Geometric model incorporating prey’s turn and predator attack endpoint explains multiple preferred escape trajectories
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 Evaluated articles (eLife)
 Neuroscience (eLife)
Abstract
The escape trajectory (ET) of prey – measured as the angle relative to the predator’s approach path – plays a major role in avoiding predation. Previous geometric models predict a single ET; however, many species show highly variable ETs with multiple preferred directions. Although such a high ET variability may confer unpredictability to avoid predation, the reasons why animals prefer specific multiple ETs remain unclear. Here, we constructed a novel geometric model that incorporates the time required for prey to turn and the predator’s position at the end of its attack. The optimal ET was determined by maximizing the time difference of arrival at the edge of the safety zone between the prey and predator. By fitting the model to the experimental data of fish Pagrus major we show that the model can clearly explain the observed multiple preferred ETs. By using the same model, we were able to explain different patterns of ETs empirically observed in other species (e.g., insects and frogs): a single preferred ET and multiple preferred ETs at small (20–50°) and large (150–180°) angles from the predator. Our results open new avenues of investigation for understanding how animals choose their ETs from behavioral and neurosensory perspectives.
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Author Response
Reviewer #1 (Public Review):
It's here where my very mild (I truly liked this article  it is well done, well written, and creative) comments arise. The implications for stochastic strategies immediately emerge in the early results  bimodal strategies come about from the introduction of two variables. There is not enough credence given to the field of stochastic behavior in the introduction  the introduction focuses too much on previous models of predatorprey interaction, and in fact, Figure 1, which should set up the main arguments of the article, shows a model that is only slightly different (slight predator adjustment) that is eventually only addressed in the Appendix (see below). The question of "how and when do stochastic strategies emerge?" is a big deal. Figure 1 should set up a dichotomy: optimal strategies …
Author Response
Reviewer #1 (Public Review):
It's here where my very mild (I truly liked this article  it is well done, well written, and creative) comments arise. The implications for stochastic strategies immediately emerge in the early results  bimodal strategies come about from the introduction of two variables. There is not enough credence given to the field of stochastic behavior in the introduction  the introduction focuses too much on previous models of predatorprey interaction, and in fact, Figure 1, which should set up the main arguments of the article, shows a model that is only slightly different (slight predator adjustment) that is eventually only addressed in the Appendix (see below). The question of "how and when do stochastic strategies emerge?" is a big deal. Figure 1 should set up a dichotomy: optimal strategies are available (i.e., those that minimize Tdiff) which would predict a single unimodal strategy. Many studies often advocate for Bayesian optimal behavior, but multimodal strategies are the reality in this study  why? Because if you consider the finite attack distance and inability of fish to evoke maximum velocity escapes while turning, it actually IS optimal. That's the main point I think of the article and why it's a broadly important piece of work. Further framing within the field of stochastic strategies (i.e., stochastic resonance) could be done in the introduction.
We appreciate the comment provided by the reviewer. We changed the second paragraph of the introduction so as to focus more on the protean tactic (stochasticity). We added a new figure (Figure 1 in the new version) to conceptually show the escape trajectories (ETs) of a pure optimal tactic, a pure protean tactic, a combination of optimal and protean tactics, and an empirically observed multimodal pattern. We explained each tactic and described that the combination of the optimal and protean tactics still cannot explain the empirically observed multiple preferred ETs.
The revised paragraph (L4966) is as follows: Two different escape tactics (and their combination) have been proposed to enhance the success of predator evasion [16, 17]: the optimal tactic (deterministic), which maximizes the distance between the prey and the predator (Figure 1A) [4, 14, 15, 18], and the protean tactic (stochastic), which maximizes unpredictability to prevent predators from adjusting their strike trajectories accordingly (Figure 1B) [1922]. Previous geometric models, which formulate optimal tactics, predict a single ET that depends on the relative speeds of the predator and the prey [4, 14, 15, 18], and additionally, predator’s turning radii and sensorymotor delay in situations where the predator can adjust its strike path [2325]. 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 [26]). Whereas some animals exhibit unimodal ET patterns that satisfy the prediction of the combined tactics or optimal tactic with behavioral imprecision (e.g., [27]), many animal species show multimodal ETs within a limited angular sector (esp., 90–180°) (Figure 1D) (e.g., [4, 5, 28]). To explore the discrepancy between the predictions of the models and empirical data, some researchers have hypothesized mechanical/sensory constraints [17, 29]; however, the reasons why certain animal species prefer specific multiple ETs remain unclear.
