Dynamical Behavior in a Discrete Three Species Prey-Predator System A.George Maria Selvam 1, R. Dhineshbabu 2 and V.Sathish 3

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1 IJISET - International Journal of Innovative Science, Enineerin & Technoloy, Vol. Issue 8, October 4. ISSN Dynamical Behavior in a Discrete Three Species Prey-Preator System A.Geore Maria Selvam, R. Dhineshbabu an V.Sathish, Sacre Heart Collee, Tirupattur 65 6, S.Inia. DMI Collee of Enineerin, Chennai - 6, S.Inia. ABSTRACT This paper investiates the ynamical behavior of a iscrete prey-preator system with three species. Stability analysis is performe an analytical results are illustrate with numerical simulations. Time series plots an phase portraits are obtaine for ifferent sets of parameter values. Bifurcation iarams are provie for selecte rane of rowth parameter. Keywors: Discrete prey - preator system, ifference equations, equilibrium points, stability.. INTRODUCTION Simple moels of foo webs may exhibit very complex ynamics. The Lotka Volterra equations [4] (ODE) successfully escribe an ecoloical preator prey moel an the oscillations in their populations. In recent ecaes, many researchers [,, 6, 7, 9] have focuse on the ecoloical moels with three an more species to unerstan complex ynamical behaviors of ecoloical systems in the real worl. They have emonstrate very complex ynamic phenomena of those moels, incluin cycles, perioic oublin an chaos [, ]. The ynamical behavior an stability analysis of nonlinear iscrete prey-preator an host parasite moel has been stuie. The iscrete time moels which are usually escribe by ifference equations can prouce much richer patterns [, 4, 5, 8]. Discrete time moels are ieally suite to escribe the population ynamics of species, which are characterize by iscrete enerations.. MATHEMATICAL MODEL In this paper, we consier the iscrete-time preypreator system escribin the interactions amon three species by the followin system of ifference equations: xt ( + ) = rxt ()[ xt ()] bxt () yt () yt ( + ) = ( c) yt () + xt () yt () eytzt () () zt ( + ) = ( f) zt () + ytzt () () () where xt ( ), yt () an zt () are functions of time representin population ensities of the prey an the mipreator an the top preator, respectively, an all parameters are positive constants. The parameter r is the intrinsic rowth rates of the prey population, b is the per capita rate of preation of the mi-preator, c enotes the eath rate of the mi-preator, is the rate of conversion of a consume prey to a preator, e is the percapita rate of preation of the top preator, f enotes the eath rate of the top preator, is the rate of conversion of a consume prey to a preator.. EXISTENCE OF EQUILIBRIUM The equilibrium points of () are the solution of the equations x = rx[ x] bxy y = ( c) y + x y eyz z = ( f ) z + yz. The equilibrium points are E = (,,), E =,,, r c r ( c) E =,, E * * * = x, y, z. b b an ( ) where, * bf * f x = +, y =, an r * c bf z = +. Interior equilibrium point E e re correspons to the coexistence of all species. 4. DYNAMICAL BEHAVIOR OF THE MODEL In this section, we stuy the local behavior of the system () about each equilibrium points. The stability of the system () is carrie out by computin the Jacobian matrix corresponin to each equilibrium point. The Jacobian Matrix J for the system () is r rx by bx J ( x, y, z) = y c x ez ey +. () z f + y The eterminant of the Jacobian Jxy (, ) is Det = ( f + y)[ r( x)( c + x) by( c)]. 5

2 IJISET - International Journal of Innovative Science, Enineerin & Technoloy, Vol. Issue 8, October 4. Hence the system () is issipative ynamical system when ( f + y)[ r( x)( c + x) by( c)] <. Theorem : The equilibrium point E asymptotically stable if r <, < c < an < f <, otherwise unstable equilibrium point. Proof: In orer to prove this result, we etermine the eienvalues of Jacobian matrix J at E. The Jacobian matrix evaluate at the equilibrium point E has the form r J( E ) = c. f Hence the eienvalues of the matrix J( E) are r =, = c an = f. Thus E is stable when r <, < c < an < f <. Otherwise E is unstable equilibrium point. Theorem : The equilibrium point E asymptotically stable if < r <, r < c an < f <, otherwise unstable equilibrium point. Proof: The Jacobian matrix J for the system evaluate at E is iven by the equilibrium point r b r J( E) = c+. r f Hence the eienvalues of the matrix J( E ) are = r, = c+ an = f. r Hence E asymptotically stable when < r <, r < c an < f <, an unstable r >, when r > c an f >. ISSN Theorem : The equilibrium point E ( bf + ) asymptotically stable if < r <, otherwise c ( c) unstable equilibrium point. Proof: The Jacobian matrix evaluate at E is iven by rc bc r( c) e re( c) J( E ) =. b b b r( c) f + b b Hence the eienvalues of the matrix J( E ) are = r f ( c) b + b an =, cr 4 cr( c ) c r 4c ± + +. Hence E asymptotically stable when ( bf + ) < r < c ( c) r c > ( bf + ) or r >. ( c), an unstable when 5. LOCAL STABILITY AND DYNAMICAL BEHAVIOR AROUND INTERIOR FIXED POINT E We now investiate the local stability an bifurcation of interior fixe point E. The Jacobian matrix J at E has of the form a a a J( E) = a a a. () a a a bf b bf = r+, a = + b, r f fe a =, a =, a =, a =, ( c) ( + bf ) a =, a =, a =. e re where, a Its characteristic equation is + A + B+ C = (4) 6

