observations of the cosmic microwave background ( cmb ) have proved to be extremely valuable for testing and constraining cosmological models .the majority of models predict that the anisotropies in the cmb signal are gaussian and their statistics isotropic across the sky .the angular power spectrum therefore provides a natural connection between theory and observation and a variety of methods have been explored to compute the power spectrum from sets of observations .maximum - likelihood methods provide an optimal estimate of the cmb power spectrum which has made them an invaluable tool for analysing the cmb for single - dish experiments and interferometers .brute force implementations of the method can only be applied to small data sets as the required computation scales as , where is the number of pixels in a cmb map ( see for a review ) . fora number of special cases one can construct maximum - likelihood estimators that perform more favourably , although their lack of generality limits their applicability and , as their computational demands scale as , even they can not be applied directly the largest contemporary ( _ wmap _ ) or future ( _ planck _ ) data .alternatively one can resort to approximate pseudo- methods , .these scale as the map - making process and are fast even for the largest data sets .hybrid methods combine a maximum - likelihood approach on large angular scales with a fast pseudo- estimator on small scales .  to compare theoretically predicted power spectra and those estimated from a set of observations it is necessary to construct a likelihood function .maximum - likelihood and pseudo- methods can only provide approximations to this likelihood .an alternative framework has been developed where one explores the full posterior distribution of the power spectrum with monte carlo samples .this method is not only exact but scales like the pseudo- methods . under the assumption of positioninvariant , circularly symmetric beams and uncorrelated noise , one can perform the beam convolution in the spherical harmonic domain and evaluate the likelihood of the data in the map domain , and the method scales as .the favourable scaling has enabled the method to be applied to the _ wmap _ data .the approach relies on the availability of an efficient method for sampling from high - dimensional distributions .previous implementations use a gibbs sampler but this restricts the applicability of the method to gaussian noise and cmb .we propose the use of a hamiltonian monte carlo ( hmc ) sampler .as opposed to the majority of markov - chain monte carlo ( mcmc ) methods , hmc scales well with problem size .few requirements are made on the distribution to be sampled , thus giving us the opportunity for great flexibility .hmc has been widely applied in bayesian computation and has also been employed for cosmological parameter estimation .  in this workwe begin , in section [ sec : psest ] , by outlining the procedure for estimating power spectra with sampling . in section [ sec : hmc ] we describe the hmc method and a technique for determining the convergence of samples drawn with a hmc sampler .a summary of the process of applying hmc to the power spectrum estimation problem can be found in section [ sec : hmcpsest ] and we provide a prescription for setting the many tuneable parameters of the sampler . in section [ sec : lowres ] we apply the method to low - resolution simulations and compare the hamiltonian and gibbs samplers .section [ sec : wmap ] details our application of the method to simulated _ wmap _ observations .our conclusions are presented in section [ sec : conc ] .suppose the true cmb sky , divided for convenience into pixels , is represented by the temperature vector .the sky is observed and the resultant data vector , in any domain , is the sum of contributions due to the underlying cmb signal in that domain and the corresponding noise .moreover the signal is usually linearly related to the true cmb sky .thus we have where the matrix represents the linear mapping from the true cmb sky to the corresponding cmb signal in whatever domain the data resides .  in the following discussion, we need not assume a particular domain for the generic data vector .nevertheless , it is most common for to represent the pixelised cmb map convolved with the instrument beam and our work , so far , has used solely this form for the data vector .  the temperature field is related to the spherical harmonic coefficients of the field by where is a single pixel in the map vector and the are the spherical harmonics .although formally one may take the upper limit of the summation to be infinite , it is more typical to choose a finite value for appropriate to the beam size .we have not considered the effect of the mono- and dipole contributions , the handling of which , within this framework , is discussed in . in this notation we may write our model for the data in the form where describes the application of the spherical harmonic transform and we represent the spherical harmonic coefficients by a real vector .  