = 0. (3.). Th stat in which prdator survivs and th prys ar washd out. Th quilibrium point is a( k) = 0 ; = (3.) 3. Th stat in which, only th pry survivs and th prdators ar washd out. Th quilibrium point is a( k) = ; = 0 (3.3) 4. Th co-xistnc stat (normal study stat). Th quilibrium point is a ( k) a( k) a( k) + a ( k) = ; = + + sinc k3( z) dz = k4( z) dz = and this stat can xist only whn 0 0 a( k) < a ( k ) (3.4) (3.5) 4 Th Stability of th Equilibrium Stats Lt = (, ) T = + U (4.) T whr U = ( u, u ) is a small prturbation ovr th quilibrium stat, = ( ) T. Th basic quations (.5), (.6) ar quasi-linarizd to obtain th t quations for th prturbd stat. (By trying th solutionsu = c λ and u = c for th quations (.5) and (.6)) du = A [ U ] (4.) dt whr λ t
A Pry-Prdator Modl with an Altrnativ Food 74 k4( z) λz dz A = 0 k3( z) λz (4.3) dz 0 dt A λi =. (4.4) Th charactristic quation for th systm is [ ] 0 Th quilibrium stat is stabl only whn th roots of th quation (4.4) ar ngativ in cas thy ar ral or hav ngativ ral parts in cas thy ar complx. 4. Stability of th quilibrium stat I Th trajctoris for both washd out stat ar a( k)t u = u0 (4.5) a( k)t u = u0 (4.6) Th solution curvs ar illustratd in figurs and : Cas : In this cas th prdator dominats th pry in natural growth as wll as in its initial population strngth. i.. u 0 < u 0 and a( k) < a ( k ) as shown in Fig. Cas : Th prdator dominats th pry in natural growth rat but its initial strngth is lss than that of pry. i.. u 0 > u 0 and a ( k ) > a ( k ) as illustratd in Fig.. In this cas, th pry out numbrs th prdator till th timinstant t = t* = aftr that th prdator out numbr th pry. ln { u0 / u0} [ a( k)- a( k) ] (At t = t *, populations of both th spcis ar sam, and from (4.5)&(4.6) u = u) 4. Stability of th quilibrium stat II Th trajctoris for only pry washd out stat ar: t u = u0 λ and 0 (- ) * 3 ( ) u = ua k k λ t λ λt u0a (- k) k * 3 ( λ) a( k) t + { u { λ + a ( k )} 0 } (4.7) { λ + a ( k )} whr k 3* ( λ) is Laplac transformation of k ( z) and 3 k3( z) dz = k4( z) dz = 0 0 a( k) and λ = a( k) - (4.8) Th solution curvs ar illustratd in Fig. 3 & 4.
75 K. L. arayan and. CH. P. Ramacharyulu Cas : Initially th pry dominats th prdator and it continus throughout its a( k) growth u 0 > u 0 and a( k) > a( k) as illustratd in Fig. 3. Cas : Initially th prdator dominats th pry i.. u 0 < u0 and a( k) a( k) > a( k). In this cas, th prdators out numbr th pry till th tim-instant a( k) u0k * 3 ( λ) u0 { λ + a( k)} t = t = ln λ * a( k) (4.9) + a( k) u0k3 ( λ) u0 { λ + a( k)} aftr that, th pry out numbr th prdator. This is illustratd in Fig. 4 4.3 Stability of th quilibrium stat III Th trajctoris for only th prdator washd out stat ar: 0 ( ) * 4 ( ) u = u a k k λ λ t λ t u0a ( k ) k * 4 ( λ ) - a( k) t + { u0 + } { λ + a( k )} { λ + a( k)} (4.0) t u = u0 λ, (4.) whr k 4* ( λ) is Laplac transformation of k ( z) and ( ) 4 λ = ( )+ a a k k Cas : Th initial strngth of th pry is gratr than that of th prdator. i.. u 0 > u 0 Initially th pry out numbr th prdator and this continus up to th tim instant, u0a (- k) k * 4 ( λ) u0 + { λ + a( k)} t = t* = λ * a( k), (4.) + u0a (- k) k4 ( λ ) u0 + { λ + a( k)} aftr which th prdator out numbrs th pry. And also th prdator spcis is notd to b going away from th quilibrium point whil th pry-spcis would bcom xtinct at th instant (t*) of tim givn by th positiv root of th quation λt a( k) t u0{ a( k) + a ( k)[ + ]} + = u0a ( k) This is illustratd in Fig. 5 Cas : Th prdator dominats th pry in its initial strngth. i.. u 0 < u 0 (4.3)
