When a prey spots a predator about to pounce, it turns swiftly and accelerates away to avoid being captured. The initial direction the prey chooses to take – known as its escape trajectory – can greatly impact their chance of survival.

Previous models were able to predict the optimal direction an animal should take to maximize its chances of evading the predator. However, experimental data suggest that prey actually tend to escape via multiple specific directions, although why animals use this approach has not been clarified. To investigate this puzzle, Kawabata et al. built a new mathematical model that better represents how prey and predators interact with one another in the real world.

Unlike past models, Kawabata et al. incorporated the time required for prey to change direction and only allowed the predators to move toward the prey for a limited distance. By including these two factors, they were able to reproduce the escape trajectories of real animals, including a species of fish, as well as species from other taxa such as frogs and insects.

The new model suggests that prey escape along one of two directions: either by moving directly away from the predator in order to outrun its attack, or by dodging sideways to avoid being captured. Which strategy the prey chooses has some elements of unpredictability, which makes it more difficult for predators to adjust their capturing method.

These findings shed light on why escaping in multiple specific directions makes prey harder to catch. The model could also be extended to test the escape trajectories of a wider variety of predator and prey species, which may avoid capture via different routes. This could help researchers better understand how predators and prey interact with one another. The findings could also reveal how sensory information (such as sound and sight) associated with the threat of an approaching predator is processed and stimulates the muscle activity required to escape in multiple different directions.

When exposed to sudden threatening stimuli such as ambush predators, most prey species initiate escape responses that include turning swiftly and accelerating away from the threat. The escape responses of many invertebrate and lower vertebrate species are controlled by giant neurons that ensure a short response time (Bullock, 1984). Many previous studies have focused on two behavioral traits that are fundamental for avoiding predation: when to escape (i.e., flight initiation distance, which is measured as the distance from the predator at the onset of escape) and where to escape (i.e., escape trajectory [ET], which is measured as the angle of escape direction relative to the stimulus direction) (Cooper, Jr and Blumstein, 2015). Previous studies have investigated the behavioral and environmental contexts affecting these variables (Meager et al., 2006; Arnott et al., 1999; Bateman and Fleming, 2014; Hein et al., 2018; Broom and Ruxton, 2005; Cooper et al., 2003), because they largely determine the success or failure of predator evasion (Walker et al., 2005; Shifferman and Eilam, 2004; Camhi et al., 1978; Kimura and Kawabata, 2018; Dangles et al., 2006), and hence the fitness of the prey species. A large number of models on how animals determine their flight initiation distances have been formulated and tested by experiments (Cooper, Jr and Blumstein, 2015). Although a number of models have also been developed to predict animal ETs (Arnott et al., 1999; Weihs and Webb, 1984; Domenici, 2002), there are still some unanswered questions about how the variability of the observed ETs is generated.

Two different escape tactics (and their combination) have been proposed to enhance the success of predator evasion (Jensen, 2018; Domenici et al., 2011a): the optimal tactic (deterministic), which maximizes the distance between the prey and the predator (Figure 1A; Arnott et al., 1999; Weihs and Webb, 1984; Domenici, 2002; Soto et al., 2015), and the protean tactic (stochastic), which maximizes unpredictability to prevent predators from adjusting their strike trajectories accordingly (Figure 1B; Humphries and Driver, 1970; Jones et al., 2011; Richardson et al., 2018; Moore et al., 2017). Previous geometric models, which formulate optimal tactics, predict a single ET that depends on the relative speeds of the predator and the prey (Arnott et al., 1999; Weihs and Webb, 1984; Domenici, 2002; Soto et al., 2015), and additionally, predator’s turning radii and sensory-motor delay in situations where the predator can adjust its strike path (Howland, 1974; Corcoran and Conner, 2016; Martin et al., 2022). The combination of the optimal tactic (formulated by previous geometric models), which predicts a specific single ET, and the protean tactic, which predicts variability, can explain the ET variability within a limited angular sector that includes the optimal ET (Figure 1C). However, the combination of the two tactics cannot explain the complex ET distributions reported in empirical studies on various taxa of invertebrates and lower vertebrates (reviewed in Domenici et al., 2011b). Whereas some animals exhibit unimodal ET patterns that satisfy the prediction of the combined tactics or optimal tactic with behavioral imprecision (e.g., Cooper, 2006), many animal species show multimodal ETs within a limited angular sector (esp., 90–180°) (Figure 1D) (e.g., Arnott et al., 1999; Bateman and Fleming, 2014; Domenici and Blake, 1993). To explore the discrepancy between the predictions of the models and empirical data, some researchers have hypothesized mechanical/sensory constraints (Domenici et al., 2011a; Domenici et al., 2008); however, the reasons why certain animal species prefer specific multiple ETs remain unclear.

Figure 1

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Multiple preferred ETs of prey can result from situations in which animals choose one behavior from multiple options. Previous work carried out in the field of human and animal psychology on the choice of a particular behavioral strategy out of a number of options has proposed a principle called ‘matching law’. According to this principle, the probability of a certain behavior to occur is related to the proportion of rewards obtained (Reed and Kaplan, 2011; Poling et al., 2011; McDowell, 2013; Houston et al., 2021). This is in contrast to a purely optimal tactic, where animals should always choose the best option (i.e., the highest rewards obtained) (Houston et al., 2021; Fawcett et al., 2013). Arguably, the field of predator-prey interactions has the potential to benefit from an analytical interpretation based on the matching law, because the multiple ETs available to the prey set a scenario similar to the multiple behavioral options considered in previous work analyzed using this principle. In line with this approach, the probability with which a prey chooses a particular ET can be related to the rewards (chances of survival) of each ET option calculated from a predator-prey geometric model.

