A Model of two mutually interacting Species with Mortality Rate for the Second Species

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1 Avilble online t Advnces in Applied Science Reserch, 0, 3 (): ISS: CODE (USA): AASRFC A Model of two mutully intercting Species with Mortlity Rte for the Second Species B. Rvindr Reddy JTUH College of Engineering, chuplly, Krimngr, Indi _ ABSTRACT The present pper concerns with model of two mutully intercting species with limited resources for first species nd unlimited resources for second species with mortlity rte. The model is chrcterized by coupled system of first order non-liner ordinry differentil equtions. In ll two equilibrium points re identified. If the deth rte of the second species is greter thn its birth rte, it is found tht there re three equilibrium points. The criteri for symptotic stbility hve been estblished for ll the equilibrium points. The co-existent stte is lwys stble. The solutions of the linerised bsic equtions re obtined nd their trends re illustrted. Key words: Equilibrium points, Mutulism, Coexistence stte, Stbility. _ ITRODUCTIO Mthemticl modeling is n importnt interdisciplinry ctivity which involves the study of some spects of diverse disciplines. Biology, Epidemiodology, Physiology, Ecology, Immunology, Bio-economics, Genetics, Phrmocokinetics re some of those disciplines. This mthemticl modeling hs rised to the zenith in recent yers nd spred to ll brnches of life nd drew the ttention of every one. Mthemticl modeling of ecosystems ws initited by Lotk [8] nd by Volterr [4]. The generl concept of modeling hs been presented in the tretises of Meyer [9], Cushing [], Pul Colinvux [0], Freedmn [3], Kpur [5, 6]. The ecologicl interctions cn be brodly clssified s Prey-Predtion, Competition, Mutulism nd so on..c. Srinivs [3] studied the competitive eco-systems of two species nd three species with regrd to limited nd unlimited resources. Lkshmi ryn [7] investigted the two species prey-predtor models nd stbility nlysis of competitive species ws investigted by Archn Reddy []. Locl stbility nlysis for two-species ecologicl mutulism model hs been presented by the present uthor et l. [, ]. Recently, stbility nlysis of three species ws crried out by Siv Reddy [8]. Further Srilth et l. [5, 6] nd Hri Prsd et.l [7] studied stbility nlysis of four species. The present investigtion is devoted to the nlyticl study of model of two mutully intercting species with mortlity rte for the second species. Before describing model, first we mke the following ssumptions: is the popultion of the first species,, the popultion of the second species, is the rte of nturl growth of the first species, is the rte of nturl growth of the second species, is the rte of decrese of the first species e to insufficient food, is the rte of increse of the first species e to interction with the second species, is the rte of increse of the second species e to interction with the first species. Further note tht the vribles, nd the model prmeters,,,, re non-negtive nd tht the rte of difference between the deth nd birth rtes is identified s the nturl growth rte with pproprite sign. The model equtions for two species mutulising re governed by system of non-liner ordinry differentil equtions. 757

2 B. Rvindr Reddy Adv. Appl. Sci. Res., 0, 3(): Bsic Equtions: The bsic equtions re given by d d + = (.) = + (.) Here we come cross three equilibrium sttes: = 0; = 0, I. the stte in which both the species re wshed out. II. = ; 0 =. Here the first species ( ) survives while the second species ( ) is wshed out. III. = ; =. In this stte both the species co-exist nd this cn exist only when > 0. Equilibrium stte I (fully wshed out stte): To discuss the stbility of equilibrium stte u ( ) t from the stedy stte, i.e. we write = 0 ; = 0, we consider smll perturbtions u ( t) nd = + u ( t), (.3) = + u ( t) (.4) Substituting (.3) nd (.4) in (.) nd (.), we get = u u + u u After lineriztion, we get nd = u + u u u = (.5) = u (.6) The chrcteristic eqution is (λ - )(λ + ) =0 One root of this eqution is λ = which is positive nd the other root is λ = which is negtive. Hence the equilibrium stte is unstble. 758

3 B. Rvindr Reddy Adv. Appl. Sci. Res., 0, 3(): The solutions of equtions (.5) nd (.6) re t u = u0 e (.7) t u = u e 0 where u 0, u 0 re the initil vlues of u nd u. The solution curves re illustrted in figures nd. (.8) Cse : u 0 > u 0 i.e. initilly the first species domintes the second species. We notice tht the first species is going wy from the equilibrium point while the second species pproches symptoticlly to the equilibrium point. Hence the stte is unstble. Cse : u 0 < u 0 i.e. initilly the second species domintes the first species. In this cse the second species out numbers the first species till the time, ln { u0 / u0} t = t* = ( + ) fter tht the first species out numbers the second species nd grows indefinitely while the second species pproches symptoticlly to the equilibrium point. Hence the stte is unstble. Further the trjectories in the ( u, u ) plne re given by u u 0 = u u 0 Equilibrium stte II ( exists while is wshed out): We hve = ; = 0 Substituting (.3) nd (.4) in (.) nd (.), we get u = u u + u u + u = u + uu + After lineriztion, we get u = u + (.9) = The chrcteristic eqution is ( λ + ) λ u { -[ ]} (.0) = 0 (.) One root of this eqution (.) is λ = which is negtive. 759

