You will be presented with the full text of a science research paper. Please write an abstract for this paper. Your response should include the abstract and no additional text.

Paper text:
in the third decade of the last century lotka @xcite and volterra @xcite formulated a coupled set of equations to describe an auto catalytic model and the statistics of fish catches in adriatic . since then lotka volterra model has become a central area of interest for periodic oscillations in nonlinear systems with competitive elements @xcite .
it has found its applications in population biology @xcite , ecology @xcite , mathematical biology @xcite .
nevertheless it has often been criticized for being biologically unrealistic and mathematically unstable .
plenty of modifications were made to make it more realistic . yet in last two decades area of application of lotka - volterra model
it has made its way through newly explored applications , from membrane dynamics of competing neurons @xcite , neural networks @xcite , metabolic algorithm @xcite to network - electronics @xcite and stochastic dynamics @xcite .
with this re - emergence of lotka - volterra dynamics , we study some unexplored aspect of the original model . + lotka - volterra dynamics describes a predator - prey system where the prey thrives on the naturally available resources while predator thrives only on its interaction with the prey and would be extinct in a non interacting system .
it is easy to realize that in such a system the prey population would thrive of the predators decrease while an increase in the predator number would cause the prey population to diminish .
the predator can not flourish for ever because a decreasing number of prey would lead to fewer interaction and would drive the predator towards extinction .
this produces the limit cycle like oscillations .
the lotka volterra limit cycle is however very different from the usual limit cycles in that it depends on the initial conditions .
this makes lotka volterra dynamics interesting .
the unbounded growth of the prey population in the absence of the predator depends on a constant source of nutrients . in any practical situation
, there will be some periodic fluctuations in the nutrients because of natural causes .
if it is a wolf - rabbit situation , then the rabbit s supply of grass will have a seasonal variation .
interestingly enough , the periodic variation in the growth rate of the prey has not been studied in literature
. there has been study of random variation @xcite but not periodic ones . in this work ,
we study the effect of periodic variations . in carrying out this analysis
, we noticed that the standard techniques that are used in dealing with nonlinear oscillators ( poincare lindstedt method @xcite , equivalent linearization @xcite have not been carried out for the lotka volterra model . accordingly in section 2 .
we apply the lindstedt poincare technique on the lotka volterra model and come up with an initial condition dependent oscillation period .
our analytic and numerical results agree . in section 3 ,
we use the analytic tools of section 2 to study the dynamics of the modulated system and also carry out a numerical analysis to test our predictions . since theunmodulated system shows an initial condition dependent limit cycle , the final state of the driven system is found to be initial condition dependent . in section 4 , we apply another specified technique which work for high frequency perturbation and show how the calculation numerics compare favourably .
a brief conclusion is presented in section 5 .
in this section , we focus on the limit cycle of the lotka- volterra model . as is well known , this limit cycle is unusual in the sense that it depends on the initial conditions .
we show how the poincare - linstedt technique can be applied to the model to obtain an initial condition dependent frequency for the limit cycle .
this calculation is valid for small amplitudes and our numerical calculation supports the validity of this result .
+ we take the simplest form of two species lotka - volterra system ; @xmath0 according to a linear stability analysis , the fixed point at ( 1,1 ) is a center .
we first transform eq([eq1 ] ) to the variables @xmath1 and @xmath2 to write , @xmath3 we note immediately that if we drop the nonlinear terms , then we have a simple harmonic oscillator of frequency unity .
we are now in a position to carry out a poincare - linstedt analysis .
we imagine the existence of a parameter @xmath4 multiplying the non - linear terms in eq([eq3 ] ) and introducing the frequency @xmath5 of the full dynamics , rewrite eq([eq2])in the form ; @xmath6 for @xmath71 we expand @xmath8 the right hand side of eq([eq4 ] ) has term of order @xmath4 and higher . at order unity ,
the solution is @xmath9 , @xmath10 , where a is the amplitude of motion . at @xmath11 ) , we have @xmath12 the solvability condition for an inhomogeneous second order differential equation now leads to @xmath13 .