for an isotropic gaussian cmb sky the covariance matrix ofthe has components where the set of coefficients constitute the theoretical angular power spectrum .note that , since the sky is real ,   we aim to sample from the joint distribution of the power spectrum coefficients .although this is difficult to perform directly , it is possible to sample from the joint density of the power spectrum coefficients and the signal realization and then marginalise over .the joint density can be written as the product of the appropriate conditional distributions the choice of prior is an interesting topic . some suggestions for making this choice but for the purpose of this work we set so that the maximum of our posterior will correspond directly to a maximum - likelihood estimate .given our choice of prior and assuming the noise is gaussian then the conditional distributions that make up ( [ eq : joint ] ) can be written in the form ,\ ] ] where , and where is easily constructed using ( [ eq : pstoalms ] ) .it is convenient to rewrite this in the form where is the power spectrum of the signal realization .the selection of a domain in which to represent the data is determined by the requirement that has a simple form . in this workwe make the assumption that in the map domain is well represented by a diagonal matrix . in this domain incomplete sky coverageis straightforwardly handled by setting the elements of that correspond to excluded pixels to zero .if the instrument beam is position invariant and circularly symmetric then we can compute the beam convolution quickly in harmonic space and the predicted noiseless data can be written in the form where represents the smoothing by the beam .the computational cost of evaluating the posterior ( and its gradients ) is now limited by the speed at which one can compute the spherical harmonic transform .the transforms scale as and can be efficiently parallelised .we draw samples from the joint space using a hamiltonian monte carlo sampler described in section [ sec : hmc ] .let us suppose that we wish to draw samples from a target density , where is the -dimensional vector of our parameters .conventional mcmc methods move through the parameter space by a random walk and therefore require a prohibitive number of samples to explore - high dimensional spaces . the hamiltonian monte carlo method draws parallels between sampling and classical dynamics . by exploiting techniques developed for describing the motion of particles in potentials it is possible to suppress random walk behaviour .introducing persistent motion of the chain through the parameter space allows hmc to maintain a reasonable efficiency even for high dimensional problems .  for each parameter , we introduce a ` momentum ' and a ` mass ' ; we discuss how to set the mass in the appendix .we construct a hamiltonian formed from a potential energy term and a kinetic energy term such that where our potential is related to the target density by our new objective is to draw samples from a distribution that is proportional to .the form of the hamiltonian is such that this distribution is separable into a gaussian in and the target distribution , i.e. we can then obtain samples from by marginalising over .  to find a new samplewe first draw a set of momenta from the distribution defined by our kinetic energy term , i.e. an dimensional uncorrelated gaussian with a variance in dimension of .we then allow our system to evolve deterministically , from our starting point in the phase space for some fixed time according to hamilton s equations , at the end of this trajectory we have reached the point and we accept this point with probability where after a new proposed sample is generated the momentum variable is discarded and the process restarts by randomly drawing a new set of momenta as described above .this implies that if we are able to integrate hamilton s equations exactly then , as energy is conserved along such a trajectory , the probability of acceptance is unity .in fact the method is more general as , provided one uses the metropolis acceptance criterion ( [ eq : acceptance ] ) , it is permitted to follow any trajectory to generate a new candidate point .however only trajectories that approximately conserve the value of the hamiltonian ( [ eq : ham ] ) will result in high acceptance rates .for some problems it may be advantageous to generate trajectories using an approximate hamiltonian that can be computed rapidly , and bear the cost of lowering the acceptance probability .  to integrate the equations of motionsit is common practice to use the leapfrog method .this method has the property of exact reversibility which is required to ensure the chain satisfies detailed balance .it is also numerically robust and allows for the simple propagation of errors .we make steps with a finite step size , such that , as follows , until . the interval must be varied , usually by drawing and randomly from uniform distributions , to avoid resonant trajectories .higher - order integration schemes are permitted , provided exact reversibility is maintained , although generally incur significant additional computational costs .   diagnosingthe convergence of a chain in an mcmc process is the subject of much literature ( see for comprehensive reviews ) . provides a method that uses the gradient information , which we must possess to calculate trajectories in hmc , to compute a convergence criteria .one constructs two estimates of the variance of a chain , that depend quite differently upon the distribution of samples across the target density , although the basic method is easily generalised to ( combinations of ) higher order central moments of .when the two estimates agree to within a certain accuracy the chain is assumed to have converged .we compute the variance of each parameter independently .our first estimate of the variance of the samples is calculated by where labels a sample in a chain of samples and the integral extends over the entire -space . for our second estimatewe take the expression for the variance and integrate by parts the first term of which will vanish if the marginalized distribution drops off faster than as tends to . using ( [ eq : psi ] ) we rewrite this expression as we compute ( [ eq : varint2 ] ) from the samples in our chain by to test for convergence we compute the ratio of ( [ eq : var1 ] ) and ( [ eq : var2 ] ) , and we believe the chain has converged when all the are close to unity .  