cyclical interactions , the simplest non - trivial example being the children s rock - scissors - paper game , are despite their apparent simplicity fascinating examples of evolutionary processes . examples from real life are the mating strategies of side - blotched lizards , overgrowths by marine sessile organisms , and competitions among different strains of bacteriocin - producing bacteria .moreover , studies of cyclical interactions may provide insights into the formation of defensive alliances and darwinian selection , as well as structural complexity and pre - biotic evolution .thus , cyclical interactions are rightfully acquiring a central role in the study of evolutionary processes , not just in lotka - volterra like predator - prey systems , but also in evolutionary game theory , where strategic complexity often leads to a closed loop of dominance between participating strategies .an interesting concept that has only recently begun to supplement models of evolutionary processes is the addition of stochasticity at some level of interactions .while the origin of stochasticity can sometimes be related to the finiteness in population size , uncertainties may also enter under the assumption of irrationality and errors in decision - making .moreover , the approach is viable when stochasticity in payoffs originates from the heterogeneity of the system , or if explicit payoff fluctuations are considered .the latter examples are closely related to the concept we adopt presently , as stochasticity does not originate from finite populations but is due to uncertainties arising by predator - prey interactions .irrespective of these particularities , however , stochasticity appears to posses the ability of having a profound impact on the evolutionary process .examples range from stochastic gain in population dynamics and eradication of coexistence in the cyclic lotka - volterra model , to cooperation promotion in the spatial prisoner s dilemma game .  already long before making its debut by evolutionary models , stochasticity has been identified as a potentially crucial agonist in certain types of dynamical systems .the so - called stochastic resonance , standing for the resonant noisy enhancement of the correlation between the system s response and a weak external stimulus , has been first reported for bistable systems , and latter on also for a broad variety of other physical as well as biological systems .importantly , noise can play an ordering role even in the absence of additional external signals , whereby the established term describing the phenomenon is coherence resonance . over the years a particular property of dynamical systems , termed excitability , has crystallized as being very beneficial when noise - induced phenomena are a desirable feature .excitability , uniquely comprising elements of slow and fast dynamics thus enabling weak perturbations of the system to result in large - amplitude deviations before recovery , has fueled studies reporting various effects of noise on temporal and spatially extended dynamical systems for over a decade .it is the different noise dependencies of the slow and fast dynamics , constituting a so - called threshold duality , that are responsible for the majority of noise - induced phenomena observed in excitable systems .in particular , while the slow dynamics , representing the lower threshold , is very susceptible even to weak noisy perturbations , the fast dynamics , representing the upper threshold , is not . on the other handhowever , the fast dynamics , resulting in large amplitude excitations , is sensitive to strong noise .consequently , the temporal order of these excitations exhibits a maximum when we gradually increase the strength of noise .although some correlations between the coherence resonance in dynamical systems and noise - guided evolution have recently been established in terms of proximity to special bifurcation points , the search for additional conceptual similarities linking the two avenues of research is still vibrant .this paper links these avenues by showing that a reticulate six - species predator - prey model with heterogeneous invasion probabilities offers adjustment possibilities via fine - tuning of a single parameter that takes the system from a noise frigid to a very susceptible state . in the latter state ,the studied predator - prey model incorporates two thresholds that can be affected by noise , whereby one is small , conceptually similar to the slow dynamics , and the other large , conceptually similar to the fast dynamics of an excitable system . by introducing gaussian noise ,we show that this threshold duality can be readily exploited in a resonant manner , thus indicating conceptual similarities between the presently reported phenomenon of noise - guided evolution within cyclical interactions and that of coherence resonance reported previously for excitable dynamical systems .we emphasize , however , that the outlined similarities are indeed only conceptual , as the quantity presently of interest is not temporal order of the dynamics as by the observation of coherence resonance but the survival chance of different species within the habitat , which resonantly depends on the intensity of noise .moreover , the threshold duality is not explicitly constituted by the slow and fast dynamics as by excitable systems , but as we will show below , is routed in the parameters defining the predator - prey interactions .the remainder of this paper is organized as follows . in the next sectionwe define the microscopic model and summarize the results for the noise - free case .we also describe the introduced gaussian noise and comment on its impact on the elementary process .the details of monte carlo ( ) simulation are provided as well . in sec .3 we present the effects of gaussian noise , and show that at appropriate system conditions the latter can have a large impact on the survival of different species , even reverting the evolutionary process in a resonant manner .in particular , while small and large intensities of noise fail to have a profound impact on the model , intermediate intensities induce a re - entrance effect of the seemingly defeated species , hence decisively guiding the evolutionary process .the non - monotonous effect of different noise intensities is supported by the application of the dynamical mean - field ( ) approximation . to test the generality of our findings, we also perform simulations of the model on different host lattice topologies . at the end of sec . 3, we show that the concept of threshold duality , linking the present results with those obtained previously for noise - driven excitable systems , is not limited to predator - prey cyclical interaction but can be applied to models of evolutionary game theory as well .finally , sec . 4 features a summary of main results and concludes by suggesting that the discovered mechanism for noise - guided evolution is applicable under different circumstances , and could thus prove useful in various fields of research such as ecology or economy , where evolutionary processes are often subject to unpredictable factors .the studied predator - prey model comprises six species that are initially uniformly distributed on the square lattice .the distribution function of species is given by a set of variables , where runs over all lattice sites .predator - prey relations of nearest neighbors and the corresponding invasion probabilities ( ) are defined by the food web presented in fig .