A Pry-Prdator Modl with an Altrnativ Food 76 In this cas th prdator spcis is notd to b going away from th quilibrium point whil th pry-spcis would bcom xtinct at th instant (t*) of tim givn by th positiv root of th quation λt a( k) t u0{ a( k) + a ( k)[ + ]} + = u0a ( k) As such th stat is unstabl. This is illustratd in Fig. 6 4.4 Stability of th normal study stat Th trajctoris for normal study stat ar: u0( λ + )-u 0 u = λ t u0( λ + )-u0 + λ λ λ λ t λ (4.4) u = u0( λ + )-u 0 λ t u0( λ + )-u0 + λ t λ λ λ λ (4.5) whr λand λ ar roots of th charactristic quation. Cas : Initially th pry dominats th prdator and it continus through out its growth i.. u0 < u 0 In this cas th prdator always out numbrs th pry. It is vidnt that both th spcis convrging asymptotic to th quilibrium point. Hnc this stat is stabl. This is illustratd in Fig. 7. Cas : Th pry dominats th prdator in natural growth rat but its initial strngth is lss than that of prdator. i.. u 0 > u 0.Initially th pry out numbr th prdator and this continus till th tim-instant t = t * = ( b3 a5) u0 + ( a3+ b) u 0 ln λ + λ ( b a6) u0 + ( a4 + b ) u0 (4.6) whr, a3 = λ+ ; a 4 = λ + ; a 5 = λ+ ; a6 = λ + ; b = λ ; b = λ (4.7) aftr which th prdator out numbr th pry. Th solution curvs ar illustratd in Fig. 8. Whn ( ) < 4 ( k) * * k3 ( λ) k4 ( λ) th roots ar complx with ngativ ral part. Hnc th quilibrium stat is stabl. This is illustratd in Fig. 9. 5 Thrshold Rsults Employing th principl of comptitiv xclusion (Gauss [3]), th following thrshold rsults ar stablishd.
77 K. L. arayan and. CH. P. Ramacharyulu a. a( k) a( k) a( k) a( k) Whn, > and > (5.) Both th spcis co-xist as shown in Fig. 0 b. a( k) a( k) a( k) a( k) Whn, > and < (5.) Only pry spcis survivs as illustratd in Fig. c. a( k) a( k) a( k) a( k) Whn, < and > (5.3) Only prdator spcis survivs as illustratd in Fig.. 6. Trajctoris
A Pry-Prdator Modl with an Altrnativ Food 78 7 Thrshold Diagrams
79 K. L. arayan and. CH. P. Ramacharyulu 8 Futur Works In th prsnt papr it is invstigatd that a Pry-Prdator modl with harvsting is proportional to th population sizs of th spcis with gstation priod for intraction. Thr is a scop to study th modl with constant harvsting of both spcis, or harvsting of any of th spcis. Furthr covr can b takn for th Pry to protct it from th attacks of th Prdator. On can construct Lypunov s function to study th global stability of th modl. Rfrncs [] J.. Kapur, Dlay diffrntial and intgro-diffrntial quations in population dynamics, J.Math.Phy.Sci., 4, 07-9, 980. [] J.. Kapur, Mathmatical modls in Biology and Mdicin, Affiliatd East- Wst, 985. [3] G. F. Gaus, Th struggl for xistnc, Williams and Wilkins, Baltimor, 934. [4] Michal Olinck, An introduction to Mathmatical modls in th social and Lif Scincs, 978, Addison Wsly. [5] R. M. May, Stability and complxity in Modl Eco-systms, Princton Univrsity Prss, Princton, 973. [6] V. S. Varma, A not on Exact solutions for a spcial pry-prdator or compting spcis systm, Bull. Math. Biol., Vol. 39, 977, pp 69-6. [7] Paul Colinvaux, Ecology, John Wily and Sons Inc., w York, 977. [8] H. I. Frdman, Dtrministic Mathmatical Modls in Population Ecology, Dckr, w York, 980. [9] K. L. arayan, A pry - prdator modl with covr for pry and an altrnat food for th prdator, and tim dlay. Caribb. J. Math. Comput. Sci.9, 006, 56-63.