All experiments are well controlled (I especially liked the control where you varied the cutoff distance given that it is so critical to the model). Some of the figures require more labeling and the main marquee Figure 1 needs an overhaul because (1) the predator adjustment model that is only addressed in the Appendix shouldn't be central to the main introductory figure  it's the equivalent of the models/situations in Figure 6, and probably shouldn't take up too much space in the introductory text either (2) the drawing containing the model variables could be more clear and illustrative.
(1) According to this comment and comment #11 from reviewer #2, we moved the two panels in the figure (Figure 1B and D in the original version) to Appendixfigure 1, and accordingly, we changed the first paragraph of the Model section so as to clearly describe that we focus on Domenici’s model in this study (L103108).
As for Figure 6 (Figure 7 in the new version) and related parts, we tempered our claims to clearly describe that our model has only the potential to explain the different patterns of escape trajectories observed in previous works. We would like to keep this figure in the main text because it is fundamental to explain the potential applicability of our model to other predatorprey systems.
(2) To alleviate the burden for readers, we added the model variables to the figure and made them colored (Figure 2B in the new version).
Finally, I think a major question could be posed in the article's future recommendations: Is there some threshold for predator learning that the fish's specific distribution of optimal vs. suboptimal choice prevents from happening? That is, the suboptimal choice is performed in proportion to its ability to differentiate Tdiff. This is "bimodal" in a sense, but a probabilistic description of the distribution (e.g., a bernoulli with p proportional to beta) would be really beneficial. Because prey capture is a zerosum game, the predator will develop new strategies that sometimes allow it to win. It would be interesting if eventually the bernoulli description could be run via a sampler to an actual predator using a prey dummy; one could show that the predator eventually learns the pattern if the bernoulli for choosing optimal escape is set too high, and the prey has balanced its choice of optimal vs. suboptimal to circumvent predator learning.
We thank the reviewer for this constructive comment. Actually, we are now developing a dummy prey system. We added the following sentence in the Discussion to mention future research.
The added sentence (L496499): Further research using a real predator and dummy prey (e.g., [48]) controlled to escape toward an optimal or suboptimal ET with specific probabilities would be beneficial to understand how the prey balances the optimal and suboptimal ETs to circumvent predator learning.
Reviewer #2 (Public Review):
First, it is unclear how the dummy predator is actuated. The description in the Methods section does not clearly address how rubber bands are used for this purpose.
To clearly mention how the dummy predator was actuated by rubber bands, we added a figure (Figure 3figure supplement 3B) and the following sentences.
The added sentences (L608611): The dummy predator was held in place by a metal pipe anchored to a fourwheel dolly, which is connected to a fixed metal frame via two plastic rubber bands (Figure 3—figure supplement 3B). The wheel dolly was drawn back to provide power for the dummy predator to strike toward the prey.
Second, the predator's speed, which previous research has identified as a critical factor during predatorprey interactions, is not measured from the motion of the dummy predator in the experiments. Instead, it is estimated using an optimization algorithm that utilizes the mathematical model and the preyspecific parameters. It is unclear why the authors chose this method over measuring velocity from their experiments. Since the prey fish are responding to a dummy predator moving toward them at a particular speed during the interaction, it is important to measure the speed of the predator or clearly explain why estimating it using an optimization procedure is more appropriate.
