Nash equilibria for attack and defense in a single predator and a single prey game

We first examined attack and defense behaviors between a single predator and a single prey species in a grid environment (Figs 1A, S1A). Grid worlds, commonly used in artificial intelligence and reinforcement learning, are abstract environments composed of lattice-like arrangements of squares [38], and are also referred to as lattice models in the field of ecology [39]. In this setting, agents moved between adjacent squares at a constant speed (default speed = 1), sensed the opponent within a certain range, and changed their speed accordingly. The available behaviors were decreasing the speed to 0, maintaining it at 1, or increasing it to 2, corresponding to ambush, no change, and chase behaviors for predators, and freeze, no change, and escape behaviors for prey. The predators and prey were nicknamed “wolves” (w) and “sheep” (s), respectively. The variables and measurements used in the simulations are listed in Tables 1 and 2, respectively.

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Table 1. Variables of simulations.

https://doi.org/10.1371/journal.pcbi.1013730.t001

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Table 2. Measurements of simulations.

https://doi.org/10.1371/journal.pcbi.1013730.t002

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Fig 1. Attack and defense behaviors with the sensory–motor algorithm between a single predator and a single prey.

A illustrates attack and defense behaviors between a single predator and a single prey (agents) in the grid world. Three time steps are shown (t1, t2, t3). The dotted lines illustrate their sensory abilities of detection ranges (D_w = 2 for the predator [wolf] and D_s = 2 for the prey [sheep]). At t1, when both agents detect no opponent, they move 1 grid per time step. The shaded grids represent possible positions they could move to at the next time step. If they detect the other at t2, they adjust their speeds by S_i (i = w for wolf and s for sheep), changing their speed to 1 + S_i. Aa shows a case where S_w = S_s = 1. The wolf chases the sheep (1 + S_w = 2), and the sheep escapes (1 + S_s = 2). If both occupy the same grid cell, as shown by the arrows in Aa-t2, the wolf eats the sheep at t3. For the illustrations of the wolf and sheep, we used open-source images from Openclipart (https://openclipart.org/detail/254708/wolf-silhouette-2; https://openclipart.org/detail/174830/sheep). B illustrates landscape diagrams that depict encounter probabilities based on combinations of detection distance (D_i). The diagrams show nine combinations, ranging from D_s = 0 and D_w = 2 (a) to D_s = 2 and D_w = 0 (i). In each landscape, the x-axis represents the sheep’s speed change (Ss), while the y-axis represents the wolf’s speed change (Sw). Encounter probabilities (p) associated with different combinations of D_i and S_i are color-coded within each landscape. High and low probabilities are indicated in dark brown and dark blue, respectively. C presents the Nash equilibrium behaviors on the same coordinates as the landscape diagrams. The positions of circles in each diagram indicate the prey’s and the predator’s Nash equilibrium speed changes (_s and _w) on the x and y axes, respectively (a–i). The diameter of each circle reflects the occurrence probability of the behaviors (_i). Blue circles represent cases where Nash equilibrium behavior is uniquely determined (diameter of 1), while orange circles indicate cases where behaviors are performed probabilistically.

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Each agent was characterized by its sensory ability (detection distance, D_i) (i = w and s) and motor ability (magnitude of speed change, S_i) upon detecting an opponent. However, agents had no information regarding their own heading or the relative position of the opponent, thus operating without directional information. Different combinations of D_i and S_i determined the behavioral responses of the agents. Fig 1A illustrates a representative example in which the predator chases and the prey escapes (D_s = D_w = 2 and S_s = S_w = 1). Initially (t1), neither agent detects the opponent, as they are outside each other’s detection range (indicated by the dotted lines), and they move to adjacent shaded grids, which represent their possible next positions. At the next time step (t2), both agents detect each other, increase their speed to 2, and move to new shaded grids. If both agents are located at the same grid at the final time step (a-t2), they encounter and predation occurs (t3). To simplify the analysis, we defined “encounter” as synonymous with predation. Other scenarios, including ambush-escape (S_w = −1, S_s = 1), ambush-freeze (S_w = −1, S_s = −1), and chase-freeze (S_w = 1, S_s = −1) combinations, are also illustrated (S1A Fig).

Rather than simulating agent movements over multiple time steps in a finite grid, we calculated encounter probabilities (p) based on a single-step forward transition from initial positions where interactions could occur, under different combinations of D_i and S_i. p was calculated as (number of encounters)/(total number of cases), enumerating all initial predator–prey positions and both agents’ post-first-step headings under detection-informed and uninformed behavior (S1 File). When D_i = 2, agents could detect each other at the outer edge of the detection range, and combinations in which they were adjacent were excluded. In our setting, because encountering a predator leads directly to predation, p was equivalent to the probability of predation, and 1 − p to the probability of evading predation. Landscape diagrams illustrating p values for nine combinations of D_i are shown in Fig 1B. These diagrams show that, under the sensory–motor algorithm without directionality, combinations of behaviors produce diverse encounter probabilities. Comparing the colors across small squares showed that p depended not only on the agent’s own S_i but also on the opponent’s S_i and both D_i values.

We then regarded p and 1 − p as the predator’s and prey’s payoffs, respectively, and constructed 2-player payoff matrices (S1 Data, S11 Data). Because the sum of payoffs was constant (p + (1 − p) = 1), the games were noncooperative constant-sum games. To determine game-theoretic solutions, we calculated the Nash equilibria from these payoff matrices for each D_i combination. The Nash equilibrium values of S_i are denoted as _i for each agent. The _i values for all D_i combinations were plotted at corresponding coordinates on the payoff landscapes (Fig 1C, S2 Data).

We identified a single different Nash equilibrium behavior as a pure strategy, depending on the combination of the agents’ sensory abilities. In particular, the prey’s sensory ability (D_s) had a substantial effect on the Nash equilibrium behaviors of both agents. For the predator, _w tended to depend on the prey’s D_s. When D_w = 1, predators adopted ambush (_w = −1) if D_s = 0 (Fig 1Cd), and no change (_w = 0) if D_s = 1 (Fig 1Ce). When D_s = 2, predators showed no preference, with all _w value providing equivalent payoff (Fig 1Cf and 1 Ci). For the prey, _s tended to depend on its own D_s. When D_s = 1, prey adopted escape (_s = 1) (Fig 1Ce and 1Ch). When D_s = 2, prey adopted freeze (_s = −1) (Fig 1Cf and 1 Ci). Because freezing at long range is counterintuitive when prey have a detection-range advantage, we recomputed p including adjacent configurations at D_i = 2, which is equivalent to permitting a two-step interaction (S2 File). Under this setting, the prey’s Nash equilibrium became escape when the prey’s detection range exceeded the predator’s (S1Cf and S1Ci Fig), indicating that the previous freeze was due to the single-step assumption, and that the more realistic escape equilibrium was identified once two-step interactions were allowed, consistent with the NetLogo model described in the next section. Similar to predators, when D_w = 2, prey showed no clear preference across _s values (Fig 1Ca and 1Cb).