the solution for @xmath14 and @xmath15 is found to be @xmath16 at@xmath17 @xmath18 the solvability condition requires that the @xmath19 part of the right hand side be orthogonal to the left eigenvector of l. this leads to @xmath20 the perturbative result for @xmath5 up to @xmath17 , after setting @xmath21 is @xmath22 where a is the amplitude of the limit cycle . if @xmath23 and @xmath24 be the initial values of x and y , then we can write @xmath25 we have checked this result numerically .
the result are shown in fig[[fig1 ] ] .
the good agreement between the computed frequencies and the obtained from eq([eq10 ] ) is apparent .    ) of its oscillation . ]
with the help of the amplitude dependent frequency of the previous section , we explored lotka - volterra population dynamics under external drive .
these systems have been extensively studied under effect of random perturbation @xcite whereas under periodic forcing they were studied much less @xcite .
we explored it in presence of periodic perturbation .
periodic perturbation leads to diurnal or annual influences on the predator - prey systems and leads to periodic intrinsic growth rate in the prey population . + under periodic forcing , the system [ eq([eq1 ] ) ] becomes @xmath26 shifting to the ( @xmath27 ) variables , we have @xmath28 we note that if @xmath29 then the solutions can be written as periodic trajectories around ( 1,1 ) with a frequency that is dependent on the initial conditions .
for @xmath29 we have equivalent oscillator @xmath30 with @xmath5 given by eq([eq10 ] ) . in this technique of equivalent linearisation eq([eq33 ] ) can be rewritten as ; @xmath31 eliminating @xmath32 we have @xmath33 defining @xmath34 then , @xmath35 when the driving amplitude is small,(@xmath36 ) @xmath37 using the transformation @xmath38 , we arrive at @xmath39 correct to @xmath40(@xmath41 ) .
we recognize eq([eq38 ] ) as a mathieu equation and note that there will be a periodic response at a frequency of @xmath42 provided we fulfill the condition @xmath43 this is in effect a construct on the initial conditions .
the result found in eq([eq39 ] ) implies that for every modulating frequency @xmath44 , there will be some initial conditions for which a periodic response will be possible .
for initial conditions in the range @xmath45 , the response will be unbounded while for initial conditions outside this range the response will be bounded and in general quasi - periodic - the periodic motion resulting when eq([eq39 ] ) is satisfied .
+ in fig[[fig2 ] ] , the unboundedness of the solution is shown for @xmath46 .
the width of the unbounded region is exactly @xmath47 .
the original set of equation have nonlinear terms that prevents the dynamics becoming unbounded .
nevertheless we get the signature of unboundedness through the steep increase of the width ( @xmath48 ) fig[[fig3 ] ] of the phase space trajectory in the same frequency range .    )
shows the signature of unboundedness of the linearized solution.fig([fig3 ] ) change of width ( @xmath48 ) ( a , b ) reflects the resonance in linearized rezime . as we move from linearized unbounded solutions @xmath48 decreases ( c , d ) .
solid lines ( a , c ) represent the unperturbed trajctories . ]    [ fig2 ]    ) ] campares well with the signature of resonance in original dynamics ( b ) . in ( b ) @xmath48 is plotted with initial conditions of population density ( @xmath49 ) . ]
@xmath48 is potentially very different from the width @xmath50 we discussed later .
@xmath48 is a effect of the dynamics of two competing frequencies having resonating effect at comparable values , whereas @xmath50 shows the presence of two different order of timescale .
in this section , we consider the variation in the parameter of the lotka volterra model to be very rapid , i.e. the period @xmath44 of the forcing in eq([eq41 ] ) , is much greater than the frequency of the unforced system .
the system fluctuates rapidly from the unperturbed trajectory and the phase space trajectory is broadened fig[[fig4 ] ] .
this broadening ( @xmath50 ) depends mainly on the frequency of forcing and feebly on forcing amplitude . as @xmath44 increases this broadening decreases very rapidly .    ) of high frequency perturbation changes as we vary frequency of perturbation ( @xmath44 ) from 10 ( a ) to 30 ( b ) .