[ fig : web ] .more precisely , determines the probability that an even labeled predator will invade an odd labeled prey , and vice versa for , while and determine probabilities of invasions within even and odd species , respectively .as exemplified in for lattice lotka - volterra type models , the studied six - species model can be defined by a concise reaction scheme . in particular , for species  and for species  where  ,  , and . in case of homogeneous invasion probabilities ( )the system has two equivalent three - species states [ denoted by and exhibiting a self - organizing pattern maintained by cyclic invasions .departing from a randomly distributed state where every lattice site is occupied by one of the six species , a spatial organization of growing domains begins .the domains are formed either by the members of the red alliance or the members of the blue alliance ( see fig . [fig : web ] for the color assignment ) . in accordance with the coarsening process on the spatial grid ,one of the two domains is eliminated with equal probability , and the system ends up in the absorbing state consisting of members of either the red or the blue alliance only . in order to characterize the stationary state more precisely , we may introduce the order parameter , whereby ( ) denotes the fraction of species on the spatial grid . here corresponds to the exclusive presence of the blue alliance , while indicates the absolute authority of the red alliance .the two domains are more precisely called `` defensive alliances '' because their members protect each other cyclically against the external invaders .for example , if species invades the allies , in particular by attacking species , its intention is immediately disabled by the species that is superior to both and .thus , the intruder is quickly abolished from the domain by the very same species that dominates species within the alliance .this reasoning applies for all other possible attempts of non - allied species to invade a defensive alliance .noteworthy , in this mechanism the proper spatio - temporal distribution of species plays a crucial role , meaning that the classical mean - field approximation , assuming a well - mixed state , can not reproduce this feature .an interesting possibility is now to study effects of inhomogeneous invasion probabilities , which might affect the evolutionary process in a non - trivial way , essentially forming the basis for noise - guided evolution to be reported below .  since effects of inhomogeneous invasion probabilities on the evolution of defensive alliances in the studied cyclically dominated modelhave already been presented in , we here just summarize the main findings that are essential for the present study , and refer the reader to the previous work for more details .first , it is important to note that heterogeneities in the invasion probabilities are introduced in an alliance - specific way . in particular, we can study what happens when one defensive alliance is more aggressive towards the other ( ) , or when the internal mechanism fails to assure flawless protection against the invaders ( ) . in order to address these two issues systematically , we introduce two control parameters that , due to symmetries in the food web , uniquely determine the stationary state of the system characterized by the order parameter .namely , let and where  ] .as soon as rises above zero ( keeping ) allies are favored as their members invade non - allied species more successfully ( ) , thus simulations yielding in the stationary state .however , the advantage of the blue alliance , given by , may be compensated by choosing sufficiently large values of .especially , if rises above zero the internal invasions within the alliance slow down in comparison to , thus decreasing the effectiveness of the protection shield of the blue alliance and in turn nullifying its advantage given by .surprisingly the results of simulations reveal a non - monotonous phase diagram in dependence on , as shown in fig .[ fig : phd ] .in particular , the advantage of allies again increases if approaches 1 ( the internal invasion probability vanishes ) .note that as allies essentially stop to invade each other within the alliance . via a stability analysis of the interface separating members of the two defensive allianceswe have shown that the penetration of the red alliance into the unfavorable region is a consequence of an interface driven effect .presently , we will show that the spatiality of the system is also a relevant feature when the system is driven from a noise frigid to a noise susceptible state by varying the parameter .  turning to the effects of noise ,we should point it out that the model involves stochasticity already in its present form .in particular , the invasions of predators to neighboring prey sites are not deterministic but characterized by a probability .the latter source of stochasticity is routed in the spatiality of the model , dictating a random selection of one neighbor of each species in order to carry out an elementary invasion process .stochastic effects also can not be neglected if the system size is finite . however , this is a plausible assumption for populations studied within the context of evolutionary game theory . in these systemsthe strategy adoption is also ruled by a probability function that depends on the payoff differences between the two competing strategies .motivated by bounded rationality , often an additional source of stochasticity is introduced by the so - called smooth imitation rule that allows an inferior strategy to replace a more successful one .the application of the fermi function , adopted from statistical physics , allows to measure the intensity of selection via a single parameter , termed accordingly as the temperature of selection . in the the stochastic effects can be neglected , while for limit the stochasticity is maximal . in our predator - prey model , lacking individual payoffs , the stochasticity resulting from finite can be modeled by introducing gaussian noise additively to all invasion probabilities so that the modified invasion probability is where and is a random variable being normally distributed with zero mean and standard deviation .it is worth noticing that the introduced gaussian noise is in fact multiplicatively coupled with the dynamics of , as can be inferred from the classical mean - field equations or the approximation .the gaussian noise satisfies the correlation function where indexes and mark the central location of two predator - prey pairs on the lattice , while and denote two consecutive pair interactions .thus , the stochastic disturbances are delta correlated ( uncorrelated ) in time as well as among predator - prey pairs .importantly , if an invasion probability becomes negative ( ) due to noise we allow backwards invasions ( a prey can occupy a predator s site ) , while potential values of invasion probabilities above 1 ( ) are simply treated as sure events .we thus allow noise - induced reversals of the direction of dominance depicted in fig .[ fig : web ] .each elementary step of the simulation involves two actions .first , two nearest neighbors are chosen at random , and second , if the two neighboring species form a predator - prey pair ( species directly connected by an arrow in fig .[ fig : web ] ) the prey ( or predator depending on ) is killed with the probability , where is the invasion probability specified along the connecting arrow and is the gaussian noise with properties as described above . on the other hand , if the two randomly chosen species form a neutral pair ( species not directly connected by an arrow in fig .[ fig : web ] ) , or if both are identical , the second part of the elementary step dictates no action . in accordance with the random sequential update, each individual is selected once on average during a particular step .first , we study how the phase separation line in the parameter space varies in dependence on . due to symmetries in the food web , and to ensure transparency , we perform all below calculations using and , thus designating and as the only parameters able to change and . figure [ fig : phd ] shows the results , from which it can be inferred that there exist an intermediate for which allies receive the biggest boost , or in other words , are able to compensate for the largest values of by a given .clearly , there exist a resonant dependence on as larger levels of stochasticity fail to have the same effect because the phase separation line gradually converges to the zero - noise limit as increases .  