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 last years a particular great effort has been done in order to understand the role of space given by a spatial structure and local interactions in the characterization of the dynamics of competing biological species systems @xcite . in this context
it has been studied irreversible stochastic lattice models @xcite with the purpose of mimic predator - prey systems with markovian local rules based in the lotka - volterra model lotka , volterra .
one of the problems that has been object of study is the connection between the time oscillations of population densities and spatial pattern distribution of the individuals of each species .    here
we study the coexistence of a two - species system by considering a probabilistic cellular automaton ( pca ) which is a modified version of the automaton devised in @xcite .
the model , to be called anisotropic predator - prey pca possess local rules that are similar to the ones of the cellular automaton proposed in @xcite  and was introduced by us in order to explore the effect of spatial anisotropy in the temporal oscillations .
we report dynamic mean - field approximations which take into account the spatial dependence of densities and pair correlations of sites . in the next section we present the model . in sec .
2 we show the equations for the time evolution of species densities and show the spatial dependence of these equations . in sec . 3 and 4
we perform dynamic simple and pair mean - field approximations .
last section summarize the model , method and results .
the physical space occupied by the species is represented by a regular lattice of @xmath0 sites in which each site can be in one of three states . at each site of the lattice
we attach a stochastic variable @xmath1 that takes the values @xmath2 , @xmath3 and @xmath4 according whether the site @xmath5 is empty , or occupied by a prey or occupied by a predator , respectively .
the state of the system can be represented by the vector @xmath6 .
the time evolution equation for @xmath7 , the probability of state @xmath8 at time @xmath9 , is given by @xmath10 where the sum is over the @xmath11 configurations of the system and @xmath12 is the transition probability from a state @xmath13 to state @xmath8 , given that at the previous time step the system was in state @xmath14 .
since we are considering probabilistic cellular automata , all the sites are updated simultaneously . in this case
we have @xmath15 where @xmath16 is the conditional transition probability per site .      the stochastic rules , embodied in the transition rate @xmath17 ,
are set up in order that the allowed transitions between states are only the ones that obey the cyclic order shown in fig .
prey can only born in empty sites ; prey can give place to a predator in a process where the prey dies and the predator is instantaneously born ; finally predator can die leaving an empty site .
the empty sites are places where prey can proliferate and can be seen as the resource for prey surveillance .
the death of predators complete this cycle , reintegrating to the system the resources for prey .
the anisotropic predator - prey pca has three parameters : @xmath18 , the probability of birth of prey , @xmath19 , the probability of birth of predator and death of prey , and @xmath20 , the probability of predator death .
two of the process are catalytic : the occupancy of a site by prey or by a predator is conditioned , respectively , to the existence of prey or predator in the neighborhood of the site .
the third reaction , where predator dies , is spontaneous , that is , it occurs , with probability @xmath20 , independently of the neighbors of the site .
we assume that @xmath21 with @xmath22 .
the transition probabilities of the anisotropic predator - prey pca are described in what follows :    \(a ) if a site @xmath5 is empty , @xmath23 , and if at north or east there is at least one prey , then it can be occupied by a prey , @xmath24 , in the next time step , with a probability proportional to the parameter @xmath18 and to the number of prey @xmath25 at north and east of site @xmath5 .
\(b ) if a site is occupied by a prey , @xmath24 , then the site has a probability of being occupied by a new predator , @xmath26 , in the next time step if there are prey at north or east . in this process
the prey dies instantaneously .
the transition probability is proportional to the parameter @xmath19 and the number of predators at north and east of site @xmath5 .
\(c ) if site @xmath5 is occupied by a predator , @xmath27 , it dies spontaneously with probability @xmath20 .
the anisotropic cellular automaton is a variation of the automaton introduced in @xcite .
here each site of a regular square lattice interacts with its first neighbors only at two preferential directions .
this anisotropic neighborhood consists of the northern and eastern neighbors of each site as shown in fig .
the set of transition probabilities per site is given by @xmath28 @xmath29 and @xmath30 where @xmath31 and the sum is over the neighbor sites localized at east and north of site @xmath32 and correspond to the number of neighbors of site @xmath5 occupied by prey individuals and predators individuals , respectively .