we have tested how this criterion compares to the widely used gelman - rubin statistic and have found that hanson s method tends to be , if anything , slightly pessimistic .we find that values of in the range to represent good convergence and values in the range to are acceptable .the gelman - rubin method requires multiple chains to be generated and compares inter - chain with intra - chain statistics , whereas hanson s test uses a single chain and compares two different intra - chain statistics .we use the hanson test as it is very easy to compute , scales well with problem size and requires that we only generate one chain . we plan to explore other intra - chain convergence diagnostics such as that proposed by .we use hmc to draw samples simultaneously from the joint density ( [ eq : joint ] ) .our potential is defined by such that and the gradient of the potential can be computed exactly by   the positivity requirement on the power spectrum can result in a high rejection rate and we have found it advantageous to reparametrize the problem in terms of the logarithm of the . for this reparametrization it is easy to calculate the corresponding potential and its derivatives . to enforce a flat prior on each must apply an exponential prior on .  to generate a new samplerequires us to evaluate the gradient at each point along the leapfrog trajectory and to evaluate the value of the potential once at the end of the trajectory .therefore , if we take leapfrog steps , we must perform spherical harmonic transforms , although we can reuse the gradient at the end of one trajectory for the first step of the next .we split the sampling process into a burn in phase , in which we attempt to lose any dependence on our starting point , and a sampling phase where we store the samples from the chain and we believe these samples are drawn from the target density . during burn in we are permitted to adjust the parameters of the sampler , for example to tune the acceptance rate .once burn in is complete we must fix the parameters of the sampler in order that our samples come from the desired distribution .a good starting point can significantly reduce the time required for burn in .we have explored a number of possibilities for computing a starting point for the signal given some initial guess for the power spectrum .one we have found particularly effective is to draw a single signal sample , as for one step of the gibbs sampler , from the conditional distribution .this is a computationally expensive process and is described fully in .the basic procedure involves solving the following equation for , the spherical harmonic coefficients of the mean field ( wiener filtered ) map , and a fluctuation term that corrects for the bias in  where is a set of spherical harmonic coefficients and a map both containing gaussian white noise of zero mean and unit variance .the sum of and is our starting sample .we solve for using a conjugate gradient algorithm ( see , for example ) .a preconditioner can be used to reduce the number of iterations required for the convergence of the conjugate gradient solver , however the construction of a preconditioner is itself a complex procedure , and since we only perform this step once and the accuracy of the result is of little consequence , we have not made use of one in this work . whether or not applying the conjugate gradient algorithm without a preconditioner is feasible depends on the nature of the data set under consideration .hmc has a large number of adjustable parameters , notably the masses .the distribution for the parameters is gaussian and so we attempt to set the mass associated with each such that they are inversely proportional to the variance of that .we justify this choice in the appendix .the masses for the are estimated for a fixed power spectrum for which the variance is computed by where we use our initial estimate of the power spectrum as the value of and compute the diagonal elements of the inverse noise covariance matrix in harmonic space using monte carlo simulations .  at high and with good signal - to - noisethe marginal distributions for each are close to gaussian and we can obtain masses from the standard expression for the variance ( see , for example ) where is the power spectrum of the noise in the data, is the beam transfer function and is the fraction of the sky observed . for low multipoles the distributions are significantly skewed and in low signal - to - noise the sharp cut off of the distribution at has a similar effect . in these cases we have found that setting the masses from the variances is insufficient. instead we tune these masses empirically .we aim to set the mass for each parameter to as small a value as possible while maintaining our target acceptance rate .we sample the from simple approximate likelihood function and gradually reduce the masses until the acceptance rate drops .this gives masses that are sufficient for sampling the full problem efficiently .during burn in we can further tune the masses ; the convergence criterion for each parameter providing a good indication of whether or not the mass associated with that parameter is set correctly .we must randomise the length of each trajectory and have found that drawing from a uniform distribution between and is appropriate .therefore we typically require the application of spherical harmonic transforms to generate a new proposed sample .we then tune the step size such that we obtain an acceptance rate between 70 and 90 per cent .a higher acceptance rate is used for hmc than other mcmc methods as the computational cost of a rejection is so high .once sampling we store each sample and the realization power spectrum of each signal sample .the can be used to form the blackwell - rao estimator of the posterior distribution .the posterior can be written which for a gaussian cmb can be written as where we can therefore compute the posterior probability of a set of from samples by it is also possible to construct the marginal distributions for any or subset of . for a single the marginal distribution can be approximated by where extremely large numbers of samples would be needed to make this estimator accurate at high .however even with a relatively small number of samples it forms a useful