to study the outlined resonant dependence , we plot as a function of , whereby measures the rise of the phase separation line in the vertical direction with respect to its position at at any given .results for different values of are presented in fig .[ fig : res ] . two important conclusions can be made .first , added gaussian disturbances affect the evolutionary process resonantly in dependence on , and second , the phenomenon becomes increasingly pronounced as ( ) .we should also point out that by values of close to 0 noise has absolutely no impact on the evolutionary process .thus , we demonstrate that as the parameter is varied from 0 to 1 the system goes from a noise frigid to a noise susceptible state , and moreover , noise - guided evolution sets in if is appropriately adjusted .our numerical findings can be supported by an alternative approach entailing the application of -site cluster approximation .since the classical mean - field approximation assumes well - mixed populations , it is easy to understand that it can not reproduce above - described results . as we have argued earlier, the spatiality plays a fundamental role by the explanation of the penetration of the red alliance into the unfavorable area .the shortage of the classical mean - field approximation may be eliminated by applying the approximation technique that proved to be very appropriate for obtaining qualitatively correct phase diagrams for several non - equilibrium systems . for a detailed description of this methodwe refer the reader to earlier papers .the approximation is a dynamical version of the cluster - variational method , and it involves finding a hierarchy of evolution equations for the probability distributions of configurations within a cluster of sites . if correlations in larger clusters are neglected then the level of approximation can be characterized by the value of .similarly as by the simulations , in the approach the introduction of gaussian noise can , besides altering the strength of invasions , also revert the direction of dominance .therefore , aside from handling the elementary processes displayed in fig .[ fig : web ] , we also have to consider the possibility of reverse processes . with this extension ,the four - point level of approximation can describe correctly not only the impact of the dynamical benefit of the red alliance , but also the impact of gaussian noise .indeed , displayed as an inset in fig .[ fig : res ] , the re - entrance phenomenon driven by increasing values of is well reproduced , and moreover , exhibits the same dependence on as revealed by simulations . in particular ,values of close to zero yield a virtually noise - resistant dynamics across the whole span of , whereas noise - guided evolution sets in as for intermediate levels of stochasticity . nonetheless , a slight difference between the results of simulations and the approximation can be observed .according to the latter , large values of can eliminate the dynamical benefit of allies , thus resulting in the solution predicted by the classical mean - field theory .an even higher level of approximation might still eliminate this minor inconsistency with the simulations , but unfortunately the resulting number of variables is too large to be manageable by the computer resources currently available to us . in sum ,the failure of classical noise - driven mean - field equations to reproduce the results of simulations and the necessity for the four - point cluster approximation confirm that the reported noise - guided evolution within cyclical interactions is indeed deeply routed in the short - range correlations of the spatial distribution , as conjectured already from the outlay of phase diagrams depicted in fig . [ fig : phd ] and findings reported previously in .in order to strengthen our arguments , we also examine the dynamics of the noise - driven model via simulations on two additional types of regular graphs , namely on the honeycomb and the triangular lattice .our choice is motivated by the fact that the two types of host lattices have different coordination number . while the honeycomb lattice ( ) allows manifesting the importance of spatial correlations , the triangular lattice ( ) may move the system closer towards mean - field conditions .therefore we argue that the noise driven re - entrance phenomenon will be more ( less ) pronounced when the honeycomb ( triangular ) host topology is applied .indeed , results presented in fig . [ fig : gr ] fully support this expectation as the observed phenomenon is substantially better expressed for than for , and thus additionally strengthen the fact that the reported noise - guided evolution within cyclical interactions is routed in the short - range correlations of the spatial distribution .next , it is of interest to draw some analogies between the reported noise - guided evolution within cyclical interactions and the phenomenon of coherence resonance reported previously in excitable dynamical systems .although we presently consider the survival of different alliances as being resonantly dependent on and not the temporal order of their evolution , and are thus strictly speaking unable to write about a classical coherence resonance phenomenon , we still argue a conceptual link can be established by addressing the presently reported phenomenon as an evolutionary coherence resonance .noteworthy , related conceptual differences emerge also in , where the authors employs similar arguments to address the problem . aside from this difference , however , we will show that the threshold duality , constituted by the noise susceptible slow dynamics and the noise robust fast dynamics of excitable systems , can be identified also in the present model , and is in fact crucial for the observation of noise - guided evolution .the threshold duality can be made visible by studying the properties of invasion probabilities along the phase separation line of the deterministic model shown in fig .[ fig : phd ] .we start with close to zero values of where , taking into account results presented in fig .[ fig : res ] , noise has no impact on the evolutionary process irrespective of .there , the switch between the dominance of allies and occurs at and , or equivalently , at and ( note that and are held fixed throughout this work ) .importantly , the differences and define the two thresholds gaussian noise can influence in order to guide the evolutionary process . in particular , if one of the two invasion probabilities or would become equal to 1 , the corresponding defensive alliance would benefit and eventually defeat the other , as can be inferred from the read line depicted in fig . [fig : phd ] ( consider for example the case when while ; clearly the blue alliance wins making in the stationary state ) .however , since at both thresholds have virtually identical magnitude ( and ) , and are exposed to the same intensity of temporally and spatially uncorrelated gaussian noise , the long - term probability of exceeding one threshold while leaving the other untouched is practically zero .thus , noise can not influence the evolution of the two defensive alliances irrespective of , as displayed in fig .[ fig : res ] .however , the situation changes substantially as increases .by the phase separation line is situated at in the noise - free case . the two relevant thresholds then equal and .as above , if gaussian noise would exceed only the red alliance would win ( ) , and via similar reasoning , if only would be exceeded the blue alliance would win ( ) .contrary to the case here the two relevant thresholds differ substantially .therefore it is easy to see that by an appropriate value of the threshold will be exceeded statistically far more often than , which ultimately results in noise - guided evolution , specifically giving the seemingly defeated alliance the winning edge over the allies , as indicated by the positive values of in fig .