We chose this method (optimizing predator speed from the prey’s viewpoint) because there was no significant effect of predator speed on the escape trajectory in our experiment (L203208). In other words, we considered that, at least in our case, the prey did not change the escape trajectory in response to the predator speed, and thus it would be more appropriate to use a specific predator speed estimated through an optimization algorithm from the prey’s point of view. It may be appropriate to use measured predator speed in other cases where the prey adjusts the escape trajectory in response to the change in predator speed. Therefore, we conducted a further analysis using actual predator speeds (both the predator speed at the onset of escape response, and the mean speed for the predator to cover the distance between the predator and prey). The results show that the model fit became worse when using measured predator speed per trial compared to the model using the fixed predator speed estimated through the optimization procedure (Table 3—source data 1; Figure 5—figure supplement 1). We added the above explanation in L219226.
One of the major claims of this article is that the model can explain escape trajectories observed in other predatorprey systems (presented in Figure 6). Figure 6 panels AC show the escape responses of different prey in response to some threatening stimuli. Further, panels DF suggest that the empirical data can be predicted with the model. But the modeling parameters used to produce the escape trajectories in DF are derived from the authors' experiments with fish, instead of the experiments with the species shown in panels AC.
We thank the reviewer for this comment. We agree that this part in the previous version was an overinterpretation. Therefore, we tempered our statements to simply suggest that our approach has the potential to explain multiple ETs observed in other taxa. The revised sentences are as follows.
Abstract (L2730): By changing the parameters of the same model within a realistic range, we were able to produce various patterns of ETs empirically observed in other species (e.g., insects and frogs): a single preferred ET and multiple preferred ETs at small (20–50°) and large (150–180°) angles from the predator.
Results (L395407): Potential application of the model to other ET patterns. ...(sip)... To investigate whether our geometric model has the potential to explain these different ET patterns, we changed the values of model parameters (e.g., Upred, Dattack) within a realistic range, and explored whether such adjustments can produce the ET patterns observed in the original work. ...(sip)... These results indicate that our model has the potential to explain various patterns of observed animal escape trajectories.
Discussion (L538548): We show that our model has the potential to explain other empirically observed ET patterns (Figure 7). ...(sip)... 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 predatorprey systems.

Evaluation Summary:
This article may be of interest to researchers working on predatorprey interactions in the fields of biomechanics and neurosensory biology. It presents a mathematical model that outputs possible escape trajectories given parameters relevant to the predatorprey system of interest. The premise of the modeling is attractive, as it includes the time required for prey to turn, but the methods as presently reported raise questions about the validity of some of the conclusions.
(This preprint has been reviewed by eLife. We include the public reviews from the reviewers here; the authors also receive private feedback with suggested changes to the manuscript. Reviewer #1 agreed to share their name with the authors.)

Reviewer #1 (Public Review):
This article clearly illustrates the limitations of previous predator escape models that (1) fail to incorporate the initial orientation of prey relative to predators, and (2) do not properly describe the endpoint of predator attacks, instead assuming infinite trajectories. The approach is novel and the implications for stochastic strategies are important. Some subtle rearrangements would improve the presentation of the data.
The correspondence between the presented behavioral data and model instantly validates the incorporation of predator attack distance and initial orientation of the prey into escape models. I am completely convinced that the lack of the two incorporated variables prevented the accurate reconstruction of ETs. These two variables create distributions over escape choices that are eventually …
Reviewer #1 (Public Review):
This article clearly illustrates the limitations of previous predator escape models that (1) fail to incorporate the initial orientation of prey relative to predators, and (2) do not properly describe the endpoint of predator attacks, instead assuming infinite trajectories. The approach is novel and the implications for stochastic strategies are important. Some subtle rearrangements would improve the presentation of the data.
The correspondence between the presented behavioral data and model instantly validates the incorporation of predator attack distance and initial orientation of the prey into escape models. I am completely convinced that the lack of the two incorporated variables prevented the accurate reconstruction of ETs. These two variables create distributions over escape choices that are eventually claimed to balance behavioral perfection (i.e., minimization of Tdiff) with unpredictability (i.e., the choice of slightly suboptimal ETs when the effect on Tdiff is negligible relative to predator capture times). This is a case where precision is sometimes favored over variability and other times variability over precision.
