The effect of lookahead on phase transition in migration of three species with cyclic predator–prey relations

A three-species predator–prey system with cyclic predator–prey relations (also called the rock–paper–scissors game) on a one-dimensional lattice where all individuals migrate in the same direction is studied. Each individual can look ahead within a certain range and can stop its migration when too many predators occur within its lookahead range. Simulation experiments revealed that the three species can coexist within a wide range of model parameter values, providing insights into the dynamical phase transition between coexisting and single phases.

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Introduction
One of the simplest models for predator-prey systems is a three-species system with cyclic predator-prey relations (also known as the rock-paper-scissors (RPS) game [1][2][3][4][5][6][7][8][9][10][11]) on a one-dimensional lattice.Lattice models for predator-prey systems and RPS games have already been studied several times in the literature (e.g.[12,13]).An overview of recent progress in the field of RPS and related evolutionary games is provided in [14], with a particular emphasis on pattern formation, mobility effects and the spontaneous emergence of cyclic dominance.Mean-field and zero-dimensional RPS models are discussed, along with the application of the complex Ginzburg-Landau equation.This article highlights the significance and utility of statistical physics in the successful investigation of large-scale ecological systems.Research on stochastic population dynamics in spatially extended predator-prey systems is reviewed in [15].The impact of random fluctuations and spatio-temporal correlations resulting from the underlying stochastic kinetics is discussed and their effects on species stability, model robustness and the formation of dynamic patterns in coupled non-linear population models are analyzed.The effects of heterogeneity in site-specific invasion rates on biodiversity and global oscillations in spatial models with cyclic predator-prey relations are studied in [16].By examining a model consisting of three species and mutations in one dimension (with two types of mutation cycles: to the respective prey and to the respective predator), a dynamic interplay between equilibrium and nonequilibrium processes is suggested in the stationary state [17].Asymptotically precise explanations of the resulting reactive steady state were developed in conditions characterized by both high and low mutation rates.Furthermore, the paper briefly delves into the potential associations between social system evolutionary game models and biodiversity.These models are often inspired by real-world systems in a variety of fields and can help to explain, for example, when and why (three or more) species in cyclic relationships can coexist in nature [8,[18][19][20][21].However, it is also known that only three species might fail to survive and hence do not exhibit a coexistence phase on a one-dimensional lattice [1][2][3][4].
Movement of individuals and migration are fundamental behavioral properties of many living species and an important factor for the dynamics of ecological systems (see, e.g.[22][23][24]).As models for such systems, cyclic predator-prey systems that consist of moving individuals and species that show migration effects have been investigated using, for example, spatial RPS games [25][26][27][28], the expanded game for five species [29] and related traffic models.The latter are inspired by road traffic and it was shown that traffic jams could induce phase transitions [30].Movement of individuals can be considered as a diffusion process and has been studied with random walk models [28].Some predators use olfactory information (such as pheromones) released by the prey, to locate it and trigger directed movement.The population dynamics of a three-species system with cyclic predator-prey relations and the use of such pheromone information for movement on a one-dimensional lattice was investigated in [31].
In this study, we investigate the effect of lookahead on the individuals of a threespecies system that has migration and cyclic predator-prey relations (i.e. the RPS game) on a one-dimensional lattice.In this system, each individual can move only rightward (unidirectional migration) and only if the number of predators that occur within a lookahead range is less than a certain threshold value.We investigate the influence of the size of the lookahead range and of the corresponding threshold value on the predatorprey dynamics.The obtained experimental results reveal how the dynamical phase transition between the single phase (where only one species survives) and the coexistence phase depends on the size of the lookahead range and the threshold parameter when species migrate.