When both agents detected each other two grids apart (D_w = D_s = 2), they adopted mixed strategies, changing their speeds probabilistically rather than deterministically (Fig 1Cc). The occurrence probabilities (_i) for each behavioral choice were indicated by the diameters of the orange circles. Specifically, prey escaped (_s,1 = 1) with a probability of 0.363, maintained speed (_s,₂ = 0) with 0.499, and froze (_s,₃ = −1) with 0.139. Similarly, predators chased (_w,₁ = 1) with 0.363, maintained speed (_w,₂ = 0) with 0.499, and ambushed (_w,₃ = −1) with 0.139. In contrast, when neither agent detected the opponent (D_w = D_s = 0), no meaningful behavior change occurred and _i could take any value (Fig 1Cg).

Nash equilibria for attack and defense based on multiple predators and prey interactions

The game in the grid world did not incorporate agent densities, costs dependent on sensory and motor traits, or costs dependent on the resulting speed. Therefore, drawing on concepts from evolutionary game theory [40–42], we developed an IBM that considers the effects in which multiple agents move, eat food, reproduce, pay costs, and die [35]. Using a programming language and an integrated development environment called NetLogo [43], we created a simulator to empirically solve the game (Fig 2A and 2B, S1 Movie, S3 File). This approach is called empirical game-theoretic analysis [44].

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Fig 2. Attack and defense behaviors with the sensory–motor algorithm between multiple predators and prey.

A illustrates the agents’ behaviors—predators (wolves, w) and prey (sheep, s)—in the NetLogo world. Both agents move at a default speed of 1 per time step. When they detect the opponent within a detection distance (D_i) (i = w or s), they change their speed in a new speed of 1 + S_i. D_w and D_s are shown as red and blue circles, respectively. In this illustration (S_w > 0 and S_s > 0), they increase their speeds, corresponding to chase in predator and escape in prey, respectively. For the illustrations of the wolf and sheep, we used open-source images from Openclipart (https://openclipart.org/detail/254708/wolf-silhouette-2; https://openclipart.org/detail/174830/sheep). B shows the NetLogo world, where green grass grows on brown soil. Wolves (in black) move, lose energy, gain energy by eating prey, reproduce, and die if their energy reaches zero. Sheep (in white) move, eat grass to gain energy, reproduce, and die either from predation or when their energy reaches zero. C represents landscape diagrams showing encounter frequency (E) between predators and prey with the values of the behavioral cost coefficient (c_b = 0.02) and reproduction coefficient of the predators (r = 3.6). Sixteen landscapes depict different combinations of agents’ detection distance, D_i (= 0, 1, 2, 3). In each landscape, the x-axis represents the prey’s speed change (S_s) and the y-axis represents the predator’s speed change (Sw). E is color-coded. Darker brown represents higher frequencies and darker blue indicates lower frequencies. D and E show landscape diagrams for prey and predator payoffs (P_s and P_w), respectively, following the same D_i and S_i combinations, and color codes as in C. F depicts Nash equilibrium behaviors. The circle positions represent the Nash equilibrium speed change of prey (_s) and predators (_w) on the x and y axes, respectively, in each diagram. Circle diameters reflect the occurrence probabilities (_i) of these behaviors. Blue circles indicate uniquely determined behaviors (_i = 1). Orange circles represent probabilistic behaviors (0 < _i < 1) which result in four behavioral pairs located at the corners of the landscape: chase and escape (1), ambush and escape (2), ambush and freeze (3), and chase and freeze (4) in Fg.

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In the IBM, the agents had a sensory trait (d_i) and a motor trait (S_i), both defined without directional information. The sensory ability (detection distance D_i) was determined as D_i = d_i × f, where f is a constant coefficient representing the sensory sensitivity (factor sensitivity, f = 10), and D_i took values of 0, 1, 2, and 3. We introduced the coefficient f to normalize trait values to a common range (0–0.3) between the sensory–motor (d_i) and non-sensory motor algorithms (K_i, where K_i represents the probability of speed change), enabling direct comparison. The motor traits, representing changes in speed, S_i, took values from −1–1 in increments of 0.2 (−1, −0.8, −0.6, −0.4, −0.2, 0, 0.2, 0.4, 0.6, 0.8, 1). The cost dependent on the sensory and motor traits was the sum of the sensory trait (d_i) and the absolute value (abs) of the S_i multiplied by the behavioral cost coefficient (c_b); that is, (d_i + abs(S_i)) × c_b, which corresponds to the cost associated with implementing neuronal circuit mechanisms for the behaviors. The cost dependent on the resulting speed was (1 or 1 + S_i) multiplied by a constant, that is, the movement cost coefficient (c_m). Therefore, different values were subtracted from the energy of the agent at each time point when the agent changed or did not change its speed.

Combinations of D_i and S_i determined the agents’ behaviors. By design, predator and prey agents were homogeneous and shared identical sensory and motor traits. These behavioral combinations influenced the population-level number of predator–prey encounter events per time step (encounter frequency, E), which in turn affected the numbers of agents (N_i). We assumed interactions occurred in a homogeneous, well-mixed population. Under this assumption, we approximated per-agent encounter number by the population-level E as an intermediate variable and applied E uniformly when evaluating payoffs. Using E and N_i, we then computed the encounter probability at each time step.

In the NetLogo program, E and N_i were recorded after agent interactions. The number of predators (N_w) included both encountered and unencountered predators, whereas the number of prey (N_s) excluded prey that were removed due to predation. Thus, predator and prey encounter probabilities (p_w and p_s) were calculated as E/N_w and E/(E + N_s), respectively.

In general, predation consists of both pre- and post-encounter processes [35], but in the IBM, these processes were simplified by treating the encounter itself as predation. In this framework, predators gain energy by eating prey, and prey gain energy by eating grass after evading the encounter in the IBM (S1 Table). Under the assumption of uniform interactions in a well-mixed environment, we approximated per-agent payoffs by the population-level payoffs based on the encounter probabilities. Accordingly, we defined agents’ payoffs (P_i) as follows: predator payoffs (P_w = p_w), and prey payoffs (P_s = 1 − p_s). Because P_i fluctuated over time, we averaged P_i to obtain statistically stationary values after the transient in the simulation had settled. These calculations mapped behavioral combinations (D_i and S_i) to the outcomes (P_i).