4 B. Rvindr Reddy Adv. Appl. Sci. Res., 0, 3(): Cse A: When >, the other root of eqution (.) is λ = The trjectories re given by where u = γ - u0 e λ t + { u0γ - u 0 } e t λ t which is positive. Hence the equilibrium stte is unstble. (.) u = u0e (.3) γ = [ + ] The solution curves re illustrted in figures 3&4. u < u, we hve CASE : For 0 0 In this cse the second species is noted to be going wy from the equilibrium point while the first species would become extinct t the instnt + t * = 0 0 (λ ) ln u u γ u As such the stte is unstble. u > u, we hve 0 CASE : If 0 0 Here the first species out numbers the second species till the time, t = t * = u0 (λ + ) u0 ln λ + u0[ (λ + ) ] there fter the second species out numbers the first species. And lso the second species is noted to be going wy from the equilibrium point while the first species would become extinct t the instnt u t * = 0 u0 γ ln u 0 (λ + ) As such the stte is unstble. Cse B: When < One root of the eqution (.) is λ = which is negtive. λ = which is negtive nd the other root is As the roots of the eqution (.) re both negtive, the equilibrium stte is stble. The trjectories in this cse re the sme s in (.) nd (.3). 760

5 B. Rvindr Reddy Adv. Appl. Sci. Res., 0, 3(): Tht is where u = γ λ u = u e t 0 - u0 e λ t + { u0γ - u 0 } e t γ = [ + ] The solution curves re illustrted in figures 5&6. u < u i.e. initilly the second species domintes the first species. CASE : 0 0 In this cse the second species lwys out numbers the first species. It is evident tht both the species converging symptoticlly to the equilibrium point. Hence the stte is stble. u > u i.e. initilly the first species domintes the second species. CASE : 0 0 Here the first species out numbers the second species till the time, t = t * = u0 (λ + ) u0 ln λ + u0[ (λ + ) ] there fter the second species out numbers the first species. As t both u & u pproch to the equilibrium point. Hence the stte is stble. Further the trjectories in the ( u, u ) plne re given by where ( q ) u q = = q cu + pu p = nd c is n rbitrry constnt. The solution curves re illustrted in figure 7. Equilibrium stte III (coexistence stte): We hve = ; ; > 0 = wherein Substituting (.3) nd (.4) in (.) nd (.), we get = u + u u u + u After lineriztion, we get nd = u + u u = u + u (.4) 76

6 B. Rvindr Reddy Adv. Appl. Sci. Res., 0, 3(): u = (.5) The chrcteristic eqution is λ λ + = 0 One root of this eqution is positive nd the other root is negtive. Hence the equilibrium stte is unstble. The trjectories re given by u = u0λ + u0 λ λ u ( λ + ) + u λ λ u = 0 0 e λ t u0λ + u0 + e λ t λ λ t e λ + u ( λ + ) + u λ λ 0 0 e λ t The curves re illustrted in figures 8&9. Cse : u 0 > u 0 i.e. initilly the first species domintes the second species. In this cse, the first species is noted to be going wy from the equilibrium point while the second species pproches symptoticlly to the equilibrium point. Hence the stte is unstble. Cse : u 0 < u 0 i.e. initilly the second species domintes the first species. In this cse the second species out numbers the first species till the time, t = t * = λ λ ( b - λ ) u + ( b ) u ln ( b λ ) u0 + ( 4 b ) u0 where b = ; b 3 = ; = λ + ; 4 = λ + fter tht the first species out numbers the second species nd grows indefinitely while the second species pproches symptoticlly to the equilibrium point. Hence the stte is unstble. Further the trjectories in the ( u, u ) plne re given by v ( )( v v ) ( u u v ) [ ] u d = v ( u vu ) where v nd v re roots of the qudrtic eqution c = nd d is n rbitrry constnt. v +bv+ c =0 with = ; ; b= 76

7 B. Rvindr Reddy Adv. Appl. Sci. Res., 0, 3(): Trjectories: REFERECES [] Archn Reddy R, On the stbility of some mthemticl models in biosciences- intercting species, Ph.D thesis, 009, JTU. [] Cushing JM, Integro differentil equtions nd Dely Models in Popultion Dynmics, Lecture otes in Biomthemtics, Springer Verlg, Heidelberg, 977, Vol. 0. [3] Freedmn HI, Deterministic Mthemticl Models in Popultion Ecology, Mrcel Decker, ew York, 980. [4] George F Simmons, Differentil Equtions with pplictions nd historicl notes, Tt McGrw-Hill, ew Delhi, 974. [5] Kpur J, Mthemticl Modelling, Wiley Estern, 988. [6] Kpur J, Mthemticl Models in Biology nd Medicine Affilited Est West, 985. [7] Lkshmi ryn K, A Mthemticl study of Prey-Predtor Ecologicl Models with prtil covers for the prey nd lterntive food for the predtor, Ph.D thesis, J..T.University, 004. [8] Lotk AJ, Elements of Physicl biology, Willims nd Wilkins, Bltimore, 95. [9] Meyer WJ, Concepts of Mthemticl Modelling, Mc Grw Hill, 985. [0] Pul Colinvux, Ecology, John Wiley nd Sons Inc., ew York,

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