Model
In our model we consider three species A, B, and C that live on a one-dimensional cyclic lattice with L sites (cells).Each site of the lattice can be occupied by only one individual.A site is in state Y where Y ∈ {A, B, C, ⊘} if the site is occupied by an individual of species A, B, C or is empty, respectively.The species have cyclic predator-prey relations where A predates B, B predates C, and C predates A. In a predation step, an individual on site i can (potentially) be predated by an individual on one of the neighboring sites i + 1 or i − 1.If an individual of type X on site i is predated by another individual of type Y on site i + 1 (or site i − 1), then the individual of type X on site i dies and an off-spring of type Y is created on site i for X = B and Y = A, or X = C and Y = B, or For the simulation experiments, the model is implemented as a one-dimensional cellular automaton.The details of the Monte Carlo simulation rules are described in the following.For initialization, all individuals are distributed randomly on the lattice.The simulation is done for all individuals in parallel.At each time step, the following two successive procedures are executed: (i) migration and (ii) predation.To elaborate, in a Monte Carlo step, all individuals initially undergo migration.Subsequently, after the migration process, all agents are revisited, and they make an attempt to capture prey, provided that it is feasible.This process collectively forms a single Monte Carlo step.
(i) Migration.From a physical perspective, migration is commonly conceptualized as a diffusion process.Consequently, physicists have frequently investigated the migration of bio-species using random walk models, where individuals move to unoccupied sites when available.It is noteworthy that, in nature, migration is not entirely random.Instead, it often exhibits a certain directionality.For instance, predators may travel in a straight line over a considerable distance [22].Similarly, herbivores do not merely engage in random diffusion but also demonstrate directed movement in specific directions.In this context, we consider unidirectional migration in a certain direction.If an individual on site k has a neighbor on site k + 1 (i.e.site is k + 1 occupied), it cannot move to site k + 1 (exclusion principle).Otherwise, if site k + 1 is empty, the individual on site k does not move in two cases: (a) There is a predator on site k + 2 and site k + 3 is occupied.(b) When site k + 1 is empty, the individual on site k counts how many predators are ahead, i.e. can be found on sites k + 2, k + 3, . . ., k + r, where r ⩾ q 0 is a parameter that determines the size of the lookahead range.Let pk (t) be the number of counted predators for site k at time t.If pk(t) ⩾ q 0 , where q 0 > 0 is a threshold parameter, the individual on site k does not move.
Apart from cases (a) and (b), the individual on site k moves rightward according to the following rule: If sites k + 1 and k + 2 are free and site k + 3 is not occupied by a predator, then the individual on site k moves to site k + 2 and otherwise it moves to site k + 1. Observe that the maximum velocity of an individual is v max = 2 cells per time step.These rules for migration are depicted in figure 1. (ii) Predation.In the predation procedure for each individual, its two neighboring sites are considered to determine whether contact with a predator has been established after the migration procedure.If this is the case, the predation occurs as described in the following.Assume that an individual of species A is on site k and let Y be the status of site k − 1 (left) and Z be the status of site k + 1 (right).Predation with respect to the individual A on site k is performed according to the following rules: (1) If Y = Z = A there is no predation, because both neighbors are of the same species A. Analogous rules are applied if the individual on site k is from species B or C. Observe that probabilities r 1 and r 2 determine the offspring rate(s).The rules for predation for (ii) (a) Individual X at site k is prevented from moving to site k + 1 if Y on site k + 2 is the predator of X and Y cannot move to k + 3 due to the exclusion principle.It is crucial to note that X at site k is only allowed to move to site k + 1 when the occupant at site k + 2 is not its predator or when site k + 3 is unoccupied.(ii) (b) Individual X at site k counts the number of predators on sites (k + 2, k + 3, . .., k + r ).Subsequently, X is permitted to move if the number of predators is below the threshold q 0 .X moves to site k + 2 if the site is empty and otherwise x moves to site k + 1.
the case of an individual of species A on site k (in the middle) can be visualized as below (before predation procedure ⇒ after predation occurred):

AAA
no predation CAA ⇒ CCA with probability r 1 and else no predation CAB ⇒ CCB with probability r 1 and else no predation AAC ⇒ ACC with probability r 1 and else no predation BAC ⇒ BCC with probability r 1 and else no predation CAC ⇒ CCC with probability r 2 and else no predation.