Fig 2C depicts the landscapes of E with the values of c_b (= 0.02) and the reproduction coefficient of the predators (r = 3.6). When D_s = 0 or 1, the E landscape shifted smoothly with changes in S_i (Fig 2Ca, 2Cb, 2Ce, 2Cf, 2 Ci, 2Cj, 2 Cm, and 2Cn). However, when D_s = 2 or 3, E showed discontinuous drops at S_s = 0 (Fig 2Cc, 2 Cd, 2 Cg, 2Ch, 2Ck, 2Cl, 2Co, and 2Cp), indicating a marked decrease in E when prey changed speed. The P_s landscape was similar to the pattern of E with inverted colors (Fig 2D); prey obtained greater payoffs when they modified their speeds (Fig 2Dc, 2Dd, 2Dg, 2Dh, 2Dk, 2Dl, 2Do, and 2Dp). By contrast, the P_w landscapes shifted smoothly and continuously without discontinuous transitions (Fig 2E).

Although the IBM included many agents, we treated the entire population as a pool of potential players engaging in pairwise contests in a two-player game [45], rather than assuming that every individual was a player. Under a homogeneous, well-mixed, statistically stationary population, we used the population-level per-step encounter probability to define per-agent normalized payoffs P_i. Unlike the standard construction that builds a payoff matrix from an isolated single pairwise interaction, this pool-of-players approach aggregated simultaneous multi-agent interactions, thereby capturing density-dependent (population-mediated) effects in predator–prey games that the single-pair method did not reveal. The resulting bimatrix of P_i was constructed as the payoff matrix of a game in which each agent selected one of 11 motor strategies (S_i), conditioned on its specific sensory ability (D_i) (S3 Data). Then, we computed Nash equilibrium speed change _i values (S4 Data).

This analysis confirmed that Nash equilibrium behaviors depended on combinations of agents’ sensory abilities, particularly the prey’s. We plotted _i values on the payoff landscape (Fig 2F). Blue circles represent pure strategies with a probability of 1 (_i = 1). For the predator, _w tended to depend on the opponent D_s. When D_s = 0, _w > 0 (chase) was preferred (Fig 2Fa, 2Fe, and 2Fi). When D_s ≠ 0, _w < 0 (ambush) was preferred (Fig 2Fb, 2Fc, 2Fd, and 2Ff). For the prey, _s tended to depend on D_s. When D_s ≠ 0, _s > 0 (escape) was adopted (Fig 2Fb, 2Fc, 2Fd, 2Ff, 2Fh, 2Fj, 2Fk, 2Fl, 2Fn, 2Fo, and 2Fp). When the predator did not detect prey or detected prey only at close ranges, or the prey detected predators from afar (D_w = 0, 1, 2, and D_s = 3, or D_w = 0, 1, and D_s = 1, 2), the predator was extinct; therefore, the _w was not determined and assumed any value (Fig 2Fh, 2Fj, 2Fk, 2Fl, 2Fn, 2Fo, and 2Fp). When the prey did not detect the predator or change its speed (D_w = 1, 2, 3, and D_s = 0), the prey adopted _s = −1, −0.8, 0.8, or 1, where high costs were required for the motor traits (Fig 2Fa, 2Fe, and 2Fi). Therefore, depending on D_i, the _i value was determined; the effect of D_s on _i is noticeable.

When both agents detected each other from afar (D_w = D_s = 2), sets of probabilistic behavior switching constituted Nash equilibria located at the diagram corners (Fig 2Fg, orange circles). Four types of behavioral pairs identified probabilistically: chase×escape, ambush×escape, ambush×freeze, and chase×freeze. The predators chased (_w,1 = 1) with a probability of 0.462 and ambushed (_w,2 = −1) with 0.538. The prey escaped (_s,1 = 1) with 0.605 and froze (_s,2 = −1) with 0.395 (Table 3). The effects of Nash equilibrium behavioral switching were further examined in a subsequent section.

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Table 3. Occurrence probabilities of behavioral switching in multi-agent predator–prey interactions when both D_w and D_s are 2.

https://doi.org/10.1371/journal.pcbi.1013730.t003

When both agents failed to detect opponents and did not change their speeds (D_w = D_s = 0), _i values were located at diagram corners with large absolute values of motor traits (Fig 2Fm). This was likely because the high cost of the motor traits reduced E (Fig 2Cm), indirectly increasing P_s (Fig 2Dm). Furthermore, Nash equilibrium results for all c_b (0, 0.001, 0.02, 0.04) and r (3.2, 3.6, 4.0) are shown in S2–S5 Figs, demonstrating that distinct Nash equilibrium combinations were identified under varying conditions.

The Nash equilibrium behavioral payoffs in attack and defense.

To examine effects of the Nash equilibrium behaviors on agents, we assigned each agent sets of detection distance D_i and Nash equilibrium speed change _i (S5 Data) to perform the Nash equilibrium behaviors in the IBM (S4 File) and measured encounter frequency (E), numbers of agents (N_i), and the Nash equilibrium payoffs (_i) (S6 Data).

In this framework, the game based on the multiple predators–prey interactions is constant-sum only under specific conditions: when the agents do not encounter each other (E = 0) or when the numbers of agents before interaction are equal (N_w = E + N_s). Under these conditions, the sum of payoffs is exactly one (P_s + P_w = 1 − E/(E + N_s) + E/N_w = 1). However, in general, when encounters occur (E ≠ 0) and N_w ≠ E + N_s, the sum deviates from one (P_s + P_w ≠ 1). Moreover, when the total number of prey and encounters (E + N_s) is greater than the number of predators (N_w), the sum of payoffs becomes greater than one (P_s + P_w > 1). This positive-sum condition arises because E/N_w − E/(E + N_s) > 0.

Actual measurements confirmed that the game is non-constant and positive-sum in most cases. Fig 3Aa presents a scatter plot showing N_s on the x-axis and N_w on the y-axis, with a reference line indicating N_s = N_w. Most data points were located below the line, indicating that prey generally outnumber predators (N_s > N_w). Fig 3Ab, 3Ac, 3Ad, and 3Ae present scatter plots with E on the x-axis and, respectively, N_s, _s, N_w, and _w on the y-axis. As E increased, N_s and _s decrease (Fig 3Ab and 3Ac). However, because N_w was proportional to E (Fig 3Ad), _w did not change markedly as E increased (Fig 3Ae). We scatter-plotted all the data of _i for all c_b and r values for _s and _w on the x and y axes, respectively, with a reference line at _s + _w = 1 (Fig 3Af). Most of the points were located in the upper-right area, indicating that the sum of _i is greater than 1 (_s + _w > 1), as predicted.

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Fig 3. Effects of Nash equilibrium attack and defense behaviors with the sensory–motor algorithm on encounters, agent numbers, and payoffs.