In previous geometric models, the prey was assumed to instantaneously escape in any direction, irrespective of the prey’s initial body orientation relative to the predator’s approach path (hereafter, initial orientation) (Arnott et al., 1999; Weihs and Webb, 1984; Domenici, 2002). However, additional time is required for changing the heading direction (i.e., turn); therefore, a realistic model needs to take into account that the predator can approach the prey while the prey is turning (Kimura and Kawabata, 2018). Additionally, in previous models, attacking predators were assumed to move for an infinite distance at a constant speed (Arnott et al., 1999; Weihs and Webb, 1984; Domenici, 2002). However, the attacks of many real predators, especially ambush ones, end at a certain distance from initial positions of the prey (Webb and Skadsen, 1980; Fouts and Nelson, 1999; Anderson, 1993). Therefore, we constructed a geometric model that incorporates two additional factors: the time required for the prey to turn and the endpoint of the predator attack. First, using a fish species as a model, we tested whether our model could predict empirically observed multimodal ETs. Second, by calculating the chances of survival of each ET option from our model, we investigated how the prey fish chose a given ET from multiple options. Third, by extending the model, we tested whether other patterns of empirical ETs could be predicted: unimodal ETs and multimodal ETs directed at small (20–50°) and large (150–180°) angles from the predator’s approach direction. The biological implications resulting from the model and experimental data are then discussed within the frameworks of predator-prey interactions and behavioral decision-making.

Model

We revised the previous model proposed by Domenici, 2002; Paglianti and Domenici, 2006 (Figure 2A) and the model proposed by Corcoran and Conner, 2016 (Appendix 1—figure 1A). Other previous models (Arnott et al., 1999; Weihs and Webb, 1984; Soto et al., 2015; Martin et al., 2022) made predictions similar to those of Domenici’s model or those of Corcoran’s model, although they used different theoretical approaches. In Domenici’s model, the predator with a certain width (i.e., the width of a killer whale’s tail used as a weapon to catch prey) directly approaches the prey, and the prey (the whole body) should enter the safety zone before the predator reaches that entry point. In this model, the prey can instantaneously escape in any direction, and the predation threat moves linearly and infinitely. Corcoran’s model is based on the same principle as Domenici’s model, but includes the concept that the predator (i.e., a bat) can adjust the approach path up to its minimum turning radius. Thus, Domenici’s model can be regarded as a special case of Corcoran’s model when the turning radius of the predator is infinitely large. These models are based on the escape response of the horizontal plane, which is realistic for many fish species as well as terrestrial and benthic species that move on substrates. They can also be applied to aerial animals such as moths escaping from bats because many predator-prey interactions are approximately two-dimensional in a local spatial scale (Corcoran and Conner, 2016; Fabian et al., 2018). Hereafter, we explain the modification of Domenici’s model (a special case of Corcoran’s model) because the data on previously published predator-prey experiments on the same species of prey and predator in our experiment (Kimura and Kawabata, 2018) show that the predator does not adjust the strike path during the attack (Figure 2—figure supplement 1, adjusted angle = 1.0 ± 6.6° [mean ± s.d.], n=5), and thus the number of parameters to estimate can be reduced. See Appendix 1 for details of the modified version of Corcoran’s model.

Figure 2 with 2 supplements see all

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In our new model (Figure 2B), two factors are added to the previous Domenici’s model: the time required for the prey to turn and the endpoint of the predator attack. We assume that a prey with a certain initial orientation β (spanning 0–180°, where 0° and 180° correspond to being attacked from front and behind, respectively) evades a sudden predation threat. Most prey species respond to the attack by turning at an angle α, and the ET results from the angular sum of α and β. ETs from the left and right sides were pooled and treated as though they were stimulated from the right side (Figure 2—figure supplement 2; see ‘Definition of the angles’ in Materials and methods for details).

When the prey’s CoM at the onset of its escape is located at point (0, 0), the trajectory of the CoM ($X_{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{y}}$ , $Y_{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{y}}$) is given by:

(1) ${\mathit{Y}}_{\mathbf{p}\mathbf{r}\mathbf{e}\mathbf{y}}={\mathit{X}}_{\mathbf{p}\mathbf{r}\mathbf{e}\mathbf{y}}\mathbf{tan}(\mathit{\alpha }+\mathit{\beta })$

The edge of the safety zone is determined by the half-width of the predator capture device (e.g., mouth) D_width, the distance between the prey’s initial position and the tip of the predator capture device at the end of the predator attack D_attack, and the shape of the predator’s capture device at the moment of attack, which is approximated as an arc with a certain radius R_device. The projection of the predator’s capture device edge along the edge of the sideways safety zone $D_2$ can be expressed as:

(2) ${\displaystyle {\mathit{D}}_2={\mathit{R}}_{\mathbf{d}\mathbf{e}\mathbf{v}\mathbf{i}\mathbf{c}\mathbf{e}}\left\{1-\mathbf{c}\mathbf{o}\mathbf{s}\left({\mathbf{s}\mathbf{i}\mathbf{n}}^{-1}\frac{{\mathit{D}}_{\mathbf{w}\mathbf{i}\mathbf{d}\mathbf{t}\mathbf{h}}}{{\mathit{R}}_{\mathbf{d}\mathbf{e}\mathbf{v}\mathbf{i}\mathbf{c}\mathbf{e}}}\right)\right\}}$