[ fig : res ] .however , by larger values of the upper threshold will be exceeded often as well , thus again nullifying the dynamical benefit coming from the faster inner cycle for the red alliance , which explains the depicted resonant dependence of .note that although larger increase the probability of crossing the lower threshold ( ) as well , the maximal value of each invasion probability equaling 1 ( above that each event is simply treated as a sure event ) prohibits a noticeable effect of this fact , and thus results in the decline of as increases .we argue that the difference in the two thresholds ( lower threshold given by and the upper one given by ) , constituting a threshold duality in the presently studied model of cyclical interactions , is conceptually similar to the threshold duality observed in excitable dynamical systems , albeit the latter is routed in the fast and slow dynamics of the system , while the former originates directly from the parameters values defining the evolutionary process .  finally , it is straightforward to extend the above argument also to the case when , whereby the noise - free phase separation line is situated at , and hence the two relevant thresholds , equaling and , clearly differ by an order of magnitude thus constituting the vital threshold duality needed for the observation of noise - guided evolution .importantly , an additional threshold emerges related to the possibility of noise - induced negative values of .specifically , the new threshold simply equals to , and is apparently of similar magnitude as .since negative values of flip the circle of dominance within the blue alliance , thus temporarily disrupting its protective shield against the invaders , intermediate values of warrant a two - fold advantage to the red alliance , ultimately resulting in the best - expressed resonance curve in fig .[ fig : res ] . the origin of the two - fold advantage of the red allies can be studied more precisely by measuring the success rate of invasions along the two inner circles ( depicted with blue and red in fig .[ fig : web ] ) of the food web . to be more specific, we calculate the normalized difference of clockwise and anti - clockwise invasions for the red and blue alliance in dependence on .note that if all invasions occur in the anti - clockwise direction ( as depicted by arrows in fig .[ fig : web ] ) and zero if clockwise and anti - clockwise invasions occur equally often . in other words ,the value of characterizes the strength of the defensive mechanism within an alliance .results in fig .[ fig : weak ] show that the defensive shield of the blue alliance suffers virtually immediately after noise is introduced , whereas the red alliance is able to uphold a perfectly functioning protection until .naturally , as increases further the defensive capacity of the red alliance also weakens .however , the window resulting from the delayed faint of the defensive mechanism , in conjunction with the previously described mechanism warranted by the threshold duality via and , enables a resonant strength of the referred domain .results displayed in fig .[ fig : res ] support this argument since limits the peak of the resonance curve obtained for , thus signaling the beginning of end of the noisy support for the red allies .also in accordance with the results presented in fig .[ fig : res ] , for the dependence of on is very similar for both defensive alliances ( red and blue ) , thus clearly signaling the absence of the mechanism warranting noise - guided evolution in the studied model .the simultaneous decrease of the defensive strength results in a noise - frigid state since all invasion probabilities are of similar magnitude .the above - described simple arguments provide a workable mechanism for noise - guided evolution , and at least conceptually unify the phenomenon with the coherence resonance in excitable dynamical systems .  to end this section, we would like to note that the above explanation warranting the resonant dependence of on , intimately relying on the existence of threshold duality in evolutionary processes , is valid for other systems as well . to demonstrate this, we briefly consider a spatial prisoner s dilemma game , where two cooperators receive the reward , two defectors receive the punishment , whilst a cooperator and defector receive the suckers payoff and the temptation , respectively , thus satisfying the prisoner s dilemma payoff ranking if .it is easy to identify a threshold duality in the system if only .if the payoffs become stochastic as a result of additive gaussian noise , inequalities and differing by will be violated statistically much more frequently by a given than the inequality differing by 1 .it is easy to understand that intermediate levels of stochasticity promote cooperation because the violation of and favors the cooperative strategy since it potentially nullifies the advantage defectors have over cooperators ( two cooperators might end up receiving a larger payoff each than a defector facing a cooperator , and a cooperator facing a defector might be better off than two defectors ) . on the other hand, this facilitative effect is limited by violations of , belonging to the so - called social dilemma , because then two defectors might be better off than two cooperators , which again gives the winning edge to the defecting strategy , and hence results in a resonant dependence of cooperation fitness on  .although being fairly simple , the described explanation outlines a general mechanism of cooperation promotion in the spatial prisoner s dilemma game , thus giving another example of noise - guided evolution relying on the threshold duality in evolutionary processes .in sum , we provide conclusive evidences that the threshold duality is crucial for the observation of noise - induced resonances in evolutionary processes .particularly , we show that a reticulate six - species predator - prey model with heterogeneous invasion probabilities possesses a single parameter that is able to guide the system from a noise frigid to a noise susceptible state , thereby relying exclusively on the emergence of threshold duality .the phenomenon revealed by simulations can be verified by a four - point cluster approximation , thus providing ample evidences that the reported noise - driven evolution within cyclical interactions is heavily routed in the short - range correlation of the spatial distribution .this conjecture is additionally supported by studying the evolution of species on different regular graphs , and confirmed by the fact that the phenomenon of noise - guided evolution weakens as the coordination number of the host lattice increases . while the observations in the studied model are related to the system - specific non - monotonous phase separation line between the two defensive alliances , we note that the general concept is valid also in the framework of evolutionary game theory , thus suggesting it may be widely applicable in various fields of research ranging from economy to ecology. m. p. acknowledges support from the slovenian research agency ( grant z1 - 9629 ) .research of a. s. was supported by the hungarian national research fund ( t-47003 , nkth - mfaiif05 ) .

Abstract: we study a stochastic predator - prey model on a square lattice , where each of the six species has two superior and two inferior partners . the invasion probabilities between species depend on the predator - prey pair and are supplemented by gaussian noise . conditions are identified that warrant the largest impact of noise on the evolutionary process , and the results of monte carlo simulations are qualitatively reproduced by a four - point cluster dynamical mean - field approximation . the observed noise - guided evolution is deeply routed in short - range spatial correlations , which is supported by simulations on other host lattice topologies . our findings are conceptually related to the coherence resonance phenomenon in dynamical systems via the mechanism of threshold duality . we also show that the introduced concept of noise - guided evolution via the exploitation of threshold duality is not limited to predator - prey cyclical interactions , but may apply to models of evolutionary game theory as well , thus indicating its applicability in several different fields of research .