A shows scatter plots of all data for the behavioral cost coefficients (c_b) and the reproduction coefficients of predators (r), illustrating the relationships among the number of agents (N_i), encounter frequency (E), and payoff (_i) (where i = w for wolf and s for sheep), based on actual measurements obtained by assigning Nash equilibrium behaviors. Aa shows a scatter plot of N_s on the x-axis and N_w on the y-axis, along the line N_s = N_w. Ab, Ac, Ad, and Ae show scatter plots of E on the x-axis against N_s, _s, N_w, and _w on the y-axis, respectively. Af shows a scatter plot of _s on the x-axis and _w on the y-axis, along the line _s + _w = 1. B and C show the effects of detection distances D_i on E, N_i, and _i. B shows a scatter plot of E versus _s, with light blue, blue, and dark blue points indicating D_s = 1, 2, and 3, respectively. Beige points represent D_s = D_w = 0, where neither predators nor prey attack or defend. B indicates that E decreases and _s increases as D_s increases, regardless of D_w (the dark blue points shift to the upper-left side). Ca and Cb show scatter plots of E on the x-axis and N_w in Ca or _w in Cb on the y-axis. Cc and Cd show scatter plots of N_s versus N_w and _s versus _w, respectively. In Ca–d, orange, brown, and dark brown points represent D_w = 1, 2, and 3, respectively, where D_s = 0. Ca indicates that N_w decreases as D_w increases (the dark brown points shift to the lower side). Cb indicates that _w increases as D_w increases (the dark brown points shift to the upper side). Cc indicates that N_s increases and N_w decreases as D_w increases (the dark brown points shift to the lower-right side). Cd indicates that _s and _w increase as D_w increases (the dark brown points shift to the upper-right side). D shows the effects of behavioral switching. Da shows that within the same (D_s, D_w) conditions, runs that exhibited behavioral switching showed higher encounter frequencies (E > 4) than runs without switching. Db and Dc show that the large E values depend on D_s, with E > 4 when D_s = 1 (Db), but not on D_w (Dc). Dd, De, and Df show scatter plots of behavioral switching in which point colors indicate behavioral switching with E > 4 in dark orange, behavioral switching with E ≤ 4 in orange, no behavioral switching in blue within the same (D_s, D_w) conditions, and other conditions without behavioral switching in gray. Dd shows a scatter plot of occurrence probabilities _s versus _w in which _s located on the right side of the x-axis indicate a higher probability of escape and a lower probability of freeze in the prey’s behavioral switching. Similarly, _w located on the upper side of the y-axis indicate a higher probability of chase and a lower probability of ambush in the predators’ behavioral switching. _i of behavioral switching with E > 4 are distributed near the center (dark orange), suggesting that the choice between the two behaviors is nearly random. The agents exhibit no clear preference. Regardless of the behavioral types, the blue point representing a pure strategy (i.e., no behavioral switching) is located in the upper right of the diagram, indicating a probability of 1. De and Df show scatter plots of N_s versus N_w and _s versus _w, respectively. Dark brown points shift to the upper-left side of De, indicating that behavioral switching with E > 4 led to increased N_w and decreased N_s. Dark brown points shift to the left side of Df, indicating that behavioral switching with E > 4 decreased _s. Dg shows a bar plot illustrating the effect of c_b on switching behavior, with percentages of switching for different values of c_b and r. Nash equilibrium behavior switching frequently occurs when c_b is small.

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The finding that the game is positive-sum in most cases suggests the possibility of conditions under which both agents can obtain greater payoffs together. To explore this possibility further, we investigated the effects of detection distance D_i on Nash equilibrium payoffs _i. To examine the effects of D_s, we compared _i between different values of D_s, with a constant value for D_w. As D_s increased, E decreased (the dark blue points shifted to the left side of Fig 3B), resulting in increasing _s (the dark blue points shifted to the upper side of Fig 3B). These results indicated that the prey’s Nash equilibrium defense reduced encounters and increased prey payoffs.

To examine the effects of D_w, we next compared the _i between different D_w, holding D_s constant. The effects of D_w were dependent on D_s. When D_s = 0, as D_w increased, E did not change markedly; however, N_w decreased (the dark brown points shifted to the lower side of Fig 3Ca), resulting in increasing _w (P_w = E/N_w) (the dark brown points shifted to the upper side of Fig 3Cb). Additionally, the decrease in N_w led to an increase in N_s (the dark brown points shifted to the right side of Fig 3Cc), which further increased _s (P_s = 1 − E/(E + N_s)) (the dark brown points shifted to the right side of Fig 3Cd). Consequently, as D_w increased, both _w and _s increased simultaneously (the dark brown points shifted to the upper right in Fig 3Cd).

These results suggest that when predators detect prey from a greater distance and attack according to the Nash equilibrium under conditions without prey defense, both predators’ and prey’s payoffs can increase. This outcome is due to a decrease in the number of surviving predators and an increase in prey numbers. These results suggest that prey may indirectly gain higher payoffs in the presence of predators, due to increased competition among the predators. Importantly, these findings were independent of c_b and r. Plots of the E-N_i, E-_i, N_i, and _i for different D_i values are shown in S6 and S7 Figs. For comparison, all plots of the Nash equilibrium payoffs in the grid world are shown in S8 Fig, which illustrates a noncooperative, constant-sum game.

Effects of Nash equilibrium behavioral switching

The Nash equilibrium behaviors exhibited either pure or mixed strategies, depending on the combinations of detection distances (D_s, D_w). Mixed strategies were observed for the combinations (1, 1), (1, 2), (1, 3), (2, 2), (2, 3), (3, 1), (3, 2), and (3, 3) for all c_b and r conditions. These mixed strategies resulted in probabilistic switching between behaviors; predators switched between chasing and ambushing, and prey switched between escaping and freezing (Fig 2Fg; S2Df, S2Ff, S2Fg, S3Bg, S3Dg, S3Ff, S3Fg, S3Fj, S3Fl, S4Dg, and S4Fg Figs).

To explore the functional consequences of behavioral switching, we analyzed the actual measurements of _i. Accordingly, we compared encounter frequency (E) with and without switching specifically for the detection distances combinations (D_s, D_w) that exhibited mixed strategies across all c_b and r values. Within the same (D_s, D_w) condition, runs that exhibited behavioral switching showed higher encounter frequencies (E > 4) than runs without switching (Fig 3Da). This increase in E was particularly observed when the prey’s detection distance (D_s) was small (D_s = 1) (Fig 3Db), whereas it was largely independent of the predators’ detection distance (D_w) (Fig 3Dc).

Next, we classified behaviors into three groups: behavioral switching with E > 4 (dark orange), behavioral switching with E ≤ 4 (orange), and no switching (blue), and then compared the occurrence probabilities (_i), agent numbers (N_i), and Nash equilibrium payoffs (_i) across these groups. In the scatter plots of _i (Fig 3Dd), all switching behaviors were broadly distributed whereas those with E > 4, in particular, were centered. This indicates that the switching probabilities between the two behaviors were near 0.5, suggesting a near-random choice between the two behaviors, with no clear preference. Such near-random switching likely contributed to the unpredictability of agent behaviors.

