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the dynamics of interacting species has attracted a lot of attention since the pioneering works of lotka  @xcite and volterra  @xcite . in their independent studies , they showed that simple prey - predator models may exhibit limit cycles during which the populations of both species have periodic oscillations in time . however , this behavior depends strongly on the initial state , and it is not robust to the addition of more general non - linearities or to the presence of more than two interacting species  @xcite . in many cases the system reaches a simple steady - state . a better understanding of the properties of such oscillations is clearly desirable , as such population cycles are often observed in ecological systems and the underlying causes remain a long - standing open question  @xcite . one of the best documented example concerns the canadian lynx population . this population was monitored for more than hundred years ( starting in 1820 ) from different regions of canada . it was observed that the population oscillates with a period of approximately 10 years and that this synchronization was spatially extended over areas of several millions of square kilometers  @xcite . several attempts were made to explain these facts ( climatic effects , relations with the food - web , influence if the solar cycle ) without success . more recently , blasius et al . @xcite introduced a deterministic three level vertical food - chain model . the three coupled nonlinear differential equations defining the model contain eight free parameters and two unknown nonlinear functions . the authors showed that an ad - hoc choice of the free parameters and nonlinear functions explains the experimental data for the canadian lynx .    in such mean - field type models , it is assumed that the populations evolve homogeneously , which is obviously an oversimplification . an important question consists in understanding the role played by the local environment on the dynamics  @xcite . there are many examples in equilibrium and nonequilibrium statistical physics showing that , in low enough dimensions , the local aspects ( fluctuations ) play a crucial role and have some dramatic effects on the dynamics of the system . accordingly , a lot of activities have been devoted during the past years to the study of extended prey - predator models . the simplest spatial generalization are the so called two patches models , where the species follow the conventional prey predator rules within each patches , and can migrate from one patch to the other  @xcite . other works have found that the introduction of stochastic dynamics plays an important role  @xcite , as well as the use of discreet variables , which prevent the population to become vanishingly small . these ingredients are included in the so called individual based lattice models , for which each lattice site can be empty or occupied by one individual of a given species or two individuals belonging to different species . it was recognized that these models give a better description of the oscillatory behavior than the usual lotka - volterra ( l - v ) equations . indeed , the oscillations in these lattice models are stable against small perturbations of the prey and predator densities , and they do not depend on the initial state . it was also found ( in two dimensional systems ) that the amplitude of the oscillations of global quantities decreases with increasing system size , while the oscillations persist on local level . it was argued that coherent periodic oscillations are absent in large systems ( although , @xcite do not discard this possibility ) . in @xcite lipowski state that this is only possible above a spatial dimension of 3 . in @xcite provata emphasize that the frequency of the oscillations are stabilized by the lattice structure and that it depends on the lattice geometry . in some papers , the stationary phase diagram was also derived , and different phases were observed as a function of the model parameters , such as an empty phase , a pure prey phase , and an oscillatory region of coexisting preys and predators . in @xcite , a coexistence region without oscillations and a domain of the control parameter space for which the stationary states depend strongly upon the initial condition , were found . however , in all the above works no systematic finite size studies have been performed , allowing to draw firm conclusions on the phase diagram of the models as a function of their sizes . it is known  @xcite , that in ecological problems the fact that a system has a finite size is more relevant than in most of the cases encountered in statistical physics , for which one concentrates on the thermodynamic limit . particularly , the size dependence of the amplitude of the oscillations , as well as a detailed description of the critical behavior near the phase transitions have not been investigated . another relevant question is how much the stationary phase diagrams of these prey - predator models have some generic properties or how much they depend upon the details of the models . the goal of this paper is to study a simple models of prey - predators on a two - dimensional lattice for which some of the above questions could be answered . our model is based on a coarse - grained description in the sense that a given cell models a rather large part of a territory and thus can contain many preys or predators . moreover , predators can not leave without preys in a given cell . those are the main differences between our model and satulovsky and tom ( st ) model  @xcite . nevertheless , it turns out that the stationary state phase diagram of the two models are quite different . our model is defined in sec .  [ sec : model ] . although governed by only two control parameters , this model exhibits a rich phase diagram . two different phases are observed : a pure prey phase , and a coexistence phase of preys and predators in which an oscillatory and a non - oscillatory region can be distinguished . in some limiting cases the model can be mapped onto another well known nonequilibrium model : the _ contact process _ ( cp )  @xcite . in sec . [ sec : mean - field ] the properties of our model are analyzed in dynamical one and two - points mean - field approximations and no undamped oscillatory behavior is found . in sec . [ sec : mc ] , extensive monte - carlo simulations are performed . it is shown that , as a function of the values of the control parameters , two types of continuous nonequilibrium phase transitions towards a prey absorbing state are present . the system size dependence of the amplitude of the oscillations is studied and several scaling relations between the amplitude of the oscillations and the correlation length are obtained . in sec . [ sec : discuss ] an underlying mechanism responsible for the spatial oscillations is proposed , which leads to a qualitative explanation of the properties of the phase diagram . in particular , we show that the spatially extended aspect of the problem is crucial to have an oscillatory region . finally , conclusions are drawn in sec . [ sec : conc ] . our system models preys and predators living together in a two dimensional territory . this territory is divided into square cells , and each of them can contain several preys and predators . in this coarse - grained description in which each cell represents a rather large territory , one can assume that each cell containing some predator will also contain some preys . hence , a three state representation is made . each cell of the two - dimensional square lattice ( of size @xmath0 , with periodic boundary condition ) , labeled by the index @xmath1 , can be at time @xmath2 , in one of the three following states : @xmath3 . a cell in state 0 , 1 or 2 corresponds respectively to a cell which is empty , occupied by preys or simultaneously occupied by preys and predators . the dynamics of the system is defined as a continuous time markov process . the transition rates for site @xmath1 are    * @xmath4 at rate @xmath5 , * @xmath6 at rate @xmath7 , * @xmath8 at rate 1 ,    where @xmath9 denotes the number of nearest neighbor sites of @xmath1 which are in the state @xmath10 . 4 is the coordination number of this two dimensional square lattice . the first two processes model the spreading of preys and predators . the two control parameters , @xmath11 and @xmath12 , characterize a particular prey - predator system . the reason for considering the sum , @xmath13 , in the first rule is simply that all the neighboring cells of @xmath1 containing some prey , ( hence @xmath14 or @xmath15 ) , will contribute to the prey repopulation of cell @xmath1 . the third process represents the local depopulation of a cell due to too greedy predators . it can be interpreted as the local extinction of the species or as the moving of them to neighboring occupied sites . spontaneous disappearance of a prey state ( @xmath16 : @xmath17 ) or that of the predators alone ( @xmath16 : @xmath18 ) is forbidden . these assumptions are reasonable because the occurrence of these processes is improbable . the rate of the third process is chosen to be 1 , which sets the time scale . as a consequence , @xmath11 and @xmath12 are dimensionless quantities . the above dynamical rules are an extension of the contact process model ( cp )  @xcite introduced as a description of epidemic spreading . the cp is a 2-state model , @xmath19 ; the status 0 and 1 represent respectively the healthy and the infected individuals . the cp dynamical rules are    * @xmath4 at rate @xmath20 , * @xmath17 at rate 1  . an epidemic survives for @xmath21 = @xmath22  @xcite and disappears for @xmath23 . the transition towards this absorbing state is of second order and belongs to the directed percolation ( dp ) universality class  @xcite . our model differs from most of the lattice models previously investigated   by the fact that on each site , each species may be represented by several individuals rather than just one . in the previously investigated models the spreading rate of the preys is simply proportional to @xmath24 . under this assumption , our model reduces essentially to the st model , in which the control parameters are defined as @xmath25 and @xmath26 . it is worth discussing first the behavior of our model in two limiting cases . in the @xmath27 limit the proportion of empty cells is negligible since the empty cells are reoccupied by preys instantly after their extinction . hence , the lattice is completely covered by preys and the @xmath28 sites behave as the infected species in the cp . namely , when decreasing @xmath12 the predator density is decreasing continuously and vanishes at the cp critical value @xmath29 = @xmath30 . one can think of the @xmath31 limit in similar terms . in this case , the proportion of the prey cells ( @xmath32 ) should be negligible since the high productivity of the predators , while the prey - predator cells should behave as the infected species in the cp . this is indeed the case if @xmath33 , but when @xmath11 gets smaller than @xmath30 , the prey density increases again instead of being zero , as we shall see later . although apparently simple , there is no way to solve analytically the model defined above . however , analytic solutions can be obtained by making some approximations . the simplest one is the one - point mean - field approximation in which all spatial fluctuations are neglected . thus , the system is characterized by the densities of prey , @xmath34 , and predator , @xmath35 , sites a=1l^2 _ i ( _ _ i , 1 + _ _ i , 2 )  ,   b=1l^2 _ i _ _ i , 2  , which values satisfy the @xmath36 conditions by definition . in terms of these densities , the mean - field dynamical equations read : = _ a a(1-a)-b [ mfa ] and = _ b b(a - b)-b [ mfb ] note , that for @xmath37 ( @xmath38 ) initial condition the predator ( and prey ) densities remains 0 . the ( [ mfa],[mfb ] ) equations clearly differ from the usual l - v ones . the main difference lies in the interaction terms as , although a larger prey density increases the predator growth rate , the rate of the predated preys only depends on the predator density . this is a simple consequence of the fact that there are no pure predator sites without preys in this model . thinking of a real prey - predator system it makes sense , as a predator has to consume a certain amount of preys in a given time to survive , independently of the number of preys around it . the @xmath39 term in the first equation plays the role of a simple verhlaust factor which assures an upper limit for the prey density @xmath40 , and similarly the @xmath41 term in the second equation do not let the density of predators exceed that of the preys . the stationary states are obtained by setting the left hand sides of eqs . ( [ mfa ] , [ mfb ] ) to zero . contrary to the simplest l - v equations , qualitatively different stationary states are obtained varying the parameters , @xmath11 and @xmath12 , as illustrated on fig . [ fig : phase_mf ] .    for @xmath42 and @xmath43 , the stationary state is a pure prey absorbing state @xmath44 . for @xmath45 the stationary state is also a prey state , @xmath46 , however , the value of @xmath47 depends upon the initial state .    in the rest of the plane @xmath48 , the stationary solution is a^s= and b^s = a^s-  , which describes a coexistence of preys and predators ( coexistence phase ) . for @xmath49 the @xmath34 and @xmath35 densities are approximately the same a^s = b^s + o ( ) = \ { 1 - + o ( ) & _ a > 1 + o ( ) & _ a = 1 + o ( ) & _ a < 1 + . and as a function of @xmath11 they show a mean field cp behavior as it is expected from the argument given in sec .  [ sec : model ] .    in the @xmath50 limit ( and for @xmath51 ) the system is  full of preys , namely a^s = 1 - ( 1- ) + o ( ) [ aapp ] and the predator density reads b^s = ( 1-)(1- ) + o ( ) [ bapp ] and , as expected , its @xmath12 dependence agrees with the prediction of the mean - field approximation for cp . this approximation predicts a second order phase transition along the whole @xmath52 line , as in the @xmath53 limit @xmath34 and @xmath35 approach linearly the values @xmath54 and @xmath55 respectively a^s & = & 1 - + o((_b-1)^2)b^s & = & _ a + o((_b-1)^2 )  . the behavior of the densities is rather surprising at the @xmath45 boundary of the coexistence phase . for @xmath56 and for @xmath51 a^s = + _ a ( + ) + o(_a^2 )  , [ a : la<<1 ] while the stationary solution , @xmath47 , for @xmath57 depends on the initial state . thus the mean field approximation predicts a discontinuity of the prey density along this boundary . however , the density @xmath58 is proportional to @xmath11 and continuous in @xmath57 . important quantities are the fluctuations of the prey and the predator densities ( mean square deviations ) , which are normalized to be size independent for large systems _ = l^2 ( -)^2 , = a b  , [ defchi ] and @xmath59 means the time average in the stationary state . for @xmath60 the majority of the sites are in state @xmath15 , with a few holes in it , hence one can suppose that the holes are independent . consequently , the number of the holes follows a poisson distribution , from which the average hole number equals to the mean square deviation . there are @xmath61 holes made of sites in the state @xmath62 and @xmath63 holes made of sites in the states @xmath62 or @xmath54 . thus _ a 1-a^s _ b 1-b^s  , [ appchi ] which is in good agreement with the simulations in a region ( non - oscillatory part ) of the coexistence phase ( see fig .  [ fig : chi ] ) . the stability of the stationary state can be analyzed by linear stability . one has to investigate the eigenvalues , @xmath64 , of the jacobian matrix related to the mean field equations ( [ mfa ] , [ mfb ] ) at the stationary densities . ( _ a a & _ b a + _ a b & _ b b + ) |_s= ( _ a(1 - 2a^s ) & -1 + _ b a^s -1 & 1 - _ b a^s )  . it turns out that the real parts of the eigenvalues are always negative , assuring the stability of the solutions . this mean field approximation do not predict limit cycles , which would correspond to having an eigenvalue , @xmath65 , with a zero real part . however , in some part of the coexistence phase the imaginary part is nonzero , so the stationary solution is approached in spirals ( poles ) , instead of straight lines ( nodes ) ( see fig .  [ fig : phase_mf ] and [ fig : flow ] ) , as it was also observed in the st model  @xcite . note , that an unexpected node region appears for @xmath66 . one can consider the presence of poles as a hint for the appearance of oscillations beyond the mean - field approximation . notice , that in this pole case , the damped oscillations are strong along the @xmath12 axes ( i.e. for @xmath67 ) . the strength of them can be characterized by the ratio of the imaginary and real part of the eigenvalues , which has a singularity in the @xmath68 limit . using ( [ a : la<<1 ] ) , we obtain || = 4 _ a^-1/2 + o ( _ a^1/2 ) for @xmath67 and @xmath51 . in this limit one can derive an expression also for the frequency , @xmath69 , of the damped oscillations = | ( ) | = 2 _ a^1/2 + o ( _ a^3/2 )  .    the mean field results can be interpreted in the following way . the approximation predicts two distinct phases : the pure prey phase and the coexistence one . it also gives some hints for a possible presence of oscillations in some parts of the coexistence phase . the phase boundaries of the two phases are described by two lines : the @xmath52 and the @xmath45 . several quantities show a power - law behavior close to these boundaries , like @xmath35 and @xmath70 at the @xmath52 boundary , and @xmath35 , @xmath69 and the strength of the damped oscillations at the @xmath45 boundary . this implies that the transitions are of second order , and the predator density , @xmath35 , seems to be a good candidate for the order parameter . the order parameter goes to zero at the phase boundaries as @xmath71 and @xmath72 with a mean - field exponent @xmath73 . we performed also a pair approximation , in which the nearest neighbor correlations are also considered as parameters . it turns out that the results differ only quantitatively from that of the one point approximation . contrary to  @xcite , our system does not show limit cycle behavior on the pair approximation level either . on general grounds , one expects that the fluctuations will play an important role in low dimensions . our model is supposed to describe a two dimensional world and accordingly , we have performed extensive monte - carlo ( mc ) simulations for systems of sizes @xmath0 , @xmath74 varying between 100 and 1000 . we used periodic boundary conditions . although our model is formulated as a continuous - time process , an equivalent ( at least for not very short times ) discrete time formulation is more suitable for numerical simulations . in one elementary time step one lattice site is chosen randomly and its state evolves according to the rules defined is sec . [ sec : model ] using rescaled rates ( all less than 1 ) as transition probabilities . one mc step is defined as the time needed such that all the sites have been , on the average , visited once . in this paper we always use the original time units defined by the model , which can be obtained simply by rescaling the time measured in mc steps .    for sufficiently large system , the stationary state does not depend on the initial conditions . usually we filled up the lattice completely with preys as an initial state and put a few predators on it . to obtain the stationary phase diagram and the stationary values of the quantities of interest the number of mc steps performed varied in the range @xmath75 to @xmath76 for systems of linear size @xmath77 to 1000 respectively . the corresponding phase diagram is depicted on fig . [ fig : phase ] . two different phases are present as a function of the two control parameters @xmath11 and @xmath12 : a pure prey phase , a prey and predator coexistence phase with an oscillatory and a non - oscillatory region . in the oscillatory region , oscillations with a well defined frequency were observed in the prey and the predator densities ( see fig .  [ fig : dens_time ] ) . although theoretically possible , we never observed an empty lattice absorbing state . the reason for that is simply that even one surviving prey fill up the system with preys in the absence of predators . as fig . [ fig : phase_t ] shows , the locations of the different regions of the phase space differ essentially from those obtained for the st model . the phase boundaries of the prey phase ( see fig .  [ fig : phase ] ) were obtained in the following way . simulations were started at parameter values for which the coexistence is maintained practically forever ( up to the maximal number of mc steps investigated ) , and we decreased one of the parameter values by @xmath78 . if the predators were still alive after a given time , @xmath79 , we decreased the parameter further . the extinction of the predators defines the phase boundary . @xmath78 was chosen to be in the range 0.005 to 0.04 , while @xmath80 mc steps . the result was very similar with @xmath81 and @xmath82 mc steps . the definition of the boundary between the oscillatory and the non - oscillatory region of the coexistence phase will be described later .    on fig . [ fig : phase ] , the boundary of the prey phase is displayed for different system sizes ( @xmath83 ) . apparently , in the @xmath84 regime the size dependence is negligible , but relevant for @xmath85 . note that this strong size dependence of the boundary coincides with the presence of oscillations . decreasing @xmath12 at any fixed value of @xmath11 , a second order phase transition takes place between the coexistence and the prey absorbing phases along a transition line @xmath86 . as for the mean field case , the predator density is considered to be the order parameter . as @xmath87 , the order parameter , @xmath35 , and @xmath70 go to zero as b ~1-a ~(_b - _ b^*(_a))^_1  . [ dp_op ] as seen on fig . [ fig : phase ] , the values of @xmath86 obtained by fitting the data with eq . [ dp_op ] are in very good agreement with the phase boundary obtained previously . fitting the data leads @xmath88 ( with satisfactory precision for @xmath89 ; see fig . [ fig : dens_dp ] ) .    in the same limit the fluctuations of the predator density also follow a power law behavior , @xmath90 . the exponent has been determined to a good precision as @xmath91 for several values of @xmath11 between 1 and 50 . the same behavior has been obtained ( only for @xmath92 and 3 ) for the prey fluctuations , @xmath93 , with the exponent @xmath94 . the critical behavior seems to be the same when the transition line is crossed while decreasing @xmath11 at fixed values of @xmath12 . the two exponents , @xmath95 and @xmath96 , are compatible with those obtained for dp in @xmath97 dimension  @xcite . thus we conclude , that this absorbing state phase transition belongs to the dp universality class , as expected on general grounds  @xcite . note that dp type phase transition in a similar lattice prey - predator system has already been observed in 1 dimension  @xcite .    for @xmath68 , the transition line , @xmath86 , ends in a special point , ( @xmath98 ) , where all the three phases meet . for @xmath99 , the mc results for the finite size dependent phase boundary suggest us that the transition happens at @xmath100 in the @xmath101 limit . approaching this transition line , @xmath68 , the prey density , @xmath34 , does not go to 1 but to a finite value depending on @xmath12 ( see fig . [ fig : dens_a ] ) . however , according to the results depicted on fig . [ fig : dens_b ] , the predator density , @xmath35 , always goes to zero in this limit as a power of @xmath11 with an exponent @xmath102 . surprisingly , this second order phase transition to the prey absorbing phase does not belong to the dp universality class . the presence of power law behavior , however , confirms that for an infinite system the transition occurs at @xmath100 for @xmath103 . this means that for this range of @xmath12 , and for any arbitrary small @xmath11 , the coexistence of the species is possible providing that the system is large enough . for @xmath99 the fluctuations of the two densities , @xmath93 and @xmath104 , behaves similarly . for a given @xmath12 , there is a clear crossover at @xmath105 from a mean field like behavior to a regime where the correlations are more important . for @xmath106 the behavior of @xmath93 and @xmath104 agrees with that predicted by mean - field theory , reflecting the fact that in this range of @xmath11 the dominant behavior comes from the noise . as the @xmath107 condition coincides with the presence of oscillations , the crossover point , @xmath105 , is taken as the definition of the border between the oscillatory and the non - oscillatory region . after a proper normalization , the relative fluctuations collapse on a single curve for @xmath108 ( see fig .  [ fig : chi ] ) . namely , k_1(_b )  , where the numerical factor , @xmath109 , depends only on @xmath12 . however , the precision of the simulation results was not satisfactory enough to obtain the functional form of @xmath109 ( and of the forthcoming @xmath110 for @xmath111 , 3 and 4 either ) . the simulation showed that @xmath112 ( @xmath113 or @xmath35 ) is size independent as it was expected from its definition ( [ defchi ] ) . as a consequence , the deviation from the average density , @xmath114 @xcite , is smaller for larger systems and evidently scales with @xmath115 . certainly , this deviation increases with the intensity of the oscillations . the above finite size behavior is in agreement with the results of earlier simulations which claimed that the oscillations in the global densities disappear with increasing system size  @xcite . our simulations predict more pronounced oscillations for smaller @xmath11 and for larger @xmath12 . the oscillations have to show up also in the correlation functions , c_a(i , ) & = & ( 1 - _ _ j(t ) , 0 ) ( 1 - _ _ j+i(t+ ) , 0 )  , c_b(i , ) & = & _ _ j(t ) , 2 _ _ j+i(t+ ) , 2  , where @xmath116 labels a lattice site distant of @xmath1 lattice spacing from the site @xmath117 . @xmath118 ( @xmath113 or @xmath35 ) depends only on @xmath1 and @xmath119 due to the homogeneity of the system in space and time . for @xmath120 the correlation function , @xmath121 , obtained numerically could be fitted by an exponential @xmath122 . in the oscillatory region the correlation lengths of preys and predators differ only through a @xmath12 dependent factor , @xmath123 , and they turned out to be proportional to the fluctuations of the prey density , @xmath124 . it means that a more correlated system displays stronger oscillations . the reason for that is simply that the dynamics of the different sites shows some synchronization within a correlation length , which results in larger oscillations ( see sec . [ sec : discuss ] for more details ) .    in order to determine the characteristic frequency , @xmath125 , and the amplitude , @xmath126 ( @xmath113 or @xmath35 ) , of the oscillations , we measured the fourier spectrum of the time dependent densities s _ ( ) = _ t 1 t | _ t=1^t ( t ) ( it ) |^2  . [ defps ] the presence of oscillations is reflected as a peak at nonzero frequency in the fourier spectrum . extracting this peak from a background noise , enable us to define @xmath127 and @xmath128 as the zeroth and the first momentum of this distribution . this analysis shows clearly that the frequency of the oscillations is independent of the system size ( see fig . [ fig : freq ] ) , and is the same for preys and predators . moreover , for a wide range of the parameters in the oscillatory phase the frequency , @xmath129 , is well approximated by @xmath130 . this linear behavior differs from the mean field prediction .    in the oscillatory region the oscillations are present for arbitrary large systems , however , their amplitude decreases with increasing system size , as @xmath131 . at this point it is important to emphasize that this fact does not imply that only small oscillations are present in large systems . indeed , for a large system the amplitude of the oscillations can be made larger by decreasing @xmath11 . on the other hand , when increasing @xmath11 the amplitude goes to zero as a power law which makes difficult to define a phase boundary for the oscillations in this way . however , there is a simple scaling relation between the amplitude and the correlation length for the preys in the oscillatory region _ a^2 2 _ a l^2 a_a  , [ xca ] as it can be observed on fig . [ fig : scale ] . the analogous expression for the predators is slightly more complicated _ b^2 ( ) ^2 k_3(_b)_b k_4(_a ) l^2 a_b  , [ xcb ] with appropriate @xmath132 and @xmath133 values . another quantity which characterizes the oscillations is the time dependent local correlations , @xmath134 . a similar investigation was made in  @xcite with time dependent correlations of the average local densities . in the oscillatory region @xmath135 displays damped , size independent oscillations . more precisely , the time correlations are size independent for any @xmath136 , while for any @xmath137 the system evolves to the prey absorbing state . clearly , this critical system size is proportional to the correlation length , @xmath138 . the size independence of @xmath139 is a simple consequence of the fact that areas which are further than @xmath140 apart are uncorrelated . the investigation of the time dependent correlations , however , provides a rather ambiguous way to define the boundary of the oscillatory region . indeed , one can observe local oscillations everywhere in the coexistence phase simply because , due to the cyclic dominance nature of the model , each site has to evolve in a loop ( @xmath141 ) . thus , according to the value of the damping factor , it is somehow arbitrary to decide if the state is oscillatory or not . it is worth noting , that at some particular values of @xmath12 ( @xmath142 or 20 ) and for small @xmath11 values ( @xmath143 or 0.4 respectively ) , where the correlation length is comparable to the system size ( @xmath144 ) , the system can evolve to a stripe like state . in this state 3 stripes of size @xmath74 , made of predator , prey and empty cells , are drifting through the system . however , for given @xmath11 and @xmath12 values , this behavior disappears when increasing the size of the system . the comparison of the mc results with the mean - field prediction shows that the later gives a qualitatively correct description of the phase diagram ( see fig . [ fig : phase_mf ] ) , as well as of the discontinuity in the prey density , @xmath34 , along the @xmath45 boundary . a qualitative understanding of the phase diagram is possible . if the birth rates are much larger than the death rate ( @xmath145 and @xmath146 ) the system is full of preys and predators ( @xmath147 ) , while for small values of @xmath12 the system evidently reaches the pure prey absorbing state .    