phase space plots ( b , d ) also show these through its change of width .
the solid lines represent the unperturbed trajectory . ]
the analysis of the system follows a procedure explained by landau and lifshitz .
we split the variables @xmath51 and @xmath32 into two parts , @xmath52 where @xmath53 , @xmath54 carry the rapid variations ( scale of @xmath44 ) and the averaged quantities @xmath55 and @xmath56 carry the slow variation ( order unity ) of the original model . substituting in eq([eq41 ] ) ; @xmath57 in this section , we do not restrict @xmath41 to be small . instead , we note that since @xmath58 , typically @xmath59 .
we choose the dynamics of @xmath60 to be @xmath61 in the above @xmath55 and @xmath56 may be treated as constants on the scale of variation over a period @xmath62 .
in that situation , eqs([eq44 ] ) and ( [ eq45 ] ) reduce to @xmath63 since @xmath64 , we write the approximate solution of eq([eq46 ] ) as @xmath65 we now average eqs([eq42 ] ) and ( [ eq43 ] ) over the fast variations , drop terms that are @xmath66 and obtain @xmath67 which is the unforced system in the coarse - grained variables .
the picture , which emerges is as follows : under a high frequency modulation , the motion can be explained as a rapidly oscillatory motion which is virtually identical to the trajectory of the unforced system .
the amplitude of the fast variations decays according to @xmath68 as shown in eq([eq47 ] ) .
numerical simulation of the system bears out these expectations.fig[[fig5 ] ]     is plotted against the frequency of modulation ( @xmath44 ) .
the fitting function ( f(x ) ) supports the form of analytical result . ]
in this paper we have shown that under lotka - volterra dynamics populations of predator and prey depend on their initial population densities .
this dependence of initial population brings forth changes in dynamics but helps us formulate an effective mathew - hill type equation for this 2-species system under periodic seasonal changes .
this also indicates slow seasonal changes have more effects in the variation of population than its faster counterpart .
99 a.j lotka , j. am .
chem soc . * 42 * , 1595 ( 1920 ) .
v. volterra , atti r. accad , naz .
lincei , mem .
fis , . mat .
* 2 * , 31 ( 1926 ) .
h. haken , _ synergetics _ , 3rd ed .
( springer - verlag , new york , 1983 ) .
d. neal , _ introduction to population biology _
( cambridge university press , cambridge , u.k .
r. m. may , _ stability and complexity in model ecosystems _
( princeton university press , princeton , 1973 ) . j. maynard smith , _ models in ecology _ ( cambridge university press , cambridge , u.k . 1974 ) .
j. d. murray , _ mathematical biology _ , 3rd ed .
( springer - verlag , new york , 2002 ) , vols .
t. fukai and s. tanaka , neural comput .
* 9 * , 77 ( 1997 ) .
zhang yi , k . k. tan , phys .
e , * 66 * , 011910 ( 2002 ) .
luca bianco , federico fontana , vincenzo manca j.j .
hopeld , proc .
usa * 81 * , 3088 ( 1984 ) .
k. gopalsamy and x.z .
he , physica d * 76 * , 344 ( 1994 ) .
m. mobilia , t. goeorgiev , u. c. tuber , phys .
e , * 73 * , 040903(r ) ( 2006 ) a. r. balsara , k. lindenberg and k. e. shuler , j. stat . phys,*27 * 787 ( 1982 ) a. b. budgor , b. j. west phys .
a , * 17 * 370 ( 1978 ) mikhail f. dimentberg , phys . rev .
e , * 65 * 036204 ( 2001 ) sabin g.c.w , summer d , math biosci .
* 113 * 91 ( 1993 )
lotka volterra model and its modified forms have long become a major area of interest for periodic motions in nonlinear systems with competitive species . the model given by volterra shows that its periodicity is dependent on initial condition . 
 this characteristics allows us to calculate the effect of periodic seasonal changes on population densities of different species .