Results and discussion
Computer simulations are carried out for a lattice of size L = 600 with predation probabilities r 1 = 0.8 and r 2 = 1.0.The density of a species X ∈ {A, B, C} is defined as the number of individuals of species X divided by the total number of cells L. The density of species X at time step t is denoted by ρ X (t).The total density at time step t is defined as ρ T (t) = ρ A (t)+ρ B (t)+ρ C (t), where T stands for 'total density'.Since in our model the total density does not change, we simply write ρ T for the total density.In each simulation run, the initial densities of the three species are equal, i.e. ρ A (1) = ρ B (1) = ρ C (1) = ρ T /3.All average values in the paper are averages over 50 runs.
Figure 2 shows the average densities of the species with highest (purple), medium (green) and lowest (blue) density at the respective time step for threshold q 0 = 2, size of lookahead range r = 6 and total densities ρ T = 0.6 (in (a)) and ρ T = 0.85 (in (b)).
In the present study, it is worth noting that the error bars for all figures are narrower than the symbols.It can be seen in figure 1(a) that the densities change at an early stage.After about 1650 time steps they reach an equilibrium state and the densities remain constant.Figure 2(b) shows that the densities of the two species approach zero successively and only one species survives with density ρ T .
Figure 3 depicts the spatial and temporal diagrams for a single simulation run with threshold q 0 = 2 and r = 6.Subfigure (a) shows the first 500 time steps and the corresponding steady state situation is shown in subfigure (b) (both subfigures are for total density ρ T = 0.6).The colors black, violet, blue and green represent the empty site and species A, B and C, respectively.In the steady-state configuration, the individuals do not interact with each other because no individual has a prey as its next individual to the right.Thus, predation does not occur and the obtained pattern in the steady-state does not change with time (i.e. a frozen state is reached).The effect of lookahead on phase transition in migration of three species