Behavioral switching with E > 4 led to increased predator numbers (N_w) and decreased prey numbers (N_s) (Fig 3De). As a consequence, prey payoffs (_s) decreased, whereas predator payoffs (_w) remained relatively stable (Fig 3Df). Thus, behavioral switching increased encounter frequency and benefited predators, suggesting an adaptive strategy favoring predators.

Finally, we examined the conditions promoting switching. Increased D_i tended to systematically promote mixed strategies (Figs 1Cc and 2Fg, and S2–4 and S9 Figs). Moreover, Nash equilibrium behavior switching frequently occurred at small values of c_b (Fig 3Dg). In addition, we observed that only one of the agents exhibited probabilistic switching while the other did not, that is, partial behavioral switching (S9 Fig).

Nash equilibrium speed changes in the non-sensory motor algorithm

To investigate how sensory input influences equilibrium strategies and payoffs, we tested a simple non-sensory motor algorithm. In this algorithm, agents change their speed stochastically without sensory information. This stochastic behavior was defined by two parameters: K_i, the probability of speed change, and S_i, the magnitude of speed change (Fig 4A). Using a NetLogo program (S5 File), we identified Nash equilibria under this algorithm. Across parameter combinations, K_i and S_i produced diverse encounter frequencies (Fig 4B), determined agent payoffs (Fig 4C and 4D), and led to Nash equilibrium speed changes (_i) (Fig 4E) (c_b = 0.02 and r = 3.6) (S7 and S8 Datas).

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Fig 4. Effects of Nash equilibrium speed changes with the non-sensory motor algorithm on encounters, agent numbers, and payoffs.

A illustrates that stochastic speed changes are defined by the probabilities (K_i) and magnitudes (S_i) of speed change (i = w for wolf and s for sheep). The agent’s life history in this model is the same as the sensory–motor algorithm model, including movement, feeding, energy gain/loss, reproduction, and death. For the illustrations of the wolf and sheep, we used open-source images from Openclipart (https://openclipart.org/detail/254708/wolf-silhouette-2; https://openclipart.org/detail/174830/sheep). B depicts landscape diagrams of encounter frequency (E) in the non-sensory motor algorithm. The landscapes represent combinations of K_i (= 0, 0.1, 0.2, and 0.3). For example, a shows K(0, 0.3), indicating K_s = 0 and K_w = 0.3, while b to p represents different combinations of K_i values. E is color-coded. Darker brown represents higher frequencies and darker blue indicates lower frequencies. C and D show the landscape diagrams of the prey’s and the predator’s payoffs (P_s and P_w), respectively, following the same K_i and S_i combinations as in B. P_i values are color-coded. E displays Nash equilibrium speed change (_i), with circles on the x and y axes indicating _s and _w at the same coordinates as the payoff landscape. F shows scatter plots of all data for the behavioral cost coefficients (c_b) and the reproduction coefficients of predators (r), illustrating the relationships among the number of agents (N_i), encounter frequency (E), and Nash equilibrium payoff (_i), based on actual measurements obtained by assigning Nash equilibrium speed changes. Fa shows N_s on the x-axis and N_w on the y-axis along the line N_s = N_w. Fb, Fc, Fd, and Fe plot E on the x-axis against N_s, _s, N_w, and _w on the y-axis, respectively. Ff plots _s on the x-axis and _w on the y-axis along the line _s + _w = 1. G and H represent the effects of K_i on E, N_i, and _i. G shows a scatter plot of E on the x-axis and _s on the y-axis, with light blue, blue, and dark blue points representing K_s = 0.1, 0.2, and 0.3, respectively. Beige points indicate K_s = K_w = 0. G indicates that E decreases and _s increases as K_s increases (the dark blue points shift to the upper-left side). Ha represents a scatter plot of E on the x-axis and N_w on the y-axis, with orange, brown, and dark brown points representing K_w = 0.1, 0.2, and 0.3, respectively, where K_s = 0. Ha indicates that N_w decreases as K_w increases (the dark brown points shift to the lower side). Hb, Hc, and Hd represent scatter plots of E versus _w, N_s versus N_w, and _s versus _w, where K_s = 0. Hb indicates that _w increases as K_w increases (the dark brown points shift to the upper side). Hc indicates that N_s increases and N_w decreases as K_w increases (the dark brown points shift to the lower-right side). Hd indicates that _s and _w increase as K_w increases (the dark brown points shift to the upper-right side).

https://doi.org/10.1371/journal.pcbi.1013730.g004

Analysis indicated that _i values depended on combinations of agents’ speed change probabilities K_i; particularly those of the prey (K_s). For predators, positive or negative _w settled irrespective of K_w but depended on K_s. When K_s = 0, 0.1, or 0.2, predators increased speed (such as chasing) (Fig 4Ea, 4Eb, 4Ec, 4Ee, 4Ef, 4Eg, 4Ei, 4Ej, and 4Ek). When K_s = 0.3, predators decreased speed (such as ambushing) (Fig 4Ed and 4Eh). For prey, positive _s settled irrespective of K_i, increasing the speed (such as escaping) (Fig 4Eb, 4Ec, 4Ed, 4Ef, 4Eg, 4Eh, 4Ej, 4Ek, 4En, and 4Eo). When K_w = 0, 0.1, and K_s = 0.3, predators went extinct; therefore, _w was not determined and did not have a single value (Fig 4El and 4Ep). When K_w = K_s = 0, _i had large absolute values of motor traits, as in the sensory–motor algorithm (Fig 4Em). Thus, the prey’s K_s had a substantial effect on the determination of Nash equilibrium strategies. All landscapes of the payoff difference between the prey and the predator and the Nash equilibria with the non-sensory motor algorithm for all c_b and r are presented in S10–S13 Figs, showing that distinct Nash equilibrium combinations are identified under varying conditions.

Next, we assigned agent sets of K_i and _i (S9 Data) to perform the Nash equilibrium speed changes (S6 File), and measured encounter frequency E, number of agents N_i, and Nash equilibrium payoff _i (S10 Data). Actual measurements showed outcomes similar to those in the sensory–motor algorithm. The game was typically a non-constant and positive-sum game. Prey’s Nash equilibrium speed changes increased their payoffs by reducing encounters. Predators’ Nash equilibrium speed changes also increased payoffs for both predators and prey when the prey did not change speed. As shown in Fig 4Fa, prey number (N_s) was usually larger than predator number (N_w). As E increased, N_s remained constant (Fig 4Fb), while _s decreased (Fig 4Fc). However, as N_w was proportional to E (Fig 4Fd), _w remained relatively constant (Fig 4Fe). The scatter-plot of _s and _w indicated that most points were located in the upper-right area (_s + _w > 1), confirming a positive-sum tendency (Fig 4Ff).