The ET toward the upper-left corner of the danger zone ${\theta }_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{n}\mathrm{e}\mathrm{r}}$ can be expressed as:

(3) ${\displaystyle {\mathit{\theta }}_{\mathbf{c}\mathbf{o}\mathbf{r}\mathbf{n}\mathbf{e}\mathbf{r}}={\mathbf{t}\mathbf{a}\mathbf{n}}^{-1}\frac{{\mathit{D}}_{\mathbf{w}\mathbf{i}\mathbf{d}\mathbf{t}\mathbf{h}}}{{\mathit{D}}_2-{\mathit{D}}_{\mathbf{a}\mathbf{t}\mathbf{t}\mathbf{a}\mathbf{c}\mathbf{k}}}}$

The x and y coordinates of the safety zone edge ($X_{\mathrm{s}\mathrm{a}\mathrm{f}\mathrm{e}}$ , $Y_{\mathrm{s}\mathrm{a}\mathrm{f}\mathrm{e}}$) are given by:

(4) $\{\begin{array}{l}{\mathit{Y}}_{\mathbf{s}\mathbf{a}\mathbf{f}\mathbf{e}}={\mathit{D}}_{\mathbf{w}\mathbf{i}\mathbf{d}\mathbf{t}\mathbf{h}},\alpha +\beta <{\mathit{\theta }}_{\mathbf{c}\mathbf{o}\mathbf{r}\mathbf{n}\mathbf{e}\mathbf{r}}\\ {\left({\mathit{X}}_{\mathbf{s}\mathbf{a}\mathbf{f}\mathbf{e}}+{\mathit{D}}_{\mathbf{a}\mathbf{t}\mathbf{t}\mathbf{a}\mathbf{c}\mathbf{k}}-{\mathit{R}}_{\mathbf{d}\mathbf{e}\mathbf{v}\mathbf{i}\mathbf{c}\mathbf{e}}\right)}^2+{\mathit{Y}}_{\mathbf{s}\mathbf{a}\mathbf{f}\mathbf{e}}^2={\mathit{R}}_{\mathbf{d}\mathbf{e}\mathbf{v}\mathbf{i}\mathbf{c}\mathbf{e}}^2,& \alpha +\beta \ge {\mathit{\theta }}_{\mathbf{c}\mathbf{o}\mathbf{r}\mathbf{n}\mathbf{e}\mathbf{r}}\end{array}$

From Equation 1 to Equation 4, the x and y coordinates of the crossing point of the escape path and the safety zone edge ($X_{\mathrm{c}\mathrm{r}\mathrm{o}\mathrm{s}\mathrm{s}}$ , $Y_{\mathrm{c}\mathrm{r}\mathrm{o}\mathrm{s}\mathrm{s}}$) are given by a function of D_width, D_attack, R_device, and α+β.

The prey can escape from the predator when the time required for the prey to enter the safety zone (T_prey) is shorter than the time required for the predator’s capture device to reach that entry point (T_pred). Therefore, the prey is assumed to maximize the difference between the T_pred and T_prey (T_diff). To incorporate the time required for the prey to turn, T_prey was divided into two phases: the fast-start phase, which includes the time for turning and acceleration ($T_1$), and the constant speed phase ($T_2$). This assumption is consistent with the previous studies (Domenici and Blake, 1991; Danos and Lauder, 2012; Fleuren et al., 2018) and was supported by our experiment (see Figure 4—figure supplement 1). Therefore:

(5) ${\mathit{T}}_{\mathbf{p}\mathbf{r}\mathbf{e}\mathbf{y}}={\mathit{T}}_1+{\mathit{T}}_2$

For simplicity, the fish was assumed to end the fast-start phase at a certain displacement from the initial position in any α (D₁; the radius of the dotted circle in Figure 2B) and to move at a constant speed U_prey to cover the rest of the distance (toward the edge of the safety zone $\sqrt{{X_{\mathrm{c}\mathrm{r}\mathrm{o}\mathrm{s}\mathrm{s}}}^2+{Y_{\mathrm{c}\mathrm{r}\mathrm{o}\mathrm{s}\mathrm{s}}}^2}-D_1$ , plus the length of the body that is posterior to the CoM L_prey). Because a larger |α| requires further turning prior to forward locomotion, which takes time (Domenici and Blake, 1991; Ellerby and Altringham, 2001), and the initial velocity after turning was dependent on |α| in our experiment (see Figure 4B), $T_1$ is given by a function of |α| [ ${\displaystyle T_1\left(|\alpha |\right)}$ ]. Therefore, T_prey can be expressed as:

(6) ${\displaystyle {\mathit{T}}_{\mathbf{p}\mathbf{r}\mathbf{e}\mathbf{y}}={\mathit{T}}_1\left(\left|\mathit{\alpha }\right|\right)+\frac{\sqrt{{\mathit{X}}_{\mathbf{c}\mathbf{r}\mathbf{o}\mathbf{s}\mathbf{s}}^2+{\mathit{Y}}_{\mathbf{c}\mathbf{r}\mathbf{o}\mathbf{s}\mathbf{s}}^2}-{\mathit{D}}_1+{\mathit{L}}_{\mathbf{p}\mathbf{r}\mathbf{e}\mathbf{y}}}{{\mathit{U}}_{\mathbf{p}\mathbf{r}\mathbf{e}\mathbf{y}}}}$

T_pred can be expressed as:

(7) ${\mathit{T}}_{\mathbf{p}\mathbf{r}\mathbf{e}\mathbf{d}}=\{\begin{array}{l}\frac{{\mathit{D}}_{\mathbf{i}\mathbf{n}\mathbf{i}\mathbf{t}\mathbf{i}\mathbf{a}\mathbf{l}}+{\mathit{D}}_2-{\mathit{X}}_{\mathbf{c}\mathbf{r}\mathbf{o}\mathbf{s}\mathbf{s}}}{{\mathit{U}}_{\mathbf{p}\mathbf{r}\mathbf{e}\mathbf{d}}},\alpha +\beta <{\mathit{\theta }}_{\mathbf{c}\mathbf{o}\mathbf{r}\mathbf{n}\mathbf{e}\mathbf{r}}\\ \frac{{\mathit{D}}_{\mathbf{i}\mathbf{n}\mathbf{i}\mathbf{t}\mathbf{i}\mathbf{a}\mathbf{l}}+{\mathit{D}}_{\mathbf{a}\mathbf{t}\mathbf{t}\mathbf{a}\mathbf{c}\mathbf{k}}}{{\mathit{U}}_{\mathbf{p}\mathbf{r}\mathbf{e}\mathbf{d}}},\alpha +\beta \ge {\mathit{\theta }}_{\mathbf{c}\mathbf{o}\mathbf{r}\mathbf{n}\mathbf{e}\mathbf{r}}\end{array}$

where D_initial is the distance between the prey and the predator at the onset of the prey’s escape response (i.e., the flight initiation distance or reaction distance), and $U_{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}}$ is the predator speed, which is assumed to be constant. From Equations 5–7, T_diff can be calculated as:

(8) ${\mathit{T}}_{\mathbf{d}\mathbf{i}\mathbf{f}\mathbf{f}}=\{\begin{array}{ll}\frac{{\mathit{D}}_{\mathbf{i}\mathbf{n}\mathbf{i}\mathbf{t}\mathbf{i}\mathbf{a}\mathbf{l}}}{{\mathit{U}}_{\mathbf{p}\mathbf{r}\mathbf{e}\mathbf{d}}}+\frac{{\mathit{D}}_2}{{\mathit{U}}_{\mathbf{p}\mathbf{r}\mathbf{e}\mathbf{d}}}-\frac{{\mathit{X}}_{\mathbf{c}\mathbf{r}\mathbf{o}\mathbf{s}\mathbf{s}}}{{\mathit{U}}_{\mathbf{p}\mathbf{r}\mathbf{e}\mathbf{d}}}-{\mathit{T}}_1\left(\left|\mathit{\alpha }\right|\right)-\frac{\sqrt{{\mathit{X}}_{\mathbf{c}\mathbf{r}\mathbf{o}\mathbf{s}\mathbf{s}}^2+{\mathit{Y}}_{\mathbf{c}\mathbf{r}\mathbf{o}\mathbf{s}\mathbf{s}}^2}}{{\mathit{U}}_{\mathbf{p}\mathbf{r}\mathbf{e}\mathbf{y}}}+\frac{{\mathit{D}}_1}{{\mathit{U}}_{\mathbf{p}\mathbf{r}\mathbf{e}\mathbf{y}}}-\frac{{\mathit{L}}_{\mathbf{p}\mathbf{r}\mathbf{e}\mathbf{y}}}{{\mathit{U}}_{\mathbf{p}\mathbf{r}\mathbf{e}\mathbf{y}}},& \alpha +\beta <{\mathit{\theta }}_{\mathbf{c}\mathbf{o}\mathbf{r}\mathbf{n}\mathbf{e}\mathbf{r}}\\ \frac{{\mathit{D}}_{\mathbf{i}\mathbf{n}\mathbf{i}\mathbf{t}\mathbf{i}\mathbf{a}\mathbf{l}}}{{\mathit{U}}_{\mathbf{p}\mathbf{r}\mathbf{e}\mathbf{d}}}+\frac{{\mathit{D}}_{\mathbf{a}\mathbf{t}\mathbf{t}\mathbf{a}\mathbf{c}\mathbf{k}}}{{\mathit{U}}_{\mathbf{p}\mathbf{r}\mathbf{e}\mathbf{d}}}-{\mathit{T}}_1\left(\left|\mathit{\alpha }\right|\right)-\frac{\sqrt{{\mathit{X}}_{\mathbf{c}\mathbf{r}\mathbf{o}\mathbf{s}\mathbf{s}}^2+{\mathit{Y}}_{\mathbf{c}\mathbf{r}\mathbf{o}\mathbf{s}\mathbf{s}}^2}}{{\mathit{U}}_{\mathbf{p}\mathbf{r}\mathbf{e}\mathbf{y}}}+\frac{{\mathit{D}}_1}{{\mathit{U}}_{\mathbf{p}\mathbf{r}\mathbf{e}\mathbf{y}}}-\frac{{\mathit{L}}_{\mathbf{p}\mathbf{r}\mathbf{e}\mathbf{y}}}{{\mathit{U}}_{\mathbf{p}\mathbf{r}\mathbf{e}\mathbf{y}}},& \alpha +\beta \ge {\mathit{\theta }}_{\mathbf{c}\mathbf{o}\mathbf{r}\mathbf{n}\mathbf{e}\mathbf{r}}\end{array}$

