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1 Available online at ISS International ejournals International ejournal of Mathematics and Engineering 8 (00) A RECURSIVE APPROACH FOR PREY-PREDATOR ECO SYSTEM ICORPORATIG DEATH RATE FOR PREDATOR B. Bhaskara Rama Sarma.Ch. Pattabhiramacharyulu and S.V..L.Lalitha 3.Department of Mathematics, BRS Classes, #-84,Vivekananda street, Hanuman nagar, Ramavarappadu, Vijayawada, A.P-,India E Mail.bbramasarma@yahoo.co.in Department of Mathematics (Retd.), IT, Warangal A.P,India. 3. Department of Electrical Engineering, Koneru Lakshmaiah College of Engineering, Vijayawada, A.P.,India ABSTRACT In the present paper a four stage recursive procedure is designed to give an approximate solution to the mathematical model equations of a two-species Prey- Predator eco-system. umerical examples are provided to explain trajectories of the solutions. The conclusions are recorded from the graphs. Key words : Recursive procedure, on-linear system, trajectories. AMS Classification : 9 D 5, 9 D 40 ITRODUCTIO The exact solutions of first order non-linear differential equations obtained in the two-species competitive eco-systems can t be obtained directly because of intractability of non-linear terms. Techniques like self portrait analysis are used to give qualitative nature of solutions in treatises of Kapoor [ ], Meyer [ ], Kushing [ ] etal. In the present investigation we present a four stage recursive procedure to give an approximate solution for a first order non-linear differential equation and the same is applied to solve a two-species Prey-Predator eco-system.in the section, general four-stage procedure is presented for solving a first order non linear differential equation. In section II the recursive procedure is applied to a two-species Prey- Predator eco-system.some numerical illustrations are given to explain the trajectories of the approximate solutions considering various situations. All the conclusions are recorded.

2 B. Bhaskara Rama Sarma,.Ch. Pattabhiramacharyulu, S.V.. L. Lalitha International ejournal of Mathematics and Engineering 8 (00) SECTIO I General Recursive procedure with four-stage approximation for first order nonlinear differential equations. Consider the system () t = A () t + f, t + h() t () d t Where () t = with initial condition Where ( 0) = 0 A () t denote the linear dependence of the system.. (.) (.) f, t corresponds to the non-linear dependence terms of the system h ()denote t replenishment / renewal rates of the system. The total procedure is represented in the following four stages. Stage I : Consider the linear system of (.) () t = A () t with same initial conditions ( 0) = 0 (.3) This system is obtained by suppressing the non-linear part in the right hand side in the absence of replenishments or renewals. () Let the solution of (.3) together with (.) be given by () t Stage II : Compute the resdenishment / renewal rate that could maintain above solution () () t r uu r () uur () () ht () () t A () t f uur =, t h in the non-linear system for the system (.) i.e. () () () () t = () t A () t f () t () h Stage III : Consider the system, t () t = f () t, t (.4) uu () r '( t) = A uur ( t) f uur, t with the homogeneous initial condition uu r (0) = 0 uur Let () () t. (.5). (.6) be the solution of system (.5) together with (.6).

3 B. Bhaskara Rama Sarma,.Ch. Pattabhiramacharyulu, S.V.. L. Lalitha International ejournal of Mathematics and Engineering 8 (00) Stage IV : An approximate solution of non-linear system (.) with (.) is given by uur uur() uur () () t = () t () t. (.7) SECTIO II. In this section we obtain the approximate solution of a typical two-species Prey-Predator eco-system with death rate for Predator and replenishments for both prey and predator.. The system under study is d = a a a + h (.) d = a + a a + h with initial conditions i (0) = i0 = i =,. (.) The computations of recursive procedure explained in section are carried out. Stage I : Taking the system (.) with linear terms along with (.) we get d d = a = a (.3) along with i (0) = i0 : i =,. (.4) The solution of (.3) with (.4) be given by = exp a t () { } 0 () = exp { a t } 0 Stage II : Basing on (.5) h, h of (.) are calculated as h t = a exp a t + a exp a + a t h () 0 ( ) 0 0 {( ) } () t = a exp( a t) a exp{ ( a + a ) t} (.5) (.6) Stage III : Using (.6) along with homogeneous initial conditions we set up the linear system d = a + h () t (.7) d = a + h () t along with (0) = (0) = 0. (.8) The solution of (.7) along with (.8) is now given as,