To further examine the effects of K_i, we compared _i between different K_i conditions. First, holding K_w constant, we varied K_s. As K_s increased, E decreased, increasing _s (the dark blue points shifted to the upper-left side of Fig 4G). Next, holding K_s constant, we varied K_w. The effect of K_w depended on K_s. When K_s = 0, as K_w increased, E did not change considerably but N_w decreased (the dark brown points shifted to the lower side of Fig 4Ha), increasing _w (the dark brown points shifted to the upper side of Fig 4Hb). By reducing N_w, N_s increased (the dark brown points shifted to the right side of Fig 4Hc), increasing _s (the dark brown points shifted to the right side of Fig 4Hd). Consequently, both _w and _s increased as K_w increased (the dark brown points shifted to the upper-right side of Fig 4Hd). The results indicate that the prey may increase their payoff in the presence of predators. These findings were independent of c_b and r. All plots of the E-N_i, E-_i, N_i, and _i for different K_i are shown in S14 and S15 Figs.

Despite the similarities, we found substantial differences between the non-sensory motor and sensory–motor algorithms. Compared with the non-sensory motor algorithm, the sensory–motor algorithm decreased encounter frequency (E) and predator number (N_w) (Fig 5A and 5B), while increasing prey number (N_s) and prey payoffs (_s) (Fig 5C and 5E). Additionally, the sensory–motor algorithm resulted in numerous instances of _w = 0, indicating predator extinctions (Fig 5D). These results indicate that the sensory–motor algorithm results in asymmetrical effects between agents, leading to an advantage for the prey. This suggests that the sensory–motor algorithm may represent an adaptive advantage for the prey.

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Fig 5. Comparison between the non-sensory motor and sensory–motor algorithms.

Outcomes of the non-sensory motor algorithm and sensory–motor algorithm are compared using histograms. A, B, C, D, and E display the proportion of instances in histograms of encounter frequency (E), the number of wolves (N_w), the number of sheep (N_s), the predator Nash equilibrium payoff (_w), and the prey Nash equilibrium payoff (_s), respectively. The values for the non-sensory motor algorithm are depicted in gray, whereas those for the sensory–motor algorithm are in green. Compared to the non-sensory motor algorithm, the sensory–motor algorithm decreases E (A) and N_w (B) while increasing N_s (C). The sensory–motor algorithm leads to a larger proportion of cases with _w = 0, indicating predator extinctions (D), and an increase in _s (E).

https://doi.org/10.1371/journal.pcbi.1013730.g005

Formulation of attack and defense behaviors with a sensory–motor algorithm

In this study, we defined attack and defense behaviors as traits by which a predator captures prey and a prey individual evades predators, respectively. Furthermore, we defined chase–ambush as the attack behavior of predators; chase is used to detect prey and increase speed, whereas ambush is used to detect prey and reduce speed [63,64]. We defined escape–freeze as the defense behavior of the prey; escape is used to detect predators and increase speed, whereas freeze is used to detect predators and reduce speed [15,19]. The models consisted of two types of agents: predators and prey (Figs 1A and 2A). We modeled the attack and defense behaviors with an algorithm using the distance from the opposing agent as sensory input [36] to change the speed as the motor output, but did not include sensory and motor directionalities. Agents had no directionality regarding the opponent’s position or their own movement direction. The formulation was based on Edelaar’s method for representing phenotypic plasticity [65]. The agents had a sensory ability, a distance to detect the opponent (D_i) (i = w and s, as predator and prey, respectively). D_i was the sensory trait (d_i) multiplied by a constant coefficient of the sensory trait (factor sensitivity, f), that is, D_i = d_i × f, where f = 10. We distinguished between the latent trait d_i (dimensionless sensory investment), which entered the cost function, and its spatial realization D_i (detection distance), which was used to compute encounter probabilities. Information-processing costs were assessed using d_i rather than D_i to avoid conflating environmental units with internal investment and to maintain numerical stability across algorithms. The factor f ensured cost comparability across algorithms and helped maintain simulation stability. This scaling allowed us to apply the same cost function to the sensory–motor (d_i) and non-sensory motor (K_i, the probability of speed change) formulations, placing them on a common cost basis and enabling a fair comparison. The bound D_i ≤ 3 was a computational convenience, not a biological limit on the trait. The agents had a motor ability, a magnitude of speed change (S_i) when they detected their opponent. The predator had the sensory ability D_w (0 ≤ D_w) and the motor ability S_w (−1 ≤ S_w ≤ 1) for its behavior, where _w indicates the predator (wolf). The predator moved in two-dimensional space at a speed of 1 and changed its speed to 1 + S_w when it found prey within a radius of D_w. When S_w was greater than 0 (0 < S_w), the predator’s speed increased (chase). When S_w was less than 0 (S_w < 0), the predator’s speed decreased (ambush). Without a prey individual, it moved at a speed of 1. Similarly, the prey had D_s (0 ≤ D_s) and S_s (−1 ≤ S_s ≤ 1), where _s indicates the prey (sheep). The prey moving at speed 1 changed its speed to 1 + S_s when it detected a predator within radius D_s. At 0 < S_s, the prey’s speed increased (escape). At S_s < 0, the prey’s speed decreased (freeze). Without a predator, it moved at a speed of 1. In addition to excluding sensory and motor directionalities, we did not consider the distance in the radius (far or close) or the sensory modality detecting the opponent [15].

Landscape diagrams of encounter probabilities for a single predator and a single prey with the sensory–motor algorithm

We created a grid world model to investigate the attack and defense behaviors of a predator and prey in a discrete space (Fig 1A). A single predator and a single prey were located on a two-dimensional grid (Fig 1A-t1) and moved one grid per time step (Fig 1A-t2). An agent detected the opponent in the D_i range (Moore neighborhood) and changed its speed to 1 + S_i (Figs 1Aa-t2, S1A). D_i assumed the values of 2, 1, or 0 (an agent detected the opponent two grids apart (= 2), detected the opponent in adjacent grids (= 1), or did not detect the opponent (= 0)). S_i assumed the values 1, 0, or −1 (an agent changed its speed to 2 (= 1 + 1), left the speed unchanged at 1 (= 1 + 0), or changed the speed to 0 (= 1 − 1)). When the predator and prey occupied the same grid, the prey was eaten; thus, the prey died (Fig 1Aa-t3, S1A Fig). We examined encounters between agents within a single time step. Because the maximum value of D_i was 2, we used a 5 × 5 grid in the x–y plane with integer coordinates, where x and y ranged from 0 to 4, inclusive. A predator was fixed at coordinate (2, 2), and the prey was placed at all possible other coordinates. When D_i = 2, agents could detect each other at the outer edge, and combinations in which they were adjacent were excluded. Encounter probability (p) was calculated in R [66] (S1 File) as (number of encounters)/(total number of cases), where “total cases” included all combinations of initial predator–prey positions and both agents’ possible headings after the first step, for both detection-informed and uninformed behaviors. This procedure evaluated a single-step forward transition from all initial configurations where interaction could occur, given (D_i, S_i), rather than multi-step simulations on a finite grid. For each D_i, p was plotted against S_s and S_w (Fig 1B), and the resulting Nash equilibrium behaviors are shown in Fig 1C.