3 with IJISET - International Journal of Innovative Science, Enineerin & Technoloy, Vol. Issue 8, October 4. A= ( a + a + a), B= a a + a a + a a a a a a C = ( a a a a ) a + a a a. * * * E ( x, y, z ) By the Routh-Hurwitz criterion, = is locally asymptotically stable if an only if AC,, an AB C are positive. 6. NUMERICAL SIMULATIONS In this section, we present the time plots, phase portraits an bifurcation iarams to illustrate the theoretical analysis an show the interestin complex ynamical behaviors of the system ()., ISSN so that,, <. Hence system () is stable (see Fiure-). Example: We shall consier r =.5, b =.5, c =., =.5, e =.4, f =.95 an =.94. At equilibrium point E, the eienvalues are =.5, =.9 an =.5 so that,, <. Hence the trivial equilibrium point is stable (see Fiure-). Fiure : Time Series Plot an Phase Portrait at E. Example: We shall consier the parameter values r =.5, b =.65, c =., =.5, e =.4, f =.5 an =.4. The equilibrium point E = (.,.8,) an the eienvalues are =.6, =.9 an =.8 so that,, (see Fiure-). <. Thus system () is stable Fiure : Time Series Plot an Phase Portrait at E. Example: We shall consier the parameter values r =., b =.65, c =., =.5, e =.4, f =.5 an =.4. The equilibrium point E = (.9,,) an the eienvalues are.9 =.945an =.5 = Fiure : Time Series Plot an Phase Portrait at E. 7

4 IJISET - International Journal of Innovative Science, Enineerin & Technoloy, Vol. Issue 8, October 4. ISSN Example4: When r =.65, b =.45, c =., =.8, e =.5, f =.5 an =.. The eienvalues are =.88 an, =.9686 ± i.549 so that < an, =.6 >. The system () is unstable (see Fiure-4). Fiure 5: Time Series Plot an Phase Portrait at E. 7. BIFURCATION ANALYSIS VIA NUMERICAL SIMULATIONS Bifurcation is a chane of the ynamical behaviors of the system as its parameters pass throuh a bifurcation (critical) value. Bifurcation usually occurs when the stability of an equilibrium chanes. In this section, we focus on explorin the possibility of chaotic behavior for the prey an the mi-preator an the top preator respectively. Fiure 4: Time Series Plot an Phase Portrait at E. While with r =.5, e =.55 an keepin all other parameters same, we obtain E = (.58,.454,.9) an the eienvalues are =.88 an, =.9667 ± i.9 so that < an =.994 <. We observe the system () is stable, (see Fiure-5). Fiure 6: Bifurcation iarams for (a) prey population, (b) mi-preator population an (c) top preator. Fiure (6) presents the bifurcation iaram for prey an the mi-preator an the top preator ensities of the system () with initial conitions x =.6, y =. an z =.4 as above an we consier the parameters values b =.5, c =., =.8, e =.55, f =.95, =.85 an r =.5:.:4.The bifurcation iarams imply the existence of chaos. Also these results reveal far richer ynamics of the iscrete-time moels. 8

5 IJISET - International Journal of Innovative Science, Enineerin & Technoloy, Vol. Issue 8, October 4. References [] Ab-Elalim A. Elsaany, Dynamical complexities in a iscrete-time foo chain, Computational Ecoloy an Software,, ():4-9. [] H.N. Aiza, E.M. Elabbasy, H. EL-Metwally, A.A. Elsaany, Chaotic ynamics of a iscrete prey preator moel with Hollin type II, Nonlinear Analysis: Real Worl Applications (9) 6-9. [] H. I. Freeman an P.Waltman, Persistence in moels of three interactin preator-prey populations, Mathematical Biosciences, vol. 68, no., pp., 984. [4] Leah Eelstein-Keshet, Mathematical Moels in Bioloy, SIAM, Ranom House, New York, 5. [5] Marius Danca, Steliana Coreanu an Boton Bako, Detaile Analysis of a Nonlinear Prey-preator Moel, Journal of Bioloical Physics : -, 997. [6] R. K. Naji an A. T. Balasim, Dynamical behavior of a three species foo chain moel withbeinton-deanelis functional response, Chaos, Solitons & Fractals, vol., no. 5, pp , 7. [7] Rai Kamel Naji an Alaa Jabbar Baai, A three species ratio-epenent foo web moel ynamics, Journal of Basrah Researches (Sciences) Volume 7, Number 4.D,. [8] M.Reni Saayaraj, A.Geore Maria Selvam, R.Janaaraj.an D.Pushparajan, Dynamical Behavior in a Discrete Prey- Preator Interactions, International Journal of Enineerin Science an Innovative Technoloy (IJESIT) Volume, Issue, March, ISSN: , pp 6. [9] M. Reni Saayaraj, A. Geore Maria Selvam, R. Janaaraj, an D. Pushparajan, Dynamical Behavior in a Three Species Discrete Moel of Prey-Preator Interactions, International Journal of Computational Science an Mathematics. ISSN Volume 5, Number (), pp. -. ISSN