many biological or chemical processes involve the dynamics of discrete `` particles '' ( e.g. , molecules or organisms ) that diffuse and interact with each other and/or with an external environment  .in population dynamics , for instance , individuals spread in space , interact , reproduce and die .if the number density of particles is very large , the macroscopic description in terms of continuous fields is typically appropriate . a well established approach to model the spatio - temporal evolution of the population density field , , is in terms of a reaction - diffusion equation that , in one spatial dimension , reads in the above equation , is the diffusion coefficient and rules the chemical kinetics or , in the language of population biology , the local rate of population growth .a well known growth term is the logistic model whose quadratic term models the competition for resources , being the local carrying capacity ( i.e. the maximal population density that can be sustained ) while denotes the intrinsic growth rate .conversely , if the density of particles is not very large the discrete nature of the population can not be neglected  , new effects arise and the continuous description becomes inaccurate . in order to account for the new effects several approachesare possible .for instance , for a population of individuals , a possibility is to introduce a cutoff at the value of the normalized density for the continuous field equations  ( see for a review ) .another possibility is to supplement eq .( [ eq : rd ] ) with a noise term accounting for microscopic fluctuations originated by the finite number of individuals  .a further modeling step is to consider a lattice where each site is occupied by an integer number of individuals  , or a contact process ( see , e.g. ,  and references therein ) , where each site is occupied at most by an individual .in the present work , we consider a system of particles diffusing in space and interacting when they get within a given interaction distance .in particular , revisiting the continuous kiss model , after kierstead , slobodkin and skellam , we investigate the effects induced by discreteness on the dynamics of a population inhabiting a favorable region ( `` an oasis '' ) surrounded by a deadly environment ( `` a desert '' ) .accounting for such effects is important to assess the role of demographic and environmental stochasticity on extinction dynamics  .several efforts have already been undertaken along this direction within the framework of the kiss model .some model modifications that include effects of intrinsic variability on the critical habitat size ensuring population persistence have been studied in relation to environmental stochasticity . as for demographic stochasticity due to population discretenessit has been investigated either using a cutoff on the continuous field equations , in the same spirit of ref . , or within a stochastic framework using field theory techniques  . in our work, we adopt a discrete particle model focusing on demographic stochasticity effects .moreover , our study is not restricted to steady state properties but also deals with dynamical features of extinction phenomena .the article is organized as follows . in section [ sec : surv ] we recall the continuous kiss model  and introduce its discrete analogue .in section [ sec : qss ] we numerically study the long time properties of the discrete model , characterizing the so - called quasi - stationary state , and comparing it with the properties of the continuous model .section [ sec : mte ] is devoted to the problem of determining the critical habitat size for population persistence and to the estimation of extinction times in the discrete particle system . discussions and conclusions are presented in sec .[ sec : concl ] . as a side discussion , in the appendix we briefly consider the effects of changes of the microscopic rules of the discrete model .determining the conditions for persistence or , conversely , estimating the probability of extinction of species and populations is a question of paramount importance in population biology .the theoretical treatment of the problem was initiated by kierstead , skellam and slobodkin who considered a population that , ruled by eqs .( [ eq : rd]-[eq : logistic ] ) , grows with rate and diffuses with diffusion coefficient within a patch of size surrounded by a deadly environment . for a given growth rate ,higher diffusivities imply larger fluxes across the boundaries , so that larger patches are necessary to compensate the population loss in order to allow stable persistence .it is then natural to determine the minimal ( critical ) patch size , , ensuring population persistence .at least dimensionally , the critical patch size can be obtained by balancing the growth rate with the exit rate from the patch due to diffusion  .this way one derives that that is the correct result up to an order one constant ( see below ) .  in one - dimensional space , for simplicity ,it is enough to consider eqs .( [ eq : rd]-[eq : logistic ] ) in the interval , representing the favorable patch , while the outside hostile environment results in the boundary conditions   detailed analysis of the kiss model can be found in , for generalizations of the model including different kind of population growth terms see .here , we only recall the main results on the critical patch size .we notice that , close to the extinction , the population density will be very small ( ) so that we can linearize the growth term ( [ eq : logistic ] ) by posing .starting from a generic initial condition , population density will thus grow ( decay ) depending on the positive ( negative ) sign of the largest eigenvalue of the ( linear ) operator with boundary conditions ( [ eq : ckiss_bc ] ) .standard computation shows that the eigenvalues are , \quad n=1,2 , ... \label{eq : ckiss_ev}\ ] ] with .consequently , the population can persist only if the favorable habitat is larger than the critical patch size .however , while in the linear case also the eigenfunctions and thus the stationary population density can be easily derived , for the logistic case only approximate solutions are known ( see , e.g. , ref . ) .it is worth remarking that proper estimations of , with suitable generalizations of the above model , are used in the design of protected areas for endangered species , in the context of conservation biology  .  in the following we focus on the logistic growth case . for later convenienceit is useful to formulate the continuous model ( [ eq : rd]-[eq : logistic ] ) in non - dimensional variables . from the above analysis it is natural to measure time in units of the inverse growth rate and space in units of . finally , normalizing the population density to the carrying capacity , that is introducing , we can rewrite eqs .( [ eq : rd]-[eq : logistic ] ) as with , where now is the non - dimensional domain size . with these units the critical patch size is .   in recent years, several efforts have been devoted to the extinction problem in populations with a finite number , , of individuals , using non - spatial models  .the factors influencing extinction , through fluctuations and decline of a population , are in principle extremely varied and attributable to a wide class of biological and environmental causes , among which demographic stochasticity  , associated to the unavoidable random occurrence of birth and death events , is one of the most important ones .  in the context of spatially extended systems, some effects of demographic stochasticity on steady patterns of populations have been analyzed using different models  .in particular , in ref . the discreteness was modeled using reaction - diffusion equations with a cutoff on the growth term , and it was found that the critical patch size for survival is larger than in the model without cutoff . a similar effect was found also in ref . , using a stochastic discrete model , where it is shown that the population can go extinct above the critical patch size of the continuous model and an `` effective '' ( finite ) critical patch size can be introduced only when the competition term is weak enough . in our study , similarly to , the discrete population can go extinct for habitat sizes guaranteeing the persistence in the continuous case .however , we will always refer to the critical patch size of the continuous model because , as we will see , this value still marks a transition in the system behavior .we now introduce a discrete version of the kiss model , generalizing the stochastic particle model of ref .  . as an essential prerequisite , when the number of particles is large, the model must reproduce the fkpp ( fisher - kolmogorov - petrovskii - piskunov ) dynamics ( i.e. , eq . ( [ eq : ckiss_lo_nd ] ) in the dimensional version ) . the fkpp equation can be derived from two coupled reaction - diffusion equations for the concentrations and of the species undergoing the autocatalytic reaction summing the dynamical equation for ( ) with that for ( ) gives the fkpp equation , provided and ( i.e. local mass conservation ) .we discretize these reaction - diffusion equations passing from the concentrations to a particle description .thus , we consider particles of type and of type , with fixed .all the particles undergo independent diffusive processes with the same diffusion coefficient , , within the favorable patch $ ] .their positions ( ) diffuse with diffusion coefficient .as for the interaction between particles , a simple way to implement the interaction term is to impose that a particle of type changes into with a probability rate depending on the intrinsic rate , , and on the number of particles within an interaction distance .this is mathematically formalized by the expression where denotes the number of -particles within a distance from the given -particle located at , and the average number of particles ( regardless their type ) within the interaction interval of length ( being the particle density ) . as discussed in ref . , whenever the local particle density is large enough , the probabilistic rule above described converges to the fkpp equation. finally , we have to fix the behavior of - and - particles at the boundaries . in order to reproduce the boundary conditions of the kiss model and locally ensuring mass conservation, we impose the following rules : particles hitting the boundaries are reflected ( nutrients are present only within the patch ) , if a particle hits the boundary it is absorbed ( as it can not survive outside the favorable patch ) and it is replaced in - place by an -particle .the particles are used as `` auxiliary '' particles in order to recover as a limit the continuous fkpp model . from an ecological perspective , considering the particles as nutrient , it would be interesting to study the case in which and particles diffuse with different diffusion coefficients .clearly , in this case the fkpp can not be expected to be the proper continuous description . herewe limit our discussion to the case of same diffusion coefficients for the two species of particles . in the appendinx ,however , as an example we briefly consider the case of non - diffusing particles .  from the algorithmic point of view, the above model consists of two basic steps : the diffusive step in which all particle positions ( ) evolve according to the dynamics , with independent normal random variables ; and the reaction step in which , for each -particle , one counts all the -particles within a distance and changes in with probability . whenever the time step is small enoughthe order in which the two steps are performed is irrelevant .the main difference between the continuous model and the discrete one is that while the ( continuous ) stationary state becomes unstable above the critical patch size , the ( discrete ) absorbing state can always be reached due to demographic stochasticity . in the following sections we will investigate in depth this issue . here, we simply observe that the presence of an absorbing state forces us to perform ensemble averages in numerical simulations .therefore , our simulations consist of a large number , , of statistically independent realizations . as for initial conditions ,each realization starts with particles of each type uniformly distributed within the patch .each realization is then followed until the absorbing state ( ) is reached .the interaction distance , , is chosen to be small compared to the size of the favorable patch but large enough to contain some particles on average ( ) , which provides a lower bound on the minimum number of particles that one can consider , typically . in all simulations we set , and so that we have as reference in the continuum limit the non - dimensional equation ( [ eq : ckiss_lo_nd ] ) .finally , the time step , , is chosen small enough such that the diffusive step is much smaller that the interaction distance ( i.e. ) . here ,we have chosen .we notice that the model is robust with respect to small changes of , while if becomes comparable with the patch size relevant differences arise , as discussed in the appendix .there , we also briefly consider the subtle effects caused by modifications to the rules specifying the dynamics of the individual - based model .at stationarity for the continuous kiss model ( black solid curve ) compared with the local density of -particles , averaged over realizations , in the discrete model ( empty circles ) .the main parameters are : , , .the ( blue ) dashed curve represents the local density of -particles obtained with a different model with fixed particles ( see appendix for details ) . ]  as an example , we show in fig . [ fig:1 ] the good agreement , at least for large , between the stationary population density , , of the continuous kiss model and the long time ( but finite , see sect .[ sec : qss ] ) particle density , obtained by ensemble averaging the spatial distribution of particles in the discrete model .in the discrete model , the presence of an absorbing state ( ) makes population extinction possible for any finite , even when .nevertheless , the time taken by the system to be absorbed can be very long ( for and large enough ) and , in that case , the system reaches a quasi - stationary state .the quasi - stationary distribution is defined as the probability distribution conditioned on non - extinction . in general, it provides interesting information about the properties of systems with an absorbing state but , except in some rare cases , it can not be evaluated explicitly .  : ( a ) and ( b ) , with particles .top panels show the survival probability , see text .bottom panels show the time evolution of the continuous ( ) and discrete ( ) biomasses , the error bars on the latter display the standard deviation , . the ( blue ) dotted curves denote the minimum and maximum values of observed in all realizations .the inset in ( b ) zooms the framed area in the main figure to better show that at long times .[ fig:2],title="fig : " ] : ( a ) and ( b ) , with particles .top panels show the survival probability , see text .bottom panels show the time evolution of the continuous ( ) and discrete ( ) biomasses , the error bars on the latter display the standard deviation , . the ( blue ) dotted curves denote the minimum and maximum values of observed in all realizations .the inset in ( b ) zooms the framed area in the main figure to better show that at long times .[ fig:2],title="fig : " ]  as shown in fig .[ fig:1 ] , the quasi - stationary distribution , apart from fluctuations , essentially reproduces the results obtained with the continuous model , at least when and are sufficiently large .it is then natural to study under which conditions this correspondence holds . in particular , in this section , we focus on the behavior of the biomass per unit length defined as ( see refs . ) the label simply recalls that this quantity is defined in the continuum limit , and allows to distinguish it from the analogous quantity in the discrete case , for which no labels will be used .we observe that asymptotically vanishes when , due to population extinction , and approaches a positive steady value , indicating survival , when .the asymptotic value can be used as an order parameter to define the extinction / survival transition .  in the discrete model , the biomass per unit length ( [ eq : biom_cont ] )is nothing but the ratio between the total number of -particles and the total number of particles , .since is a time - fluctuating quantity , we consider its ensemble average where the brackets denote averaging over different realizations with the same initial conditions and conditioning on survival ( i.e. at each time only the realizations with are considered ) .when conditioning on non - extinction , at long times , approaches a definite value ( fig .[ fig:2 ] ) which is the average over the quasi - stationary distribution .  