as already discussed in sec . [ sec : model ] , in the @xmath27 the system is full of preys ( @xmath148 ) and the predators behave like the infected species in the cp . it means that they could survive only for @xmath149 , where a dp like second order transition occurs . this is in agreement with the mean field results and with the simulation for @xmath150 ( see fig .  [ fig : dens_dp ] ) . one can also derive an approximate formula for the position of the phase boundary between the non - oscillatory and the prey phase @xmath86 . for @xmath151 , the system is almost full of preys ( @xmath152 ) and , in some sense , the dynamics of the predators is close to that of the cp . the predators die at rate 1 and spread at rate @xmath12 , but they can not enter into the empty sites . one can introduce an effective @xmath153 and describe the process as a cp , namely , the predators can enter any neighboring site at this rate . as the number of empty sites is proportional to leading order to @xmath154 , the effective parameter should be @xmath155 , where @xmath156 is a fitting parameter . as this cp displays a phase transition at @xmath157 , in terms of the original parameter the transition occurs at @xmath158 . this conjecture is in excellent agreement with the simulation data for @xmath159 with @xmath160 ( see fig .  [ fig : phase ] ) . for @xmath146 the new prey sites are usually immediately occupied by predators as well . however , with a small but finite probability , a predator site can disappear before the predators spread to the new born prey site , and in this way , a prey site can be left alone and grow ( similarly to the eden model  @xcite ) . this rare event is negligible when the predator density is large enough and a prey island can not grow for long periods of time . practically this is the case for @xmath33 . in this case , the number of prey sites is negligible small , and the predator sites behave as the infected species in the cp . one can see on the fig . [ fig : dens_a ] and [ fig : dens_b ] , that for @xmath161 the two densities ( @xmath162 ) are equal to that of the cp if @xmath33 . however , in the vicinity of @xmath30 the densities are low , which allows an isolated prey island to grow for a long time . if @xmath163 the predator islands are shrinking and , if @xmath11 and @xmath74 are not too small , they can survive until a growing prey island reaches one of them . at this moment , the predators invade very quickly the prey territory and increase their population size ( see fig .  [ fig : conf100 ] ) . these new predator sites start to die out leaving a few prey sites alone , and the whole procedure starts again . this mechanism insures the survival of the predators much bellow the cp critical density and results in oscillations in the population sizes .    for @xmath103 , but not too large , the qualitative picture is slightly different . as one can observe on fig . [ fig : conf5 ] , groups of predators are wandering through the system towards prey - dense areas . if two fronts of predators meet they usually stop moving and the local population of predators starts shrinking . the oscillations are maintained in a somewhat similar way than for the @xmath146 case : these predators can only survive if the preys become dense around them . this is more probable for larger values of @xmath11 , and it is also clear that the predators have a better chance to survive in larger systems .    according to the above statements , the key point in the underlying mechanism of oscillations is the existence of blocked predator islands which are located and trapped in sparse prey areas . indeed , blocked predators in sparse prey areas result in growing prey populations ; however , the resulting dense prey population allows predators to move and predate again . this mechanism drives back the system to the beginning of the loop . clearly , predators can only be trapped in sparse prey areas if @xmath11 is smaller or of the order of the death rate , 1 . this explains the location of the oscillatory region . note , that the above argument is based on the spatial nature of the system , suggesting that the spatially extended character is fundamental for the existence of such prey - predator type of oscillations . this mechanism also provides a qualitative understanding of the key properties of the system . the trapped predators can invade the prey area only when the preys are dense enough again , which takes a time proportional to @xmath154 , and leads to @xmath164 . according to the simulations the correlation length , @xmath140 , increases with decreasing @xmath11 . indeed , as @xmath11 decreases , the trapped predators have to wait longer to escape , hence fewer groups of predators survive . this increases the distance between the groups , resulting in larger prey islands , which average size is proportional to @xmath165 . when the correlation length is of the order of the system size , there are islands of preys of typical size l , extruding the predators out of the system . hence , the condition @xmath166 characterizes the phase boundary between the oscillatory and the prey phase . on the other hand , a correlation length of order one ( @xmath167 ) , means that the noise dominates the system . thus , @xmath167 characterizes the boundary between the oscillatory and the non - oscillatory region of the coexistence phase .    