Geometric model incorporating prey’s turn and predator attack endpoint explains multiple preferred escape trajectories
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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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.)
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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.
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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.
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