in the bottom panels of figure [ fig:2]a ,b we compare the time evolution of and for two different values of . for the discrete case we also show , as error bars , the standard deviation of the ratio , in the top panels , we show the probability , ( measured as the fraction of realizations such that at time ) , that the population survived up to time .  forany given , when is large enough ( fig .[ fig:2]a ) , essentially coincides with the continuum - limit and all realizations survive ( in the observation window ) for long time . in these circumstances , the minimal values of ( lower blue dotted curve in fig .[ fig:2]a ) are well above zero , implying that very unlikely fluctuations are needed to reach the absorbing state .conversely , as approaches the critical value ( fig .[ fig:2]b ) this is no longer true . as shown in the top panel of fig .[ fig:2]b , exponentially decays to zero at a relatively short time .more importantly , the agreement between and the continuum limiting value gets poorer . in particular , at long times , approaches a quasi - stationary value which is larger than the corresponding continuum one ( fig .[ fig:2]b , inset ) .this is clearly a consequence of conditioning on non - extinction which leads to an overestimation of the biomass , for these values of the patch size .   andaverage biomass in the quasi - stationary state , , as a function of for different values of as in legend .the solid curve displays that corresponds to the behaviour eq .( [ eq : biom_l ] ) . the factor0.75 can not be catched by the argument exposed in the main text .inset : standard deviation of the biomass ( [ eq : bfluct ] ) in the quasi - stationary regime , , as a function of for two values ( and ) of the patch size .the number of realizations used for the averages ranges from to depending on the value of .[ fig:3 ] ]  the latter observation becomes more quantitative when looking at fig .[ fig:3 ] where we compare , as a function of , the limiting value with the quasi - stationary mean value , estimated by averaging over the time interval in which it fluctuates around a constant value ( see fig .[ fig:2 ] ) .we have also computed using the algorithm proposed in ref . , which directly probes the quasi - stationary distribution , obtaining ( not shown ) indistinguishable results .the plateau , observed when , is essentially due to conditioning on non - extinction .the effect is stronger the smaller is the number of particles .the dependence of on the number of particles clearly demonstrates the departure from the continuum limit when approaches the critical value .however , for any there exists a value of ( which diverges as ) above which tends to the continuous value .  when the population survive , it is possibile to show , by using a meanfield - like argument , that the biomass of the continuous field can be approximated by as confirmed in fig .[ fig:3 ] .the idea is to approximate the right hand side of eq .( [ eq : ckiss_lo_nd ] ) as , where denotes the steady state spatially averaged value .the first eigenvalue of this problem , , is linked to the eigenvalue of the linear kiss model ( [ eq : ckiss_ev ] ) via . to have a stationary solution one has to impose , so that .but at stationarity , which implies ( [ eq : biom_l ] ) . of course this kind of argument can not be expected to work when is close to 1 .it is worth observing that , for , eq .( [ eq : biom_l ] ) implies that the continuous biomass is linear in .   and three values of as in label .the solid curves display the gaussian distribution obtained using the mean and standard deviation values measured from data .we observe a fairly good agreement at least for large enough .when and/or is not large enough the agreement ceases to be so good , as one can observe from the left tail at . [ fig:4 ] ]  the inset of fig .[ fig:3 ] shows the standard deviation of the biomass in the quasi - stationary state , , at varying , for two values of . at least for and large enough ,the numerical results indicate that .we notice that the inverse square - root dependence of on implies for the standard deviation of the number of -particles , suggesting that the distribution of particles is gaussian , as confirmed in fig .[ fig:4 ] .this observation will be exploited in the next section .in the continuous kiss model , the critical patch size discriminates between populations able to persist ( ) and those doomed to extinction ( ) . for its ecological importance, many efforts have been devoted to understanding how this critical length is modified in less idealized situations , e.g. taking into account more complex heterogeneous environments  and/or the presence of external factors like fluid advection  . on the other hand ,as previously mentioned , in the case of the individual - based model extinction can happen even when .it is then natural to wonder how the survival / extinction transition of the continuous kiss model translates in the behavior of the average time to extinction  , which is linked to the survival probability as  .computations of mean extinction times have been mainly carried on in non - spatial models , where the dependence of the extinction time on the number of individuals at varying the population growth rate has been obtained using diffusive approximations .these results have been influential for their rather simple mathematical formulation and generality .however , it has been recently recognized that the diffusive approximation can give wrong answers .  in what followswe examine the dependence of on the population size in the extinction ( ) , critical ( ) and persistent ( ) regions of the continuous kiss model .the different regions , indeed , are characterized by unalike functional dependencies of the time to extinction on as shown in fig .[ fig:5 ] .we observe that quantitatively depends on the initial condition , however the dependence on and , which is here explored , is qualitatively independent of the initial condition , provided the initial population is not too small in order to avoid spurious fast extinctions .therefore , we always considered the initial population . an alternative strategy could be to start from the quasi - stationary state .however , when is small and/or , as seen in the previous section , the quasistationary state is not so well defined from a physical point of view .  , vs population size , , for different patch sizes .panel ( a ) from bottom to top , encompassing the extinction ( empty red symbols ) and persistence ( blue filled symbols ) regions , as well as the critical point ( black half filled symbols ) of the continuous model .the continuous red lines display the prediction ( [ eq : te_ext ] ) with fitted from data ; the dashed blue lines correspond to the exponential behavior ( [ eq : te_pers ] ) with fitted .for we found a power - law behavior , , with best fitting exponent ( see text ) .panels ( b ) and ( c ) display a zoom of the main figure , in appropriate semilog scale , of both the logarithmic ( [ eq : te_ext ] ) and the exponential ( [ eq : te_pers ] ) regime , respectively .we emphasize that the slope in ( b ) is not fitted but obtained from the first eigenvalue .depending on the value of the number of realizations ranges from to .[fig:5 ] ]  let us first consider the extinction region ( ) . in this case , for the continuous model , extinction is signaled by the density field exponentially going to zero with rate given by the first eigenvalue , . in particular , at sufficiently long times we have that the biomass per unit length behaves as , with some constant . in the discrete model ,it is natural to assume that extinction will take place at a time such that , implying , apart from a constant , the logarithmic dependence on  which is consistent with the numerics , at least for large enough ( fig .[ fig:5 ] ) .we observe that the logarithmic behavior ( [ eq : te_ext ] ) is also found in non - spatial models .conversely , for , displays an exponential dependence on ( fig .