tool for the analysis of large angular scales .it is worth noting that the expression ( [ eq : brfromsamples ] ) , or its one - dimensional marginalized version ( [ eq : brmarginal ] ) , do not depend on the -samples , but only on the realization power spectra of the -samples .to compare the hamiltonian and gibbs samplers we applied them both to a set of low resolution simulations .we produced a map of the cmb with a healpix ( pixels ) .our cmb simulation is a realization of a cosmology with the best fitting parameters from the 5-year _ wmap _ observations and includes multipoles up to .we smoothed the map with a -degree gaussian beam and added isotropic noise with an rms amplitude of per pixel .we chose the noise level so that we could explore how the sampler behaved as a function of the signal - to - noise ratio .we degraded the _kp2 mask such that any ( large ) pixel in our final mask is excluded if any of the ( small ) subpixels in the original mask are excluded .this has the effect of enlarging the kp2 mask to remove around percent of the sky : a large contiguous area along the galactic plane and a number of small regions around the locations of bright point sources .  for each samplerwe take 20000 burn in samples and then record the next 50000 samples .a large number of samples helps to estimate correlation lengths accurately ; far fewer samples are required to explore the distribution . the marginal distributions of a selection of the are shown in fig .[ fig : lowresmarginal ] . for most data is too noisy to constrain the value of the however we do see good agreement between the results from the gibbs and hamiltonian samplers .the hmc samples have also been used in conjunction with the blackwell - rao estimator to generate a smooth approximation to the marginal distributions .this estimator appears to agree well with the histograms across this range of , but more samples are likely to be needed if we were to calculate the joint distribution of the .    in order to characterise the performance and efficiency of the samplers we considered the correlation of the samples . assuming that the are independent we can examine the auto - correlation function , we show the auto - correlation function for a selection of multipoles in fig .[ fig : lowrescorcoeff ] . as the signal - to - noise ratio for a single , defined as the ratio of the signal and noise power spectra at that , decreases with increasing samples become more highly correlated ; it takes more steps of the samplers to generate independent samples .this feature is a well known limitation of the gibbs sampler caused by the fact that drawing the power spectrum from the conditional distribution is limited to the size of the cosmic variance while the joint distribution may be much wider .similar behaviour is observed with the hamiltonian sampler although the cause is now related to the difficulty in sampling the highly skewed distributions that occur when the signal - to - noise ratio is low .the correlation length for each parameter can be estimated using where we truncate the summation at some maximum lag at which the auto - correlation function becomes noisy .[ fig : lowrescorlen ] shows how the measured correlation lengths for the power spectrum parameters from the gibbs and hamiltonian samplers depend on the signal - to - noise ratio for each parameter , again estimated assuming the parameters are independent .we see that in the high signal - to - noise regime the gibbs sampler performs exceptionally well whereas the hamiltonian sampler produces samples with typical correlation lengths of around four steps .once the data becomes noise dominated the picture is less clear with the hamiltonian sampler generally performing marginally better than the gibbs sampler . as the signal - to - noise ratio drops below about 0.01both samplers perform poorly .it is worth noting that hamiltonian sampler requires around an order of magnitude fewer spherical harmonic transforms ( the computationally intensive step in the process ) per sample than a gibbs sampler that uses no preconditioner and around a factor of 3 - 4 fewer transforms than is reported for gibbs samplers with carefully tuned preconditioners .furthermore we have found that the correlation lengths of the hamiltonian sampler strongly depend on the masses one uses , offering the opportunity for significant improvements given a more sophisticated prescription for setting the masses .  ) as a function of the signal - to - noise ratio of each parameter .the red points show the results from the gibbs sampler the blue points those from the hamiltonian sampler ., scaledwidth=50.0% ]we produce a cmb simulation as for section [ sec : lowres ] but with ( pixels ) and including multipoles up to .the map was then smoothed with a -arcmin gaussian beam , which is similar in size to the beam of the _ wmap _ _ w_-band .we then added anisotropic uncorrelated noise by making use of the published and noise variance for the 5-year _ wmap _ combined _ w _ band map .the map was cut with the kp2 mask which excludes of the sky .we included multipoles up to in our analysis .this gives us a total of around parameters in our sampling space .  to generate a good signal starting point , using a single gibbs sample , required iterations ( minutes on the hardware described below ) of the conjugate gradient to solve ( [ eq : meanfield ] ) and ( [ eq : fluctuation ] ) such that the rms residual was less than .  for these simulations we made a total of burn in samples and recorded samples from the post burn - in phase .it takes seconds to generate a single sample using two dual core intel xeon 5150 processors and the mpi parallelised healpix spherical harmonic transforms , resulting in a total processing time of around hours .  for comparison we applied the master method to the same data set .our peak likelihood sample and 68 per cent confidence intervals , binned with the _ wmap _ team s scheme , are shown alongside the results of the master method in fig .