Because $\frac{D_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}}}{U_{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}}}+\frac{D_1}{U_{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{y}}}-\frac{L_{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{y}}}{U_{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{y}}}$ are independent of α and β, we can calculate the relative values of $T_{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}}$ ($T_{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}}`$) in response to the changes of α and β, from:

(9) ${\mathit{T}}_{\mathbf{d}\mathbf{i}\mathbf{f}\mathbf{f}}\mathbf{\prime }=\{\begin{array}{ll}\frac{{\mathit{D}}_2}{{\mathit{U}}_{\mathbf{p}\mathbf{r}\mathbf{e}\mathbf{d}}}-\frac{{\mathit{X}}_{\mathbf{c}\mathbf{r}\mathbf{o}\mathbf{s}\mathbf{s}}}{{\mathit{U}}_{\mathbf{p}\mathbf{r}\mathbf{e}\mathbf{d}}}-{\mathit{T}}_1\left(\left|\mathit{\alpha }\right|\right)-\frac{\sqrt{{\mathit{X}}_{\mathbf{c}\mathbf{r}\mathbf{o}\mathbf{s}\mathbf{s}}^2+{\mathit{Y}}_{\mathbf{c}\mathbf{r}\mathbf{o}\mathbf{s}\mathbf{s}}^2}}{{\mathit{U}}_{\mathbf{p}\mathbf{r}\mathbf{e}\mathbf{y}}},& \alpha +\beta <{\mathit{\theta }}_{\mathbf{c}\mathbf{o}\mathbf{r}\mathbf{n}\mathbf{e}\mathbf{r}}\\ \frac{{\mathit{D}}_{\mathbf{a}\mathbf{t}\mathbf{t}\mathbf{a}\mathbf{c}\mathbf{k}}}{{\mathit{U}}_{\mathbf{p}\mathbf{r}\mathbf{e}\mathbf{d}}}-{\mathit{T}}_1\left(\left|\mathit{\alpha }\right|\right)-\frac{\sqrt{{\mathit{X}}_{\mathbf{c}\mathbf{r}\mathbf{o}\mathbf{s}\mathbf{s}}^2+{\mathit{Y}}_{\mathbf{c}\mathbf{r}\mathbf{o}\mathbf{s}\mathbf{s}}^2}}{{\mathit{U}}_{\mathbf{p}\mathbf{r}\mathbf{e}\mathbf{y}}},& \alpha +\beta \ge {\mathit{\theta }}_{\mathbf{c}\mathbf{o}\mathbf{r}\mathbf{n}\mathbf{e}\mathbf{r}}\end{array}$

Because $X_{\mathrm{c}\mathrm{r}\mathrm{o}\mathrm{s}\mathrm{s}}$ and $Y_{\mathrm{c}\mathrm{r}\mathrm{o}\mathrm{s}\mathrm{s}}$ are dependent on D_width, D_attack, and R_device as well as $\alpha +\beta$, and $D_2$ is dependent on D_width and R_device, we can calculate $T_{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}}`$ in response to the changes of α and β, from D<sub>1</sub>, D<sub>width</sub>, D<sub>attack</sub>, R<sub>device</sub>, U<sub>prey</sub>, U<sub>pred</sub>, and $T_1\left(\left|\alpha \right|\right)$ . Given that the escape success is assumed to be dependent on $T_{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}}`$, the theoretically optimal ET can be expressed as:

(10) $\mathbf{T}\mathbf{h}\mathbf{e}\mathbf{o}\mathbf{p}\mathbf{t}\mathbf{i}\mathbf{m}\mathbf{a}\mathbf{l}\mathbf{E}\mathbf{T}=\underset{\mathit{\alpha }+\mathit{\beta }}{\mathbf{a}\mathbf{r}\mathbf{g}\mathbf{m}\mathbf{a}\mathbf{x}}\left({\mathit{T}}_{\mathbf{d}\mathbf{i}\mathbf{f}\mathbf{f}}\mathbf{\prime }\right)$

Our geometric model, incorporating the endpoint of the predator attack, D_attack, and the time required for the prey to turn, T₁(|α|), to maximize the difference between the prey and the predator in the time of arrival at the edge of the safety zone, T_diff, clearly explains the multimodal patterns of ETs in P. major. Figure 8 shows an example of how multiple ETs result in successful escapes from predators. Specifically, according to the model, when the prey escapes at 140° or 170°, it will not be captured by the predator. On the other hand, when the prey escapes along an intermediate trajectory (157°), it will be captured because it swims toward the corner of the danger zone to exit it, and therefore it needs to travel a longer distance than when escaping at 140° or 170°. This example illustrates that the multimodal patterns of ETs are likely to be attributable to the existence of two escape routes: either moving sideways to depart from the predator’s strike path or moving opposite to the predator’s direction to outrun it. Interestingly, both components of the predator-prey interaction (i.e., D_attack and T₁(|α|)) added to the previous model (Domenici, 2002) are important for accurate predictions of the ET distribution because when they are considered by the model separately, the predictions do not match the experimental data (Figure 5—figure supplements 4 and 5; Table 3).