4 B. Bhaskara Rama Sarma,.Ch. Pattabhiramacharyulu, S.V.. L. Lalitha International ejournal of Mathematics and Engineering 8 (00) a a ( ) 0 0 = exp( a t) ( exp( a t) ) + ( exp( a t) ) 0 a a ( ) a 0 a0 = 0 exp( at) ( exp( at) ) ( exp( at ) ) a a Stage IV : An approximate solution to the system (.) is now provided as () () = () ( ) () t = () ( ) () t = a 0 a i.e. a 0 a. umerical computation : (.0) 0 0 () t = exp( a t) ( exp( a t) ) ( exp( a t) ) 0 0 () t = exp( a t) ( exp( a t) ) + ( exp( a t) ) a a a a (.9) (.0) (t), (t) are obtained numerically for a sampled initial values 0 =, 0 = 0.5 and with species competing parameters. a = a = a = 0.5,.0,.5, a = 0.5,.0,.5,.0 a = 0.,,.5 All the graphs showing trajectories are illustrated in Fig. Fig.. Fig. Fig.

5 B. Bhaskara Rama Sarma,.Ch. Pattabhiramacharyulu, S.V.. L. Lalitha International ejournal of Mathematics and Engineering 8 (00) Fig. 4 Fig. 3 Fig. 5 Fig. 6

6 B. Bhaskara Rama Sarma,.Ch. Pattabhiramacharyulu, S.V.. L. Lalitha International ejournal of Mathematics and Engineering 8 (00) Fig. 7 Fig. 8 Fig. 9 Fig. 0

7 B. Bhaskara Rama Sarma,.Ch. Pattabhiramacharyulu, S.V.. L. Lalitha International ejournal of Mathematics and Engineering 8 (00) Fig. Fig. 3.3 Conclusions from the graphs: Following conclusions are derived from the graphs (Fig. Fig.). If Growth rate of second species S decreases the curves are with decreasing Steepness.. Both, decrease with t as it is expected in a competing process because of utilization of energy during the predation. 3. For a <, weak competition of S over S, falls with slower rate than. However with increasing growth rate a of S steepness increases for the curves and falling rate is decreased with comparison with. slowly falls down. With increasing a steepness decreases. 4. For a =, equal competition of S & S the steepness of increase with a. However falling of is faster along with increasing a. 5. For a >, S is stronger than S it is observed that is increasing. This tendency is observed at slow rate in case of a =. This behaviour is expected of S because of resource limiting term and death rate are present. 6. Inspite of death rate of S, falls much faster than. However this fall becomes slower in case of a >. 7. As the resource limiting coefficients a of S increases falls down rapidly. Trend index: a decreases steepness increases a decreases steepness falls a decreases steepness increases

8 B. Bhaskara Rama Sarma,.Ch. Pattabhiramacharyulu, S.V.. L. Lalitha International ejournal of Mathematics and Engineering 8 (00) REFERECES [] Bhaskara Rama Sarma.B &. Ch. Pattabhiramacharyulu: Stability Analysis of a two-species competitive eco-system-,international journal of logic based intelligent systems,volume-, o., 008,PP [] Bhaskara Rama Sarma.B,. Ch. Pattabhiramacharyulu & S.V..L.Lalitha : A recursive procedure for general prey-predator eco-system ;Communicated to Bulletin of Marathwada Math.Association, Aurangabad, Maharastra (April,009) [3] Bhaskara Rama Sarma.B,. Ch. Pattabhiramacharyulu &S.V..L.Lalitha : A recursive procedure for two-species competitive eco-systemwith decay and replenishment for one of the species Communicated to Acta Ciencia Indica, ew Delhi (October,008). [4] Lakshmi arayan. K: A Mathematical study of a prey- predator ecological model with a partial cover for the prey and alternate food for the predator, Ph.D Thesis, J..T.U. (005) [5] Srinivas..C : Some Mathematical Aspects of Modelling in Bio-Medical Sciences - Ph.D. Thesis Submitted to Kakatiya University (99).