[ fig : binnedps ] . for most of the range of angular scalesthe two estimates and their errors agree well . on the largest angular scales the master estimate tends to underestimate the uncertainities and the symmetric errors are far from representative of the posterior .percent confidence intervals as compared to the results of an application of the master method to the same simulated _ wmap _ data .the black solid line shows the power spectrum from which the simulation was generated while the grey shows the power spectrum of the realization .the grey squares and error bars show the master results .the black circles and error bars show the peak and 68 per cent confidence intervals found from samples generated with the hmc sampler.,scaledwidth=50.0% ]  in fig . [fig : binnedr ] we show a summary of the convergence statistics , using hanson s diagnostic , see section [ sec : converge ] , demonstrating that we have fully explored the distribution across the entire range in . for all multipolesthe value is within the range to .samples used to produce the power spectrum in fig .[ fig : binnedps ] .although convergence is judged from the for every parameter we show here only the average for in each bin for the ( blue line ) and ( red line ) ., scaledwidth=50.0% ]we have introduced the hmc sampler for cmb power spectrum estimation and demonstrated its performance both on low - resolution simulations and simulations of 5-year _ wmap _ data .we find that the hamiltonian sampler has similar or shorter correlation lengths when compared to the gibbs sampler except in the regions of the highest signal - to - noise .bearing in mind the reduced computational cost and greater flexibility of the hamiltonian sampler we believe it is an attractive method for performing the analysis .  for high - resolution data sets of size ( , ) we can generate a sample in seconds on a high - end desktop .this is a significant gain over the reported performance of gibbs samplers .hmc requires that we are able to compute the logarithm of the target density and its gradients .even if exact gradients are not available we can generate approximate trajectories and these will still result in samples drawn from the required distribution .the generality of the approach removes the requirement for strictly gaussian signal and noise and therefore promises to be an interesting method for tackling a wide range of related problems .we are currently testing the performance of the method on high - resolution _ planck _ simulations and working on extending the method to include polarization .we also intend to apply the technique to the _ wmap _ data .we thank morgan french for his contributions to the early development of our sampler .jft acknowledges a stfc ( formerly pparc ) studentship .maja is a member of the cambridge planck analysis centre , supported by stfc grant st / f005245/1 .this work was conducted in cooperation with sgi / intel utilising the altix 3700 supercomputer at damtp cambridge supported by hefce and stfc .we acknowledge the use of the legacy archive for microwave background data analysis ( lambda ) . support for lambda is provided by the nasa office of space science .some of the results in this paper have been derived using the healpix package .hamiltonian monte carlo can be extremely sensitive to the choice of masses .when sampling from an approximately isotropic distribution this does not affect the performance significantly but when the marginal distributions of different parameters show considerable variation in width the masses must be set correctly to sample efficiently . suggests that one should set the mass associated with each parameter to be approximately equal to the variance of that parameter in the target density .this is an attempt to circularise the trajectories in the space .we take an alternative approach , where the mass for a parameter is inversely proportional to the width of the distribution , as suggested in . in order to justify this approach we have generalised the framework in to describe the application of the leapfrog method .consider the problem of sampling from an -dimensional gaussian distribution in with covariance matrix .our hamiltonian is quadratic in and  where is a mass matrix , and the trajectory will be determined by hamilton s equations  we integrate the equation of motion with the leapfrog method  a single application of the leapfrog method can be written in the form  where is the identity matrix .we can rewrite this in a matrix form   = { \mathbfss{t } }  \left[\begin{array}{c }    \bmath{x}\left(t \right)\\    \bmath{p}\left(t \right )   \end{array }  \right],\ ] ] where .\ ] ]  if the method is to be stable under the repeated application of then we require its eigenvalues to have unit modulus .the eigenvalues are found from the characteristic equation = 0.\ ] ]  to explore the space rapidly we wish to find the largest compatible with the condition for stability .any dependence of ( [ eq : characteristiceqn ] ) on implies no single value for will meet the requirement for every eigenvalue to have unit modulus ( unless both and are proportional to the identity matrix ) . the maximum value for should therefore be controlled by the width of the distribution for a small subset of parameters .  by setting remove the dependence of on the size of the distribution . in this situationthe characteristic equation reduces to ^n  = 0\ ] ] and the stability criterion is met by .  if the dimensionally of the problem is such that it is impractical to perform the required matrix inversion and decomposition of ( to compute the hamiltonian and to draw new values for the momentum variables respectively ) then simple approximations must be employed .typically one might construct a diagonal mass matrix with the mass associated with each parameter inversely proportional to the variance of that parameter .