we have considered this probabilistic cellular automaton with the purpose of verifying the effect of anisotropy in the properties of the time oscillations of the predator - prey system .
the rules considered are in some sense inspired in the north - east - center ( nec ) cellular automaton @xcite , which also consider an unsymmetrical neighborhood of northern and eastern sites .
the present stochastic dynamics predicts the existence of states , called absorbing states , in which the system becomes trapped .
once the system has entered such a state it can not escape from it anymore remaining there forever .
there are two absorbing states .
one of them is the empty lattice .
the other absorbing state is the one in which the lattice is full of prey .
this situation occurs if there are few predators and they become extinct .
the remaining prey will then reproduce without predation filling up the whole lattice .
the existence of absorbing stationary states is an evidence of the irreversible character of the model or , in other words , of the lack of detailed balance @xcite .
however , the most interesting states , the ones that we are concerned with in the present study , are the active states characterized by the coexistence of prey and predators .      the densities of prey , predator and empty sites are defined as @xmath33 @xmath34 @xmath35 respectively .
the lower index @xmath5 is used to denote the site and the upper index @xmath9 stands for the time .
the pair correlation of two neighbor sites @xmath5 and @xmath36 one being occupied by prey and the other by predator is defined as @xmath37 in this definition the sites @xmath5 and @xmath36 are two neighboring horizontal sites , the site @xmath36 being at the east of @xmath5 . for two neighboring sites placed
vertically we use the notation @xmath38 where @xmath39 denotes the neighbor of @xmath5 to the north .
the evolution equation for the densities can be obtained by using the rules of the automaton and the evolution equation for the probability @xmath40 . using equations ( [ master ] ) , ( [ regra1 ] ) , ( regra2 ) and ( [ regra3 ] ) , we obtain the time evolution equation for the density of prey at site @xmath5 , @xmath41+p_i(1)-\frac{b}{2 } \left [ p_{ij}\left (   \begin{array}{c } 2 \\   1% \end{array } \right)+p_{ik } (   \begin{array}{cc } 1 & 2% \end{array } ) \right ] ,   \label{dens1}\ ] ] where we have used prime and unprimed to denote quantities at time @xmath42 and @xmath9 , respectively .
again , @xmath43 denotes a pair correlation of a site @xmath5 which is empty and its neighbor at the north @xmath36 which is occupied by a prey and @xmath44 denotes the pair correlation of the site @xmath5 which is is empty and its neighbor to the east @xmath39 which is occupied by a prey .
note that site @xmath5 is the site to be updated .
the time evolution equation for the density of predators at site @xmath5 is given by @xmath45+(1-c)p_i(2 ) .
\label{dens2}\ ] ]    the evolution equations for correlations of two neighbor sites are given by equations which involves clusters of two , three and four neighbors to the north and east and they are too cumbersome .
we will write them in the pair approximation in the next subsection
.    we will assume in the present analysis that the densities and the pair correlations are homogeneous so that the correlations of sites placed horizontally or vertically are the same . in other words the systems exhibits a specular symmetry along the southwest - northeast line ( see fig .
[ nec ] ) .
therefore , @xmath46 and @xmath47
we will first analyze the equations ( [ dens1 ] ) and ( [ dens2 ] ) for the densities of prey and predators by means of a simple mean field approximation @xcite . in this approximation
we write the probability of a given cluster of sites as the product of the probabilities of each site . in our analysis
we maintain the space dependence of each probability .
this is necessary since the rules that define the model are not isotropic .
we will use the notations @xmath48 where the index @xmath49 stands for a site at the @xmath49 layer .
a layer is defined as the sites belonging to a line perpendicular to the southwest - northeast axis , as shown in fig .