as shown by the study of the time dependent correlations , domains separated by a distance larger than @xmath165 oscillate asynchronously around a constant value with the same frequency , @xmath168 . according to this picture , one can derive a more quantitative description for the oscillatory region . let us assume that for @xmath169 , the global densities of each species can be written as the sum of local coarse - grained densities at a typical length scale @xmath165 . moreover , we assume that all these local densities oscillate with the same frequency but a different phase , @xmath170 . in general , the amplitude of the local oscillations should depend on the parameters @xmath11 and @xmath12 . however , as one can observe on fig . [ fig : conf100 ] and [ fig : conf5 ] , the predators can only enter an almost fully dense prey area and the predator fronts leave an almost empty field behind them . hence , as suggested by the numerical simulations , everywhere in the oscillatory region , the local amplitude for the prey density can be considered as a constant , @xmath171 . = a^s + d ( ) ^2 _ l=1^()^2 ( t + _ l )  . supposing that the @xmath170 values change much more slowly than @xmath69 , @xmath172 shows a simple @xmath173 behavior for long periods of time ( see fig . [ fig : dens_time ] ) . thus , for @xmath172 one can derive the value of the density fluctuations , @xmath93 , and the amplitude of these oscillations , @xmath174 , using ( [ defchi ] ) and ( [ defps ] ) , and take the average over all the possible @xmath170 configuration taken from a flat distribution . this procedure reproduces the result of eq . ( [ xca ] ) up to a multiplicative factor in front of the correlation length . we have studied a two dimensional prey - predator model , ( size @xmath0 ) , which exhibits a rich stationary state phase diagram . a particular attention has been payed to the study of finite size effects , and we were able to draw clear cut conclusions concerning the behavior of the model both for @xmath74 finite as well as for the limit @xmath175 . three kinds of stationary states can be reached according to the values of the control parameters : a pure prey state , and two coexisting prey - predator ones with and without oscillations . two different kinds of second order transitions were found when going into the prey absorbing phase . the transition between oscillatory and non - oscillatory coexistence phase is , in fact , a cross - over between two asymptotic regimes characterized by a very small and a large correlation length respectively . in the oscillatory regime , scaling relations were established between several physical quantities . a qualitative explanation for the existence of such oscillatory regime is given , pointing out the crucial role of the spatial extension of the system . indeed , the frequency of the oscillations is determined locally due to the dynamics related to blocked predator islands in sparse prey areas . regions of linear size @xmath165 oscillate with the same frequency but with different phases . this explains the decreasing amplitude of oscillations with increasing system size . on the other hand , slowly changing phases result periodic oscillations in the overall prey density for long periods of time . moreover , for suitable choices of the control parameters one can have synchronized oscillations with finite amplitude across arbitrary large systems . thus we think that our simple model could offer a qualitative explanation for the behavior of the lynx population problem described in sec . [ sec : intro ] .
a coarse grained description of a two - dimensional prey - predator system is given in terms of a 3-state lattice model containing two control parameters : the spreading rates of preys and predators . 
 the properties of the model are investigated by dynamical mean - field approximations and extensive numerical simulations . 
 it is shown that the stationary state phase diagram is divided into two phases : a pure prey phase and a coexistence phase of preys and predators in which temporal and spatial oscillations can be present . 
 the different type of phase transitions occuring at the boundary of the prey absorbing phase , as well as the crossover phenomena occuring between the oscillatory and non - oscillatory domains of the coexistence phase are studied . 
 the importance of finite size effects are discussed and scaling relations between different quantities are established . 
 finally , physical arguments , based on the spatial structure of the model , are given to explain the underlying mechanism leading to oscillations . 
 = 10000    2