When D_s = 2, the prey’s Nash equilibrium strategy became freeze (_s = −1). It seemed unrealistic for the prey to freeze when the prey had a greater detection distance than the predator (Fig 1Cf and 1 Ci, D_s = 2 and D_w = 1 or 0). This arose from a simplifying assumption. We recalculated p including combinations in which the agents were adjacent when D_i = 2 (S2 File). This program corresponds to calculating encounter probabilities for a two-step interaction. Under these conditions, the prey’s Nash equilibrium strategy became “escape” when it had a greater detection distance than the predator (S1Cf and S1Ci Fig). These results indicated that the “freeze” strategy observed at long range arose because only a single-step interaction was considered. When a two-step interaction was considered, the more realistic “escape” strategy was the Nash equilibrium, consistent with results from the NetLogo model.

The grid world does not include the effects of agent density or the costs associated with their behavior.

Payoff landscapes derived from encounter probabilities in multi-agent predator–prey interactions using the sensory–motor algorithm

We constructed a model based on modifications to the Wolf Sheep Predation model from the NetLogo Model Library [43] (Fig 2A, S3 File). The model was composed of agents (predators and prey) and patches that represent the environment. The agents moved and interacted in a continuous two-dimensional space. They could occupy any real-valued (x, y) coordinates, ensuring smooth movement. The patches served as a discrete spatial structure for managing environmental states, such as grass regrowth, but did not constrain the agents’ movement. The agents were not restricted to moving from patch to patch. The agent’s movement direction changes at each time step, randomly within a forward-facing 90-degree range (Fig 2B, and S1 Movie). These specifications are consistent with those used in the original Wolf Sheep Predation model provided in the NetLogo Model Library. The predators moved continuously, gained energy by consuming prey when they were located in the same patch, reproduced, lost energy as they paid costs, and died when their energy reached zero. Similarly, the prey moved continuously, gained energy by consuming grass, reproduced, lost energy as they paid costs, and died when their energy reached zero or when they were located in the same patch as a predator.

We considered three types of costs: basal metabolism, cost dependent on sensory and motor traits, and cost dependent on the resulting speed. Basal metabolism remained constant (c_e). The cost dependent on sensory and motor traits was the sum of the sensory trait (d_i) and the absolute value (abs) of the motor trait (S_i) multiplied by the behavioral cost coefficient (c_b); thus, it was calculated as follows.

This corresponded to the cost associated with implementing neuronal circuit mechanisms for these behaviors. The cost of the resulting speed was calculated by multiplying the speed (1 or 1 + S_i) by a constant, the movement cost coefficient (c_m), as follows.

Thus, in the simulation, different cost values were subtracted from the energy of the agent at each time point when the agent changed or did not change its speed. When the speed of the agent changed to 1 + S_i, the following value was subtracted.

The following value was subtracted when the agent did not change its speed.

Predators and prey were predetermined to perform a single behavior in the IBMs. The sensory ability (detection distance D_i) was d_i multiplied by a constant, the coefficient of the sensory trait (factor sensitivity, f), that is, D_i = d_i × f, where f = 10; D_i had 4 different values (0, 1, 2, and 3). This design made the framework readily extensible to future cross-modality comparisons without redefining the cost function or losing comparability. By keeping cost on d_i and relating spatial range via D_i = d_i × f, the scale factor f can absorb modality-specific (e.g., olfaction, vision, audition, mechanosensation) or medium-specific transduction efficiencies (e.g., transparency of air or water). In practice, using D_i directly in the cost over-penalized large detection radii and made some runs unstable; the d_i-based scaling with f avoided this while preserving the trait-to-range mapping. The value of f was determined from preliminary simulation experiments.

The motor ability was equal to speed change S_i, which considered 11 different values ranging from −1–1 at intervals of 0.2. The combinations of D_i and S_i defined 44 different behaviors as their strategies of the agents. The interactions between these behavioral strategies resulted in 44 × 44 (= 1,936) unique strategy combinations. The following parameters were set as constants: energy gained by a predator eating prey, energy gained by prey from eating grass, reproduction coefficient of prey, time for grass to grow, c_e, c_m, and f. The following variables were used: d_i, D_i, S_i, c_b, and r. Values of the constants and variables used in the simulations with the sensory–motor algorithm are listed in S1 and S2 Tables, respectively.

An individual agent’s behavior was defined by its detection distance (D_i) and speed change (S_i). For each parameter combination, predator and prey agents shared identical sensory and motor traits (homogeneous population) and were assumed to interact in a well-mixed environment (random pairing; no spatial structure). Under these assumptions, we treated the number of predator–prey encounter events (encounter frequency, E) and agents’ number (N_i) at each time step in the population-level as intermediate variables. Both E and N_i varied over time as emergent properties.

Simulations were performed for all predator–prey behavior pairs using nlrx [67], which controls NetLogo in R. When each simulation was run from the beginning up to 10,000 time steps and the interaction between predators and prey reached a steady state, the following data were collected from 10,000–11,000 time steps: time step, E, number of wolves (N_w), and number of sheep (N_s) at each time point. As E, N_w, and N_s were reported at the end of the “to go” procedure after the agents interacted in the NetLogo program, N_w included the number of wolves both with an encounter and no encounter; on the other hand, N_s did not include the number of sheep that died of predation with an encounter but included only with no encounter. Therefore, encounter probabilities of wolves (p_w) and sheep (p_s) were defined as E/N_w and E/(E + N_s), respectively. The formula appeared asymmetrical, but the encounter itself is symmetrical, with contributions from a sheep and a wolf.