[ nec ] . using this approximation and considering equations ( [ dens1 ] ) and ( [ dens2 ] )
we obtain @xmath50 and @xmath51 where @xmath52 . due to this relation
there is no need for the equation for the density of empty sites .
the analysis of the above set of equations show that the stable solutions are of two types : a prey absorbing state , where the all lattice is full of prey ; and an active solution with prey and predators densities constant in time and space .
so , we have just homogeneous and nonoscillating solutions when we treat the anisotropic pca by means of simple mean - field approximation . at this level of mean - field approximation the space anisotropy does not play any role in determining the kind of coexistence of species .
now we consider the evolution equations for one - site correlations and two site ( pair ) correlations .
the evolution equations for the pair correlations depend on clusters of three and four sites . following the approach of the pair approximation @xcite we write the clusters of three and four sites as products of pair correlations and one - site correlations
. however , we must be careful in doing this approximation because we are maintaining the spatial anisotropy dependence of the one and two site correlations .
for example , the correlation of a site at layer @xmath49 in state @xmath2 , a site at layer @xmath53 in state @xmath2 and a site at layer @xmath54 in state @xmath3 , is written in the pair approximation as @xmath55 there are more complex clusters that appear in the evolution equation for pair correlations .
for example , the following cluster of four sites is approximated by @xmath56 observe that @xmath57 and @xmath58 are two independent variables .
now we use the notation @xmath59
@xmath60 @xmath61    using the pair approximation we derive the following equations for pair correlations , @xmath62 @xmath63 @xmath64 @xmath65 @xmath66 @xmath67 @xmath68 @xmath69 @xmath70 @xmath71 @xmath72 @xmath73 @xmath74 @xmath75 @xmath76 @xmath77 @xmath78 @xmath79    the equations for the densities are @xmath80 and @xmath81    therefore , we get a set of eight equations that we have iterated to find the solutions .
we considered periodic boundary conditions and an initial condition where half of the lattice with a set of densities and the other half with another set of densities .    for @xmath82 and great values of @xmath20
the system attains a stationary state which is the absorbing prey state .
decreasing @xmath20 there is a transition to an active state , coexistence of species , and we found very interesting solutions which are characterized by travelling waves of the densities of each species . in fig .
[ oscespac ] we show , for a particular set of the parameters @xmath82 and @xmath83 , the prey and predators oscillations as a function of space and for a fixed instant of time .
we can see that the behavior of the space oscillations are very complex and very different from a sinusoidal wave .
we remark that our assumption concerning the spatial dependence of densities and pair correlations underlying our mean - field approach could not easily be conceived a priori .
it was set up by considering the type of local dynamics with unsymmetrical rules and from the results of monte carlo simulations for the present pca @xcite . to further clarify this point we show in fig .
[ snap ] a snapshot generated by simulation of the present pca on a square lattice where each site can be occupied by prey , predators or can be empty .
periodic boundary conditions were used and the system evolved in time according to the rules defined by eqs .
( [ regra1 ] ) , ( [ regra2 ] ) and ( [ regra3 ] ) . a synchronous update was used .
we see that the distribution of individuals of each species is not homogeneous but exhibits a pattern composed by layers of prey and predators .
these layers are displayed in a manner perpendicular to the southwest - northeast direction , the direction along which the oscillations occur as can be seen in fig .
we have found that the spatial oscillations do not occur independently of the time oscillations , since the spatial layers of prey and predators are not static but are like fronts moving along the southwest - northeast direction .
the spatial pattern oscillations are intimately associated to local time oscillations of the species kelly , tome2007 .
these features were incorporated in the present mean - field analysis of the model .
we have considered a predator - prey probabilistic cellular automaton with anisotropic local stochastic rules .
we studied this model by means of dynamic mean - field approximation at two orders : simple mean - field approximation and pair mean - field approximation .
due to the anisotropy these approximations only can be performed if we consider the spatial dependence of probabilities .
the simple mean - field approximation for this automaton just provides the prey absorbing state and active spatial homogeneous solutions which are also constant in time .
the pair mean - field approximation gives much more rich results and show that the active states characterized by time oscillations of species densities are inhomogeneous in space .
therefore , we have spatiotemporal patterns of coexistence .
this is in accordance with previous monte carlo simulations @xcite where we have found local time oscillations connected to inhomogeneous spatial distributions of species individuals .
we consider a probabilistic cellular automaton to analyze the stochastic dynamics of a predator - prey system . 
 the local rules are markovian and are based in the lotka - volterra model . 
 the individuals of each species reside on the sites of a lattice and interact with an unsymmetrical neighborhood . 
 we look for the effect of the space anisotropy in the characterization of the oscillations of the species population densities . 
 our study of the probabilistic cellular automaton is based on simple and pair mean - field approximations and explicitly takes into account spatial anisotropy .