[ fig:5 ] ) , this is a quite robust feature observed in non - spatial models where it is derived either with the diffusive approximation or more rigorous large- approximations ( see ref . for a compact review ) . here , the spatial extension complicates the use of those well established techniques .however , we observe that the exponential behavior for the time to extinction is consistent , at least for and/or large enough , with the gaussian behavior of the quasistationary distribution observed in the previous section ( see fig . [ fig:4 ] ) . indeed, we can assume and it is reasonable to expect that the extinction time will be controlled by the small tail of the quasistationary distribution .roughly , the idea is that if before the extinction a well defined quasistationary state sets in , then the extinction will take place when  the extinction takes place with certainty . ] and the time to reach this state will be proportional to the inverse of its probability so that .hence , using ( [ eq : pnb ] ) we obtain where we have used that , with independent of ( as is true for large , see fig .[ fig:3 ] ) , and as from the inset of fig .[ fig:3 ] .numerical findings show that the logarithmic and exponential behaviors are separated by the power - law at the critical patch size of the continuous model .in particular , our best fit gives , which is not far from the analytical estimate found in non - spatial stochastic logistic models  .however , while not large , the difference in the value of the exponent is nevertheless clearly measurable and might be due to the spatial structure of the system under study .( color online ) mean extinction time as a function of the patch size for ( red circles ) , ( orange triangles ) and ( blue squares ) . in the inset versus compared to the asymptotic prediction ( [ eq : te_ext_l ] ) ( solid black curve ) .the vertical dashed line marks the critical value .the number of realizations used for the averages ranges from to depending on the value of . ]we conclude briefly discussing the dependence of on ( fig .[ fig:6 ] ) . when , the extinction time is mainly controlled by the eigenvalue of the continuous kiss model . indeed , from eq .( [ eq : te_ext ] ) , using the expression of we have which is in fairly good agreement with the numerical data when is large ( inset of fig . [ fig:6 ] ) .we studied an individual - based reaction - diffusion system of ecological interest .we focused on a model of a discrete population in a favorable patch , surrounded by a deadly environment , which recovers the kiss model in the continuum limit .the presence of an absorbing state together with demographic stochasticity makes population extinction possible even when the continuous model would allow for population survival .in such a case , however , the system attains a quasi - stationary state that we have investigated at varying the system and population sizes . we have shown that above the kiss critical patch size the biomass in the quasi - stationary state recovers the continuum limit value only if the population is large enough . on the other hand , when the population has a small number of individuals , the link between biomasses , defined in the continuous and particle models , ceases to exist .in particular , when the patch size tends to the critical value the number of individuals required to recover the continuum limit diverges .  moreover , we have shown that the transition from extinction to survival translates , in the discrete model , in the transition from a logarithmic to an exponential dependence of the average time to extinction , , at varying the population size , . at the transition ,these two behaviors are separated by a power - law dependence with an exponent definitively different from the analytical prediction obtained in the non - spatial logistic model .we conclude mentioning that it would be interesting in the future to investigate the effects of population discreteness in more complex heterogeneous environments and possibly in the presence of advection . in such cases , besides the survival / extinction transition , we expect that the discreteness of the population will impact in nontrivial ways on the spatial propagation properties of the population .in this appendix we consider how two different modifications in the individual - based model affect the system dynamics : the first one regards variations of the interaction distance  in order to check the robustness of the particle model , the second one regards particles motility and consists in considering fixed a particles to investigate changes in the continuum limit .   for at varying as from the legend and considering non - motile a - particles .the power law behavior , identifying the extinction / survival transition , is robust for variations of the interaction distance around the reference value ; for much larger values ( ) the critical behavior is considerably altered .the same happens if one considers a model in which particles of type do not move ( filled diamonds ) . ]the interaction distance is a crucial parameter for the individual - based model and it is associated to fluctuations of the number of individuals .a significant variation of should thus have important consequences , through its impact on the effective growth rate . in figure [ fig:7 ]we present a comparison between numerical results obtained in the reference case , used in this study , and those obtained with different values of . for illustrative purposesonly , we focus on the average extinction time as a function of by fixing at the critical value .as shown in the figure , the power - law scaling ( with ) , and hence also the value of the critical patch size in the discrete model , is robust if the interaction distance varies within the same order of magnitude ( and ) .the situation is different when becomes comparable with the patch size : is no longer described by a power law ( see the case in fig .[ fig:7 ] ) meaning that the discrete model is no longer at the critical point . indeed at large the curve bends towards a logarithmic shape as in fig .[ fig:5 ] when .actually , a similar tendency towards smaller values of characterizes the large behavior for , although the effect is so weak that it is difficult to be seen in the figure .the faster extinction is likely due to the fact that increasing causes the hostile boundary conditions to be felt deeper inside the favorable patch . in other words , while on average the number of particles within gets larger , the fraction of particles does not , due to its depletion close to the boundary .as for the modifications of the motility of particles , we study a case in which particles are fixed , and the only other change in the particle model is that , in order to preserve the total number of individuals when a particle is absorbed at the boundary , a new particle is introduced at a random position ( uniformly in the patch ) .this model has a biological justification in the case in which individuals of type ( the nutrient ) are non - motile ( e.g. a plant species ) and individuals of type belong to a motile species feeding on it .the new model shows a deep alteration in the critical behavior at ( see the filled diamond symbols in fig .[ fig:7 ] ) and the continuum limit of such a model is not given by eqs .( [ eq : rd]-[eq : logistic ] ) ( see also fig . [ fig:1 ] ) because particles of type and type do not undergo the same diffusive process .one can expect that in this case the effective growth rate is reduced due to less frequent encounters between individuals of different types particles close to the boundaries have less chances to turn into particles , and extinction times result to be smaller with respect of the original particle model .this highlights once more that the relation between individual - based and continuous reaction - diffusion models is a delicate one , which needs to be carefully taken into account when modeling the evolution of biological populations .

Abstract: understanding the conditions ensuring the persistence of a population is an issue of primary importance in population biology . the first theoretical approach to the problem dates back to the 50 s with the kiss ( after kierstead , slobodkin and skellam ) model , namely a continuous reaction - diffusion equation for a population growing on a patch of finite size surrounded by a deadly environment with infinite mortality  i.e. an oasis in a desert . the main outcome of the model is that only patches above a critical size allow for population persistence . here , we introduce an individual - based analogue of the kiss model to investigate the effects of discreteness and demographic stochasticity . in particular , we study the average time to extinction both above and below the critical patch size of the continuous model and investigate the quasi - stationary distribution of the number of individuals for patch sizes above the critical threshold .