Although predation typically involves both pre-encounter and post-encounter processes, our model intentionally idealized behavioral effects by focusing solely on the pre-encounter phase, and thus excluded the latter. In the IBM, energy was updated event-by-event: predators gained the constant wolf-gain-from-food only when they consumed a prey individual during an encounter, whereas prey gained sheep-gain-from-food from grazing in steps without a predator encounter. These constants were used solely for event-driven energy accounting in the IBM; their values were calibrated in preliminary simulations and are listed in S1 Table. For the payoff-matrix analysis used to identify Nash equilibria, we defined normalized payoffs. Accordingly, the payoff of wolves (P_w) was defined as the probability of encountering prey (P_w = p_w), and the payoff of sheep (P_s) as the probability of evading predators (P_s = 1 − p_s). P_s, P_w, and differences in payoffs (P_diff = P_s − P_w) were calculated at each time step. Under the assumption of uniform interactions in a well-mixed environment, we approximated per-agent payoffs by the population-level payoffs based on the encounter probabilities. These values fluctuated periodically with periods shorter than 1,000 time steps. We averaged E, P_i, and P_diff over time steps 10,000–11,000; these averages represented post-transient, statistically stationary values.

We examined the effects of c_b (0, 0.001, 0.02, and 0.04) and r (1.4, 1.8, 2.8, 3.2, 3.6, and 4.0) on E, P_w, P_s, and P_diff. For each combination of D_i, 121 simulations were run, in which each engaged in 11 different behaviors (11 S_w and 11 S_s). E, P_w, P_s, and P_diff were placed on the diagrams, with S_s and S_w on the x and y axes, respectively. Sixteen landscape diagrams were depicted for the combinations of D_i. The experiments were repeated six times.

Nash equilibrium behaviors in the sensory–motor algorithm

We studied the functions of these behaviors using game theory [31]. Attack and defense behaviors were considered as predator and prey strategies, respectively, and consisted of a strategic game between the two players [23]. In the grid world, encounter probability (p) was the payoff for the predator and 1 − p was the payoff for the prey. The bimatrices with p and 1 − p were the payoff matrices of constant-sum games. The S values of the Nash equilibrium (_i) were calculated under combinations of D_i by using Game Theory Explorer (GTE) (https://cgi.csc.liv.ac.uk/~rahul/bimatrix_solver/)[32,68]. A script was written in Python to automate the calculations using GTE. _i were shown at the same coordinates as those in the payoff landscape diagrams.

The NetLogo IBM included many agents. For game-theoretic analysis, we treated the population as a pool of potential players engaging in pairwise contests [45]. We defined normalized per-agent payoffs as P_w and P_s. This differed from the conventional method of constructing payoff matrices from single isolated pairwise interactions and was intended to aggregate simultaneous multi-agent interactions, thus capturing density-dependent, population-mediated effects that were absent in single-pair settings. The resulting bimatrix of P_i, derived from these behaviors, served as the payoff matrix of the game, in which each agent selected one of the eleven motor strategies (S_i) based on its specific sensory ability (D_i). This framework allowed for a systematic analysis of the strategic interactions between the predator and prey within the model [69,70]. The calculations of _i using the GTE [68] were automated using a script written in Python. _i was shown for the combinations of D_i at the same coordinates as the payoff landscape diagrams.

To examine effects of the Nash equilibrium behaviors on agents, we assigned agent sets of D_i and _i (S5 Data) to perform the Nash equilibrium behaviors (S4 File) and measured E, N_i, and Nash equilibrium payoffs (_i) (S6 Data). The experiment was repeated ten times. We examined the effects of Nash equilibrium behaviors on E, N_i, _i, switching probability, and percentages of behavioral switching by drawing scatter, violin, and bar plots.

In this study, we did not use statistical significance tests on p-values to interpret the simulation model results. Statistical methods are considered valuable for data with limited sample sizes, but not for simulation data with virtually unlimited sample sizes [71].

Formulation of agents’ speed changes with a non-sensory motor algorithm and Nash equilibrium in game theory

The agents’ movements were formulated using a non-sensory motor algorithm. The agents set the probabilities (K_i) and magnitudes (S_i) of speed change of their stochastic speed changes (Fig 4A). The predator moving with speed 1 changed its speed to 1 + S_w (−1 ≤ S_w ≤ 1) with the probability K_w (0 ≤ K_w ≤ 1). When S_w was greater than 0 (0 < S_w), the speed of the predator increased, corresponding to that of the predators’ chasing. When S_w was less than 0 (S_w < 0), the speed decreased, corresponding to that of the predators’ ambushing. With a probability of 1 − K_w, the predator moved at a speed of 1. The prey moving with speed 1 changed its speed to 1 + S_s (−1 ≤ S_s ≤ 1) with the probability K_s (0 ≤ K_s ≤ 1). When S_s was greater than 0 (0 < S_s), the prey’s speed increased, corresponding to that of the prey’s escaping. When S_s was less than 0 (S_s < 0), the speed decreased, corresponding to that of the prey’s freezing. With a probability of 1 − K_s, the prey moved at a speed of 1.

We constructed a model with the non-sensory motor algorithm in the IBM using NetLogo (S5 File). The life history of the agents was the same as that in the sensory–motor algorithm model, consisting of movement, feeding, energy gain or loss, reproduction, and death. The “costs” comprised basal metabolism, the cost of probability and motor traits, and the cost of the resulting speed. Basal metabolism and the cost of the resulting speed were the same as those in the sensory–motor algorithm model. The cost of the probability and motor traits was defined as the sum of K_i and the absolute value of the motor trait (abs(S_i)) multiplied by c_b, which corresponds to the cost associated with implementing neuronal circuit mechanisms for their stochastic speed change, as follows:

The following value was subtracted when the agent changed its speed to 1 + Si.

Conversely, the following value was subtracted when the agent did not change its speed.

The probability of speed change (K_i) had four values (0, 0.1, 0.2, and 0.3). Except for K_i, the other parameters (values of the constants and variables) were the same as those of the sensory–motor algorithm model. The values of the constants and variables used in the simulations with the non-sensory motor algorithm are listed in S3 and S4 Tables, respectively. In addition, the measurements and data collection were the same as those for the sensory–motor algorithm model. This experiment was repeated 6 times. The supplementary source data show the payoff matrices for all the experiments (S7 Data).

The Nash equilibrium speed changes were computed in the same manner as those in the sensory–motor algorithm using the GTE with the script written in Python. The results for all the experiments are shown in the Supplementary Source Data (S8 Data). The results of the non-sensory motor algorithm model were depicted in the same manner as those of the sensory–motor algorithm model; E, P_i, and _i were shown in the landscape diagrams.

To examine the effects of the Nash equilibrium speed changes on agents, we assigned agent sets of K_i and _i (S9 Data) to perform the Nash equilibrium speed changes (S6 File) and measured E, N_i, and _i (S10 Data). The experiment was repeated ten times. We examined the effects of the Nash equilibrium speed changes on E, N_i, and _i in the non-sensory motor algorithm model by drawing scatter plots. Furthermore, we compared the effects of the non-sensory motor and the sensory–motor algorithms on E, N_i, and _i by drawing histograms.