Multiple preferred escape trajectories are explained by a geometric model incorporating prey\u27s turn and predator attack endpoint

The escape trajectory (ET) of prey - measured as the angle relative to the predator\u27s 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\u27s 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 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. Our results open new avenues of investigation for understanding how animals choose their ETs from behavioral and neurosensory perspectives

Toward the quantification of a conceptual framework for movement ecology using circular statistical modeling.

To analyze an animal's movement trajectory, a basic model is required that satisfies the following conditions: the model must have an ecological basis and the parameters used in the model must have ecological interpretations, a broad range of movement patterns can be explained by that model, and equations and probability distributions in the model should be mathematically tractable. Random walk models used in previous studies do not necessarily satisfy these requirements, partly because movement trajectories are often more oriented or tortuous than expected from the models. By improving the modeling for turning angles, this study aims to propose a basic movement model. On the basis of the recently developed circular auto-regressive model, we introduced a new movement model and extended its applicability to capture the asymmetric effects of external factors such as wind. The model was applied to GPS trajectories of a seabird (Calonectris leucomelas) to demonstrate its applicability to various movement patterns and to explain how the model parameters are ecologically interpreted under a general conceptual framework for movement ecology. Although it is based on a simple extension of a generalized linear model to circular variables, the proposed model enables us to evaluate the effects of external factors on movement separately from the animal's internal state. For example, maximum likelihood estimates and model selection suggested that in one homing flight section, the seabird intended to fly toward the island, but misjudged its navigation and was driven off-course by strong winds, while in the subsequent flight section, the seabird reset the focal direction, navigated the flight under strong wind conditions, and succeeded in approaching the island

Distributions of the maximum likelihood estimates.

<p>The horizontal axes indicate the classes of each parameter, and the vertical axes show the frequencies from 200 simulations. Shaded: F14, white: F15. The arrows indicate the true parameter values.</p

GPS trajectory of an adult <i>Calonectris leucomelas</i> breeding on Sangan Island.

<p>The bold lines represent the 15 selected flight sections (F1–F15), while the gray lines represent other flying trajectories. The arrows indicate the entire observed flight direction GPS coordinates when the bird appeared to be on the sea surface are not displayed, and the two flights are connected by the thin line.</p

Example 2: homeward flights.

<p>(A) GPS trajectories of F14 and F15 (bold lines), 5 examples of simulated trajectories of each selected model (thin lines), and 100 examples of final locations of 1000 simulated trajectories produced by each selected model (green: F14, red: F15, throughout). The gray lines indicate 5 examples produced from the model that were not selected for F15 (setting the island as the focal point). The inset enlarges the takeoff section. (B, C) Heading distribution; the observed heading distribution (─•─), and mean (bold line) and 95% confidence envelopes (thin lines). (D) The red thin, red bold, thin, and dotted lines represent the regression curves, modes, medians, and 25% and 75% quartiles of the selected model for F15, respectively. Dots are points from the scatter diagram indicating the direction of heading (Θ<i>_t</i>_–1, Θ<i>_t</i>). The four rotated curves indicate the density functions of the Kato–Jones distributions for θ<i>_t</i> when θ<i>_t</i>_–1 is π/4, π/2, 3π/4, and π. (E) A scatter diagram showing the relationship between the direction of heading and observed speed (<i>T</i> = 1) and the expected speed when <i>V_t</i>_–1 = {the mean over each flight section} and Θ<i>_t</i> – Θ<i>_t</i>₋₁ = 0.</p

Examples of trajectories simulated by the movement model.

<p>(A) Examples of the transformation θ<i>_t</i> = <i>M</i>(θ<i>_t</i>_–1; α, <i>w</i>) (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0050309#pone.0050309.e009" target="_blank">equation (5</a>), α = 0). (B) Examples of the probability density functions of the von Mises distribution (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0050309#pone.0050309.e017" target="_blank">equation (7</a>), μ = 0). (C) Examples of trajectories produced by <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0050309#pone.0050309.e007" target="_blank">equation (3</a>) using the heading model (4) with the transformation in (A) and the von Mises distribution in (B). (D) An example of trajectory produced by <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0050309#pone.0050309.e007" target="_blank">equation (3</a>) using the heading model (12). In (C) and (D), speeds were fixed at 1, the grid unit is 10, and the first 50 steps are shown. The same random samples from the von Mises distribution of κ = 6 were used for the four red trajectories.</p

Example 3: searching flights.

<p>(A, C) Observed GPS trajectory of F7 (A) and F10 (C). The bold lines are observed trajectories and the thin lines are five examples of simulated trajectories for each selected model. The insets enlarge each takeoff section. (B, D) The thin, bold, dashed, and dotted lines represent the regression curves, modes, medians, and 25% and 75% quartiles of each selected model, respectively. The dots represent points from scatter diagrams between Θ<i>_t</i>_–1 and Θ<i>_t</i> for every 4 s (B) and 3 s (D). The four rotated curves are the density functions of the Kato–Jones distributions for θ<i>_t</i> when θ<i>_t</i>_–1 is −π/3, 0, π/3, and 2π/3 (B) and 0, π/3, 2π/3, and π (D).</p

Example 1: outward flight.

<p>(A) Observed GPS trajectory of F2 (bold line, the final location is indicated by the large gray circle), five examples of simulated trajectories (thin lines), 100 examples of the final locations (dots) of 1000 simulated trajectories produced by the selected model. (B) Black dots: Points from the scatter diagram showing the relationship between speed and direction of heading. White dots: Predicted speeds by the selected model from (Θ<i>_t</i>, <i>V_t</i>_–1, cos(Θ<i>_t</i> – Θ<i>_t</i>_–1)) plotted on Θ<i>_t</i>. Bold line: The expected speed as a function of the direction of heading when the previous speed was fixed as the mean over the flight section and angular velocity was 0. (C) The heading distribution: Observed trajectory (─•─), and mean (bold line) and 95% confidence envelopes (thin lines) derived from 1000 simulated trajectories.</p

The method of generating the Kato–Jones distributions.

<p>If random variables of the von Mises distributions in (A) are transformed by <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0050309#pone.0050309.e023" target="_blank">equation (10′</a>) shown in (B), the Kato–Jones distributions in (C) are obtained. In (C), the bold/thin lines are used when the transformation displayed by the bold/thin line in (B) was used, and the red/blue line was used when the von Mises distribution displayed by the red/blue line in (A) was used. The transformations in the bottom row are omitted.</p