Lotka Volterra Model with Time Delay

Transcription

1 International Journal of Mathematics Research. ISSN Volume 6, Number (4), pp. 5- International Research Publication House Lotka Volterra Model with Time Dela Tapas Kumar Sinha and Sachinandan Chanda Computer Centre, North Eastern Hill Universit, Shillong 793 Department of Mathematics, Shillong Poltechnic, Shillong 793 Abstract We model the Lotka-Volterra equations with a time dela. For this model we derive the conditions for the existence of stead state conditions, asmptotic stabilit.. Introduction Time dela is inherent in the predator-pre sstems. Both the predator and pre require their own times to grow from birth to maturit and from maturit to death. If the time spans for predator and pre are greatl different it will lead to instabilit in the predator pre sstem. Using a time dela of a predator pre sstem we derive the conditions for temporal stabilit of a predator pre sstem following the techniques given in []. Predator pre models with time dela (or lag) have been studied b a number of authors [] [8]. Time dela can induces instabilit, oscillations and bifurcation and in some cases switching of stabilities. However most authors have used a single time dela to characterize the predator pre sstem. Here we have associated unique delas with both predator and pre.. Time Dela Equations for Predator-Pre The lotka Volterra equations with time dela are given b d t t tt () dt t d t tt dt () Here t, t are dela times for predator and pre respectivel. On substituting

2 6 Tapas Kumar Sinha & Sachinandan Chanda we obtain t t t e A and t e A (3) t t t A te (4) ( ) Ae t t (5) tt Here A, A correspond to the predator, pre population at time t =. t and For solutions of t to exist, we must have, from equations (4) and (5), that tt Ae tt Ae (6) From equation (6), we get t t t a b e (7) a Where b AA (8) Note that we have defined a as the time dela (which is defined in terms of a ) has to be positive. In absence of dela cos t t ba b then the equation (7) becomes t t P, t ab e (9) Suppose that i ( > ) is a purel imaginar root of equation (9).It suffices to seek solution.putting i in (9) and separating real parts and imaginar parts, we get b cos t t () a Sin t t () Eliminating t t from Equations () & (),we get

3 Lotka Volterra Model with Time Dela 7 4 U a b b () Let be the determinant of equation (3),we get a b 4b If a b 4b,then there is no positive solution of. If a b b positive solution of 4 or if and given b a b,then there ma be a ba (3) If b, then there is one positive solution of given b ba (taking onl positive sign) (4) Thus From equation (), we get t t From equation (), we get a sin (5) cos t t a (6) cos t Equation (6) determines the sign of cos t t t must lie. From equation (5), we get and the quadrant in which sin a n ; n,,, 3,... n (7) Where t t (8)

4 8 Tapas Kumar Sinha & Sachinandan Chanda Where is given b equation (4) and where second quadrant according to the sign of cos t t sin a is chosen in the first or 3. Stabilit Analsis To analze the change of behavior of the stabilit of the stead state E with respect to t,we examine the sign of d d,where i. d If,then the sstem is unstable for some values of. d d If,then is the sstem stable for some values of. d Differentiating equation () with respect to t,we get d e d ae Hence d 4 i cos isin d at i i a cos isin (9) Using (5) & (6) in (9) and simplifing, we get a aa a 4a 4 d Re d at i aa a () Since and d, Re at ever root. dt at i Therefore ever time a root crosses the imaginar axis with increasing t it crosses from left to right, stabilit of the zero solution is lost at,where a sin () and stabilit persists for all.

5 Lotka Volterra Model with Time Dela 9 4. Conclusion We have derived the stabilit criteria for a delaed predator pre sstem. Note that is the composite dela as given b (9). Criteria for stabilit is given in (). For the case A or A one finds that. In the opposite limit where A, A A A are ver large one obtains Note that in both limits if,. AA Here is the rate of growth of predator and is the rate of death of pre. In other words in this limit there is no possibilit of an predator pre oscillations. References [] N.C. Majee and A. B. Ro Temporal dnamics of a two-neuron continuous network model with time dela Appl. Math,. Modelling,(997), [] T. K. Kar and A. Ghorai, Dnamic behavior of a delaed predator-pre model with harvesting, Applied Mathematics and Computation, () [3] Chao-Pao Ho, Yuei-Liang Ou The Influence of Time Dela on Local Stabilit for a Predator-Pre Sstem, Tunghai Science Vol. 4: 47 6 Jul, [4] M.A. Haque, A Predator Pre Model with Discrete Time Dela Considering Different Growth Function of Pre, Advances in Applied Mathematical Biosciences, Number (), pp. -6 [5] Peter J. Wangersk and W. J. Cunningham, Time Lag in Pre-Predator Population Models, Ecolog, (957) [6] Roger arditi, Jean-marie abillon, Jorge vieira da silva, The Effect of a Time- Dela in a Predator-Pre Model Mathematical biosciences (977) 33, 7- [7] T. K. Kar, Selective harvesting in a predator-pre fisher with time dela, Mathematical and Computer Modelling, (3) [8] A. Martin and S. Ruan, Predator-pre models with dela and pre harvesting, Journal of Mathematical Biolog, ()

6 Tapas Kumar Sinha & Sachinandan Chanda