Figure 8

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Two different escape tactics have been proposed to enhance the success of predator evasion (Jensen, 2018; Domenici et al., 2011a): the optimal tactic, which maximizes T_diff (i.e., the distance between the prey and the predator) (Arnott et al., 1999; Weihs and Webb, 1984; Domenici, 2002; Soto et al., 2015), and the protean tactic, which maximizes unpredictability to prevent predators from adjusting their strike trajectories accordingly (Humphries and Driver, 1970; Jones et al., 2011; Richardson et al., 2018; Moore et al., 2017). Our results suggest that the prey combines these two different tactics by using multiple preferred ETs. Specifically, when the optimal ET advantage is large (i.e., when the initial orientation is 20–60°), the prey mainly uses the optimal ET (Figures 3A and 6). However, when the optimal ET advantage over the suboptimal ET is negligible (i.e., the initial orientation is close to 0° or within the range 110‒180°), the prey uses optimal and suboptimal ETs to a similar extent (Figures 3A and 6). In such cases, the ET of the prey would be highly unpredictable for the predator. The unpredictability at initial orientations near 0° and 180° is consistent with the study that applied the conventional geometric model to the larval zebrafish Danio rerio (Nair et al., 2017), where the optimal and suboptimal ETs are approximately symmetrical to the axis of the predator attack. This phenomenon can be explained by the toward-away indecision at orientations nearly perpendicular to the threat (Domenici and Blake, 1993; Domenici and Batty, 1997). On the other hand, the unpredictability observed at initial orientations near 110–180° is related to the similarly advantageous choice between escaping with an ET at around 140° or 180°. Interestingly, at initial orientations >120°, our results show that these two ETs are reached by using toward and away responses, respectively. The overlap between the ETs of toward and away responses in the overall dataset (Figure 3) suggests that toward responses are not ‘tactical mistakes’ of the prey that turns toward a threat, but are simply related to reaching an optimal or suboptimal ET. These results suggest that the prey strategically adjusts the use of optimal and protean tactics based on their initial orientation. This allows the prey to have unpredictable ETs, thereby preventing predators from anticipating their escape behavior, while keeping T_diff large enough to enter the safety zone before the predator reaches it.

From a behavioral decision-making perspective, our results suggest that the prey follows the matching law (Reed and Kaplan, 2011; Poling et al., 2011; McDowell, 2013; Houston et al., 2021), where the probability that an optimal or suboptimal ET is chosen is proportional to its chances of survival (i.e., T_diff). As the matching law predicts (Houston et al., 2021), the prey stochastically draws from a Bernoulli distribution dictated by the optimal ET advantage for the binary choice between an optimal and a suboptimal ET, thereby introducing an element of unpredictability, which can prevent predators from learning. Because most empirical studies supporting the matching law use unnatural reinforcement learning paradigms or human behaviors (Reed and Kaplan, 2011; Poling et al., 2011; McDowell, 2013; Houston et al., 2021), this result suggests that the matching law is also applicable to animal behavior in realistic contexts. Further research using a real predator and dummy prey (e.g., Szopa-Comley and Ioannou, 2022) controlled to escape toward an optimal or suboptimal ET with various specific probabilities is required to test whether our model accurately predicts the best combination of the optimal and suboptimal ETs when accounting for the predator learning.

A relevant question from a perspective of neurosensory physiology is how the animals are able to determine their ETs within milliseconds of response time. The initial orientation of the prey has been incorporated into various neural circuit models (Eaton et al., 2001; Yono and Shimozawa, 2008; Card, 2012; Levi and Camhi, 2000), but these models assume that prey animals always escape in a 180° direction (i.e., opposite to the stimulus source), irrespective of the initial orientation. However, the present study shows that animals use suboptimal ETs as well as optimal ETs, and that these ETs may change in a nonlinear fashion, depending on the initial orientation. More specifically, the Mauthner cell and other neurons involved may be activated in accordance with the Bernoulli probabilities dictated by the model, which determine the proportions of away and toward responses and the magnitude of turn to achieve the multiple preferred ETs. Thus, we require new neurophysiological models of ETs to understand how neural circuits process the sensory cues of a threatening stimulus, resulting in muscle actions that generate multiple preferred ETs.

Our geometric model assumes that the prey determines the ETs based on a fixed predator speed. This assumption is supported by the results of our experiments, where the effects of predator speed on the mean and variability of ETs are not significant. Although we did not find any effect of predator speed, it is possible that a speed outside the range we used may affect ETs. Recent studies show that larval zebrafish exhibit less variable ETs under faster threats than they do under slower threats (Stewart et al., 2014; Bhattacharyya et al., 2017), and the difference in ET variability between fast and slow threats is dependent on whether the Mauthner cell is active or not (Bhattacharyya et al., 2017). Therefore, any differences in the ET variability of the present study compared to previous studies could be related to the different involvement of the Mauthner cells. Using the conventional geometric model (Weihs and Webb, 1984), Soto et al. showed that the choice of ET only matters to a prey when the predator speed is intermediate, because a prey that is much faster than its predator can escape by a broad range of ETs, whereas a prey that is much slower than its predator cannot escape by any ETs (Soto et al., 2015). The predator speed used in this study is in the range of the real predator speed in the previous study using the same species of both predator and prey (Kimura and Kawabata, 2018). Thus, our results are ecologically relevant, and the prey is likely to have optimized their ETs based on a fixed predator speed, where the choice of ET strongly affects their survival.

The relationship between |α| and the time required for a 15 mm displacement, T₁(|α|), (Figure 4A) indicates that the time required for a 15 mm displacement is relatively constant up to an |α| of about 45°, while a further change in |α| requires additional time. This relationship is likely to be attributable to the kinematics and hydrodynamics of the C-start escape response, because the initial velocity after the stage 1 turn increases linearly up to about 45°, beyond which it plateaus (Figure 4B). Interestingly, a recent study on swimming efficiency during acceleration found that efficiency increases linearly with yaw amplitudes up to a certain value, beyond which efficiency plateaus (Akanyeti et al., 2017).