It's here where my very mild (I truly liked this article  it is well done, well written, and creative) comments arise. The implications for stochastic strategies immediately emerge in the early results  bimodal strategies come about from the introduction of two variables. There is not enough credence given to the field of stochastic behavior in the introduction  the introduction focuses too much on previous models of predatorprey interaction, and in fact, Figure 1, which should set up the main arguments of the article, shows a model that is only slightly different (slight predator adjustment) that is eventually only addressed in the Appendix (see below). The question of "how and when do stochastic strategies emerge?" is a big deal. Figure 1 should set up a dichotomy: optimal strategies are available (i.e., those that minimize Tdiff) which would predict a single unimodal strategy. Many studies often advocate for Bayesian optimal behavior, but multimodal strategies are the reality in this study  why? Because if you consider the finite attack distance and inability of fish to evoke maximum velocity escapes while turning, it actually IS optimal. That's the main point I think of the article and why it's a broadly important piece of work. Further framing within the field of stochastic strategies (i.e., stochastic resonance) could be done in the introduction.
All experiments are well controlled (I especially liked the control where you varied the cutoff distance given that it is so critical to the model). Some of the figures require more labeling and the main marquee Figure 1 needs an overhaul because (1) the predator adjustment model that is only addressed in the Appendix shouldn't be central to the main introductory figure  it's the equivalent of the models/situations in Figure 6, and probably shouldn't take up too much space in the introductory text either (2) the drawing containing the model variables could be more clear and illustrative.
Finally, I think a major question could be posed in the article's future recommendations: Is there some threshold for predator learning that the fish's specific distribution of optimal vs. suboptimal choice prevents from happening? That is, the suboptimal choice is performed in proportion to its ability to differentiate Tdiff. This is "bimodal" in a sense, but a probabilistic description of the distribution (e.g., a bernoulli with p proportional to beta) would be really beneficial. Because prey capture is a zerosum game, the predator will develop new strategies that sometimes allow it to win. It would be interesting if eventually the bernoulli description could be run via a sampler to an actual predator using a prey dummy; one could show that the predator eventually learns the pattern if the bernoulli for choosing optimal escape is set too high, and the prey has balanced its choice of optimal vs. suboptimal to circumvent predator learning.
Overall, a very good article.

Reviewer #2 (Public Review):
The major strength of this research is its use of experimental data to validate the model. The authors use a dummy predator to elicit escape responses from prey fish, Pragus major. Data from these experiments are then used to estimate several parameters of their mathematical model. I have two concerns regarding the authors' methods.
First, it is unclear how the dummy predator is actuated. The description in the Methods section does not clearly address how rubber bands are used for this purpose. Second, the predator's speed, which previous research has identified as a critical factor during predatorprey interactions, is not measured from the motion of the dummy predator in the experiments. Instead, it is estimated using an optimization algorithm that utilizes the mathematical model and the preyspecific …
Reviewer #2 (Public Review):
The major strength of this research is its use of experimental data to validate the model. The authors use a dummy predator to elicit escape responses from prey fish, Pragus major. Data from these experiments are then used to estimate several parameters of their mathematical model. I have two concerns regarding the authors' methods.
First, it is unclear how the dummy predator is actuated. The description in the Methods section does not clearly address how rubber bands are used for this purpose. Second, the predator's speed, which previous research has identified as a critical factor during predatorprey interactions, is not measured from the motion of the dummy predator in the experiments. Instead, it is estimated using an optimization algorithm that utilizes the mathematical model and the preyspecific parameters. It is unclear why the authors chose this method over measuring velocity from their experiments. Since the prey fish are responding to a dummy predator moving toward them at a particular speed during the interaction, it is important to measure the speed of the predator or clearly explain why estimating it using an optimization procedure is more appropriate.
One of the major claims of this article is that the model can explain escape trajectories observed in other predatorprey systems (presented in Figure 6). Figure 6 panels AC show the escape responses of different prey in response to some threatening stimuli. Further, panels DF suggest that the empirical data can be predicted with the model. But the modeling parameters used to produce the escape trajectories in DF are derived from the authors' experiments with fish, instead of the experiments with the species shown in panels AC.
The conclusions of this work, in its current form, are not fully supported by the results and analyses. Therefore, the general utility of the model needs to be demonstrated.