J. Stat. Mech. (2024) 023212
This is because when the individual is at site k the site k + 1 is free and the number of predators (i.e.species C (green color)) on sites k + 2, . . ., k + 7 is less than 2.However, at t = 96, when the individual of species A is at site k = 104 there are three predators ahead on sites k + 5 = 109, k + 6 = 110 and k + 7 = 111.Thus, the individual cannot move rightward at this time step.Moreover, as it does not have a neighboring C on sites k − 1 = 103 and k + 1 = 105, it is not predated and continues surviving.At t = 96, an individual of species C on site k = 101 becomes a neighbor of the individual of species A on site k = 104 in the migration stepin this case, C moves two sites ahead according to migration rule (c).In the corresponding predation step, the individual of species C eats the individual of species A on site 104 and produces an offspring of species C on the same site.Subfigure (d) shows for a relatively high total density ρ T = 0.85 that predation takes place very often during the first time steps.This is due to many contacts between the individuals.Then, only one of the three species can survive after reaching the steady-state situation (at about 1100 time steps).This phase is called a single phase.Clearly, before a simulation run starts, each of the three species has the same chance of survival.
We now investigate how the steady-state situation depends on the total density ρ T and the threshold parameter q 0 .For each combination of parameter values, the average densities of the three species in the steady state situation over 50 runs are computed.Figure 4 shows the average densities for different threshold values q 0 with r = 6.Dots stand for the average values of the highest, medium and lowest density species, respectively.It is observed that a phase transition from a coexistence phase for low total densities to a single phase for high total densities takes place for values q 0 = 1 and q 0 = 3.For q 0 = 0, no such phase transition can be observed and all species survive (coexistence phase) even for high total densities as long as the total density is smaller than 1.Observe that for total density ρ T = 1, individuals cannot move as all sites are occupied.Hence, lookahead has no influence on the migration process.Thus, only predation happens and due to the cyclic predator-prey relation only one species survives in the steady state.
Figure 5 illustrates the phase diagram for r = 6.It is observed that the critical density for the phase transition shows a sudden decrease at q 0 = 1.For values higher than q 0 = 1, the critical density increases until q 0 = 6.The corresponding critical densities q 0 = 1, . . ., 6 are approximately 0.62, 0.76, 0.81, 0.86, 0.92 and 0.98, respectively.This behavior can be explained, because when the total density ρ T is low, it is very unlikely for an individual to find q 0 or more predators ahead.Thus, all individuals can move forward with a high probability.When ρ T increases, the probability of finding q 0 or more predators ahead also increases.As a result, it becomes less likely that the individual moves forward.This leads to a decrease in contact between individuals and therefore predation becomes less likely.This fact causes the critical density to shift to higher values.
The effect of the size of the lookahead range on the critical density can be seen when comparing figures 4 (for r = 6) and 6 (for r = 4).Subfigures 4(b) and 6(a) (respectively, 4(c) and 6(b)) show that the critical density is similar for both cases of threshold values q 0 = 1 and q 0 = 3.Average densities of three species in the steady state as a function of total density ρ T for different values of threshold q 0 : (a) q 0 = 0, (b) q 0 = 1, (c) q 0 = 3 and (d) q 0 = 6, with r = 6.Shown for each value of ρ T is the average over 10 runs of the densities of the respective species with highest, medium and lowest density.
Figure 7 shows what happens in the model without lookahead, i.e. for r = 0, where each individual always moves forward if the right nearest neighboring site is empty.Then, the transition from the coexistence phase to the single phase occurs at ρ T = 0.5.In a zero-sum RPS game, i.e. in our model, the gain of one player (species) is inherently offset by an equivalent loss experienced by the other player (species) [14,26,32].Conversely, in a non-zero-sum RPS game, a potential exists for simultaneous gains or losses by both players [14,26,27].Here, we present simulation results showing how the phase diagram changes when the predation step of our model (that actually includes predation and reproduction) separates between the predation phase and the reproduction phase to turn our model into a non-zero-sum RPS game.Accordingly, after AB → A0 (or BC → B0 or CA → C0) in the predation phase, the probability that A leaves an offspring in the cell that becomes empty in the reproduction phase is given by parameter g: A0 → AA (or B0 → BB or C0 → CC) with probability g.Thus, by separating the predation and reproduction phases of our model, a non-zero-sum RPS game model is established.Note that parameter g corresponds to the probability of leaving offspring if predation has happened, and its value is taken 1.0 in our original model, i.e. in the zero-sum RPS game.The parameter values in the predation phase are again taken as r = 6, r 1 = 0.8 and r 2 = 1.0.
The influence of parameter g on the phase diagram is shown in figure 8.It is shown that for small values of threshold q 0 (i.e.q 0 ⩽ 2 when comparing g = 1 and g = 0.95), decreasing values of g lead to a broadening of the coexistence phase region.This phenomenon can be attributed to the fact that a lower g value decreases the probability of reproduction and thereby reduces the total number of individuals.This reduces the likelihood of encounters between individuals and decreases the frequency of prey-predator interactions.Consequently, this leads to an extended coexistence phase of the system.For larger values of threshold q 0 (i.e.q 0 ⩾ 3 when comparing g = 1 and g = 0.95), the decreasing density of individuals for small g values will lead to small probabilities of individuals not moving.This leads to a small likelihood of prey-predator interactions and thus a larger coexistence phase.This is different for large values of g where for high density the probability of moving is small even for large values of q 0 .This leads to a high probability of predator-prey interactions and results in a smaller coexistence phase (compare the explanation to figure 5). Figure 8 shows that-as has already been observed for g = 1-for values g < 1 the critical total density first decreases and then increases with increasing values of q 0 .As g increases, the q 0 value at which the critical total density starts to increase shifts to lower values.
In the literature, other alternative non-zero-sum RPS game models have been proposed.As an example, in [32], the role of individual mobility is investigated and the extinction probability is computed as a function of the interaction range.It is found that mobility can either promote or hinder coexistence, depending on the interaction radius.It is further reported that there is an optimal mobility value at which the coexistence probability is significantly larger than in the absence of mobility.This suggests that mobility in the continuous plane plays a positive role in promoting coexistence.It is well-known that these phenomena are absent in lattice-based models.Within our model and as a lattice-based approach, the parameter q 0 serves as a similar function to the interaction radius: figure 8 shows that the critical total density first decreases and then increases with increasing q 0 .As g increases, the q 0 value at which the critical total density starts to increase shifts to lower values.In [32] there is a region of interaction range where the probability of coexistence drops to almost zero and then increases with increasing interaction radius.Then, the probability of coexistence decreases again to zero as the interaction radius increases.However, we do not observe this behavior in our model, so the coexistence phase continues to become broader with increasing values of q 0 .