Based on the STRANGE framework for animal behavior research (Webster and Rutz, 2020), we identified potential biases that may limit the generalizability of our findings. Our empirical data are obtained from one species of hatchery-reared fish with a specific life stage, which has never experienced predators. Therefore, this study alone cannot exclude the possibility that fish of different species, origins, life stages, and rearing histories have different rules for ETs, which our model cannot explain. However, similar multiple preferred ETs have been observed in many fish species and other animal taxa, including hatcheries/wild origins and different life stages (Domenici et al., 2011b). Therefore, we believe that our model is not specific to our experiment but is applicable to other cases showing multiple preferred ETs.

We show that our model has the potential to explain other empirically observed ET patterns (Figure 7). Based on the model assuming that the predator makes an in-line attack toward the prey, which is realistic for ambush and stalk-and-attack predators (Moore and Biewener, 2015) (e.g., frogs [Camhi et al., 1978], spiders [Dangles et al., 2006], and fish [Kimura and Kawabata, 2018; Webb and Skadsen, 1980; Fouts and Nelson, 1999; Rand and Lauder, 1981]), either single or multiple ETs at around 90–150° and around 180° are predicted, as have been observed in many empirical studies of animals escaping from ambush predators and artificial stimuli (Domenici et al., 2011b). Based on the model assuming that the predator can adjust its approach path, which is realistic for pursuit predators, multiple ETs directed at small and large angles from the predator’s approach direction can be predicted, as observed in the empirical studies of prey escaping from pursuit predators (Corcoran and Conner, 2016; Bulbert et al., 2015). Further research measuring the escape response in various species and applying the data to our geometric model is required to verify the applicability of our geometric model to various predator-prey systems.

Our work represents a major advancement in understanding the basis of the variability in ETs observed in previous works (reviewed in Domenici et al., 2011b). Our results suggest that prey use multiple preferred ETs to maximize the time difference between themselves and the attacking predator, while keeping a high level of unpredictability. The results also suggest that prey strategically adjust the use of protean and optimal tactics with respect to the advantage of the optimal ET over the suboptimal ET. Because multimodal ETs similar to what we observed here have been found in many fish species and other animal taxa (Domenici et al., 2011b), this behavioral phenotype may result from convergent evolution in phylogenetically distant animals. From a neurosensory perspective, our findings open new avenues to investigate how the animals determine their ETs from multiple options with specific probabilities, which are modulated by the initial orientation with respect to the threat.

Mathematical formula for the geometric model modified from Corcoran and Conner, 2016. When the prey’s CoM at the onset of its escape is located at point (0, 0), the trajectory of the CoM ($X_{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{y}}$ , $Y_{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{y}}$) is given by:

The edge of the safety zone is determined by the half-width of the predator capture device D_width, the distance between the prey and the predator at the onset of the prey’s escape response D_initial, the distance required for the predator to react the prey’s escape response to initiate its turn D_react, the distance between the prey’s initial position and the tip of the predator capture device at the end of the predator attack D_attack, the minimum turning radius of the predator R_turn, and the shape of the predator’s capture device at the moment of attack, which is approximated as an arc with a certain radius R_device (Appendix 1—figure 1B). The edge of the safety zone can be divided into four parts: (1) the straight line before the onset of the predator’s turn, (2) the arc of the minimum inner turning radius, (3) the capture device shape at the end of the predator attack when it attacks with the minimum turning radius, (4) the involute curve where the tip of the predator capture device traverses a specific distance from the initial position, which can be described by the trace of unwrapping a taut string from the minimum turning circle (Appendix 1—figure 1B). Note that the model may lack some parts, depending on the values of parameters.

Appendix 1—figure 1

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The projection of the predator’s capture device edge along the X-axis $D_2$ can be expressed as:

The angle for the predator to traverse with the minimum turning radius γ can be expressed as:

The x and y coordinates of the change point between the first and second parts of the safety zone edge (${\displaystyle X_{\mathrm{c}\mathrm{p}1}}$ , $Y_{\mathrm{c}\mathrm{p}1}$) can be expressed as:

The x and y coordinates of the change point between the second and third parts of the safety zone edge ($X_{\mathrm{c}\mathrm{p}2}$ , $Y_{\mathrm{c}\mathrm{p}2}$) can be expressed as:

The x and y coordinates of the change point between the third and fourth parts of the safety zone edge ($X_{\mathrm{c}\mathrm{p}3}$ , $Y_{\mathrm{c}\mathrm{p}3}$) can be expressed as:

The x and y coordinates of the first part of the safety zone edge ($X_{\mathrm{s}\mathrm{a}\mathrm{f}\mathrm{e}1}$ , $Y_{\mathrm{s}\mathrm{a}\mathrm{f}\mathrm{e}1}$) are given by:

The x and y coordinates of the second part of the safety zone edge ($X_{\mathrm{s}\mathrm{a}\mathrm{f}\mathrm{e}2}$ , $Y_{\mathrm{s}\mathrm{a}\mathrm{f}\mathrm{e}2}$) are given by:

The x and y coordinates of the third part of the safety zone edge ($X_{\mathrm{s}\mathrm{a}\mathrm{f}\mathrm{e}3}$ , $Y_{\mathrm{s}\mathrm{a}\mathrm{f}\mathrm{e}3}$) are given by:

For calculating the x and y coordinates of the fourth part of the safety zone edge ($X_{\mathrm{s}\mathrm{a}\mathrm{f}\mathrm{e}4}$ , $Y_{\mathrm{s}\mathrm{a}\mathrm{f}\mathrm{e}4}$), the formula of involute curve from a circle of radius $R_{\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{n}}$ whose center is the origin (0, 0) with a tip of the string at ($R_{\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{n}},0$) is introduced as:

where θ denotes an angle for an unwrapped string from the circle in a counterclockwise direction. By moving and rotating this point (x, y), we can calculate the fourth part of the safety zone edge ($X_{\mathrm{s}\mathrm{a}\mathrm{f}\mathrm{e}4}$ , $Y_{\mathrm{s}\mathrm{a}\mathrm{f}\mathrm{e}4}$) as:

From Equations 1–11, to the x and y coordinates of the crossing point of the escape path and the safety zone edge ($X_{\mathrm{c}\mathrm{r}\mathrm{o}\mathrm{s}\mathrm{s}}$ , $Y_{\mathrm{c}\mathrm{r}\mathrm{o}\mathrm{s}\mathrm{s}}$) are given by a function of D_width, D_attack, R_device, D_initial, D_react, R_turn, and α+β.

The prey can escape from the predator when the time required for the prey to enter the safety zone (T_prey) is shorter than the time required for the predator’s capture device to reach that entry point (T_pred). Therefore, the prey is assumed to maximize the difference between the T_pred and T_prey (T_diff). To incorporate the time required for the prey to turn, T_prey was divided into two phases: the fast-start phase, which includes the time for turning and acceleration ($T_1$), and the constant speed phase ($T_2$). This assumption is consistent with the previous studies (Domenici and Blake, 1991; Danos and Lauder, 2012; Fleuren et al., 2018) and was supported by our experiment (see Figure 4—figure supplement 1). Therefore:

For simplicity, the prey was assumed to end the fast-start phase at a certain displacement from the initial position in any α (D₁; the radius of the dotted circle in Appendix 1—figure 1B) and to move at a constant speed U_prey to cover the rest of the distance (toward the edge of the safety zone $\sqrt{{X_{\mathrm{c}\mathrm{r}\mathrm{o}\mathrm{s}\mathrm{s}}}^2+{Y_{\mathrm{c}\mathrm{r}\mathrm{o}\mathrm{s}\mathrm{s}}}^2}-D_1$ , plus the length of the body that is posterior to the CoM L_prey). Because a larger |α| requires further turning prior to forward locomotion, which takes time (Domenici and Blake, 1991; Ellerby and Altringham, 2001), and the initial velocity after turning was dependent on |α| in our experiment (see Figure 4B), $T_1$ is given by a function of |α| ( $T_1($|α|)). Therefore, T_prey can be expressed as:

When the prey reaches the first part of the safety zone edge, T_pred can be expressed as:

When the prey reaches the second part of the safety zone edge, T_pred can be expressed as:

When the prey reaches the third or fourth part of the safety zone edge, T_pred can be expressed as:

From Equations 1–16, we can calculate $T_{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}}$ in response to the changes of α and β, from D₁, D_width, D_attack, R_device, D_initial, D_react, R_turn, U_prey, U_pred, and ${\displaystyle T_1\left(|\alpha |\right)}$. Given that the escape success is assumed to be dependent on ${\displaystyle T_{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}}}$, the theoretically optimal ET can be expressed as:

The datasets (Dataset1-5) of the escape response in P. major, used for statistical analysis and figures, and the R code (Source code 1-3) for the mathematical model, statistical analysis, and figures are available in Figshare: https://doi.org/10.6084/m9.figshare.17021930.v1.

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Author details

1. Yuuki Kawabata

   Graduate School of Fisheries and Environmental Sciences, Nagasaki University, Nagasaki, Japan

   Contribution

   Conceptualization, Resources, Data curation, Software, Formal analysis, Supervision, Funding acquisition, Validation, Investigation, Visualization, Methodology, Writing - original draft, Project administration, Writing – review and editing

   For correspondence

   yuuki-k@nagasaki-u.ac.jp

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0001-8267-5199

2. Hideyuki Akada

   Faculty of Fisheries, Nagasaki University, Nagasaki, Japan

   Contribution

   Investigation

   Competing interests

   No competing interests declared

3. Ken-ichiro Shimatani

   The Institute of Statistical Mathematics, Tachikawa, Japan

   Contribution

   Formal analysis, Writing – review and editing

   Competing interests

   No competing interests declared

4. Gregory Naoki Nishihara

   Institute for East China Sea Research, Organization for Marine Science Technology, Nagasaki University, Nagasaki, Japan

   Contribution

   Formal analysis, Writing – review and editing

   Competing interests

   No competing interests declared

5. Hibiki Kimura

   Graduate School of Fisheries and Environmental Sciences, Nagasaki University, Nagasaki, Japan

   Contribution

   Investigation, Visualization

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0003-3710-2564

6. Nozomi Nishiumi

   1. Graduate School of Fisheries and Environmental Sciences, Nagasaki University, Nagasaki, Japan
   2. National Institute for Basic Biology, Okazaki, Japan

   Contribution

   Investigation, Writing – review and editing

   Competing interests

   No competing interests declared

7. Paolo Domenici

   1. CNR-IAS, Località Sa Mardini, Oristano, Italy
   2. CNR-IBF, Area di Ricerca San Cataldo, Pisa, Italy

   Contribution

   Conceptualization, Formal analysis, Supervision, Writing – review and editing

   Competing interests

   No competing interests declared

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