Abstract: we present a method for fast optimal estimation of the temperature angular power spectrum from observations of the cosmic microwave background . we employ a hamiltonian monte carlo ( hmc ) sampler to obtain samples from the posterior probability distribution of all the power spectrum coefficients given a set of observations . we compare the properties of the hmc and the related gibbs sampling approach on low - resolution simulations and find that the hmc method performs favourably even in the regime of relatively low signal - to - noise . we also demonstrate the method on high - resolution data by applying it to simulated _ wmap _ data . analysis of a _ wmap_-sized data set is possible in a around eighty hours on a high - end desktop computer . hmc imposes few conditions on the distribution to be sampled and provides us with an extremely flexible approach upon which to build .   [ firstpage ]  cosmic microwave background  methods : data analysis methods : statistical


in the third decade of the last century lotka and volterra 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 .it has found its applications in population biology , ecology , mathematical biology .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 modelit has made its way through newly explored applications , from membrane dynamics of competing neurons , neural networks , metabolic algorithm to network - electronics and stochastic dynamics .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 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 , equivalent linearization 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 ; according to a linear stability analysis , the fixed point at ( 1,1 ) is a center .we first transform eq([eq1 ] ) to the variables and to write , 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 multiplying the non - linear terms in eq([eq3 ] ) and introducing the frequency of the full dynamics , rewrite eq([eq2])in the form ; for we expand the right hand side of eq([eq4 ] ) has term of order and higher . at order unity ,the solution is , , where a is the amplitude of motion . at ) , we have the solvability condition for an inhomogeneous second order differential equation now leads to .the solution for and is found to be at the solvability condition requires that the part of the right hand side be orthogonal to the left eigenvector of l. this leads to the perturbative result for up to , after setting is where a is the amplitude of the limit cycle . if and be the initial values of x and y , then we can write 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 whereas under periodic forcing they were studied much less .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 shifting to the ( ) variables , we have we note that if then the solutions can be written as periodic trajectories around ( 1,1 ) with a frequency that is dependent on the initial conditions .for we have equivalent oscillator with given by eq([eq10 ] ) . in this technique of equivalent linearisation eq([eq33 ] ) can be rewritten as ; eliminating we have defining then , when the driving amplitude is small,( ) using the transformation , we arrive at correct to ( ) .we recognize eq([eq38 ] ) as a mathieu equation and note that there will be a periodic response at a frequency of provided we fulfill the condition this is in effect a construct on the initial conditions .the result found in eq([eq39 ] ) implies that for every modulating frequency , there will be some initial conditions for which a periodic response will be possible .for initial conditions in the range , 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 .the width of the unbounded region is exactly .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 ( ) 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 ( ) ( a , b ) reflects the resonance in linearized rezime . as we move from linearized unbounded solutions 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 ) is plotted with initial conditions of population density ( ) . ] is potentially very different from the width we discussed later . is a effect of the dynamics of two competing frequencies having resonating effect at comparable values , whereas 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 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 ( ) depends mainly on the frequency of forcing and feebly on forcing amplitude . as increases this broadening decreases very rapidly .  ) of high frequency perturbation changes as we vary frequency of perturbation ( ) 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 and into two parts , where , carry the rapid variations ( scale of ) and the averaged quantities and carry the slow variation ( order unity ) of the original model . substituting in eq([eq41 ] ) ; in this section , we do not restrict to be small . instead , we note that since , typically .we choose the dynamics of to be in the above and may be treated as constants on the scale of variation over a period .in that situation , eqs([eq44 ] ) and ( [ eq45 ] ) reduce to since , we write the approximate solution of eq([eq46 ] ) as we now average eqs([eq42 ] ) and ( [ eq43 ] ) over the fast variations , drop terms that are and obtain 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 as shown in eq([eq47 ] ) .numerical simulation of the system bears out these expectations.fig[[fig5 ] ]   is plotted against the frequency of modulation ( ) .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 )

Abstract: 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 .