Conclusion
In this study, the effect of lookahead between migrating individuals of a predator-prey system with cyclic predator-prey relations (RPS game) is studied on a one-dimensional lattice.Three species are considered to be living on the one-dimensional lattice.The movement direction of individuals in migration is rightward if the next neighboring site is empty and the number of predators within the lookahead range is less than a threshold value.Simulations showed that there are two distinct phases in which three species can coexist or only one species survives.Without lookahead, the transition between the two phases happens at the critical density 0.5.With lookahead, the critical density for the transition to the single phase occurs only at higher densities.It has been shown by simulations how the critical density depends on the threshold value and on the size of the lookahead range.The region conducive to coexistence widens as the threshold value increases, indicating that biodiversity can be attained with higher total density.For future work and with respect to applications, it would be interesting to study systems of higher dimension.Clearly, this is a complex task since it has to be defined how lookahead in higher dimensions exactly works and how precisely individuals can move.

( 2 )
If Y = C and Z ̸ = C (or Y ̸ = C and Z = C ) predation of the individual on site k happens with a probability of r 1 , i.e. the individual of species A on site k is removed and a new individual (offspring) of species C is placed on site k.(3) If Y = C and Z = C predation of the individual on site k happens with probability r 2 , i.e. the individual of species A on site k is removed and a new individual (offspring) of species C is placed on site k.

Figure 1 .
Figure 1.Rules for migration.X, Y and Z denote A, B or C, and O represents an unoccupied site.(i)Individual X situated at site k is restricted from moving to site k + 1 because the site is occupied (exclusion principle).(ii) (a) Individual X at site k is prevented from moving to site k + 1 if Y on site k + 2 is the predator of X and Y cannot move to k + 3 due to the exclusion principle.It is crucial to note that X at site k is only allowed to move to site k + 1 when the occupant at site k + 2 is not its predator or when site k + 3 is unoccupied.(ii) (b) Individual X at site k counts the number of predators on sites (k + 2, k + 3, . .., k + r ).Subsequently, X is permitted to move if the number of predators is below the threshold q 0 .X moves to site k + 2 if the site is empty and otherwise x moves to site k + 1.

Figure 3 .
Figure 3. Spatial and temporal diagrams of three species for q 0 = 2, r = 6 and total densities ρ T = 0.6 (subfigures (a)-(c)) and ρ T = 0.85 (subfigure (d)).Subfigure (a) shows the first 500 time steps and subfigure (b) the steady-state situation.Subfigure (c) shows the details of moving and predation processes for sites 10 to 181 over time steps t = 88 until t = 104.Black dots refer to empty sites.

Figure 3 (
Figure 3(c)  shows in detail how moving and predation take place over time in a non-steady-state situation for density ρ T = 0.6.It can be seen that the individual of species A (violet color) which is at site k = 100 for t = 94 moves rightward until t = 96.

Figure 4 .
Figure 4. Average densities of three species in the steady state as a function of total density ρ T for different values of threshold q 0 : (a) q 0 = 0, (b) q 0 = 1, (c) q 0 = 3 and (d) q 0 = 6, with r = 6.Shown for each value of ρ T is the average over 10 runs of the densities of the respective species with highest, medium and lowest density.

Figure 5 .
Figure 5. Phase diagram for the phase transition between the coexistence phase and the single phase with r = 6.

Figure 6 .
Figure 6.Effect of the size of the lookahead range on the position of the transition point.Subfigures (a) and (b) show the same case as figures 4(b) and (c) with the only difference that the size of the lookahead range is r = 4.

Figure 8 .
Figure 8. Phase diagram depicting the influence of g on the coexistence region.