A quantitative framework for investigating risk of deadly collisions between marine wildlife and boats

Transcription

1 Methods in Ecology and Evolution 2016, 7, doi: / X A quantitative framework for investigating risk of deadly collisions between marine wildlife and boats Julien Martin 1,2*, Quentin Sabatier 1,3,4, Timothy A. Gowan 1, Christophe Giraud 5, Eliezer Gurarie 6, Charles Scott Calleson 7,JoelG.Ortega-Ortiz 1,8, Charles J. Deutsch 9, Athena Rycyk 10 and Stacie M. Koslovsky 1 1 Florida Fish and Wildlife Conservation Commission, Fish and Wildlife Research Institute, St Petersburg, FL 33701, USA; 2 Southeast Ecological Science Center, U.S. Geological Survey, 7920 NW 71st Street, Gainesville, FL 32653, USA; 3 Ecole Polytechnique, CMAP, UMR CNRS 7641, Palaiseau Cedex, France; 4 Department of Wildlife Ecology and Conservation, University of Florida, Gainesville, FL 32611, USA; 5 Laboratoire de Mathematiques d Orsay, UMR 8628, Universite Paris-Sud, F Orsay Cedex, France; 6 Department of Biology, University of Maryland, College Park, MD 20742, USA; 7 Florida Fish and Wildlife Conservation Commission, Imperiled Species Management Section, Tallahassee, FL 32399, USA; 8 College of Marine Science, University of South Florida, St Petersburg, FL 33701, USA; 9 Florida Fish and Wildlife Conservation Commission, Fish and Wildlife Research Institute, Gainesville, FL 32601, USAand 10 Department of Oceanography, Florida State University, Tallahassee, FL 32306, USA Summary 1. Speed regulations of watercraft in protected areas are designed to reduce lethal collisions with wildlife but can have economic consequences. We present a quantitative framework for investigating the risk of deadly collisions between boats and wildlife. 2. We apply encounter rate theory to demonstrate how marine mammal boat encounter rate can be used to predict the expected number of deaths associated with management scenarios. We illustrate our approach with management scenarios for two endangered species: the Florida manatee Trichechus manatus latirostris and the North Atlantic right whale Eubalaena glacialis. We used a Monte Carlo simulation approach to demonstrate the uncertainty that is associated with our estimate of relative mortality. 3. We show that encounter rate increased with vessel speed but that the expected number of encounters varies depending on the boating activities considered. For instance, in a scenario involving manatees and boating activities such as water skiing, the expected number of encounters in a given area (in a fixed time interval) increased with vessel speed. In another scenario in which a vessel made a transit of fixed length, the expected number of encounters decreases slightly with boat speed. In both cases, the expected number of encounters increased with distanced travelled by the boat. For whales, we found a slight reduction (~01%) in the number of encounters under a scenario where speed is unregulated; this reduction, however, is negligible, and overall expected relative mortality was ~30% lower under the scenario with speed regulation. The probability of avoidance by the animal orvesselwassetto0becauseoflackofdata,butweexplored the importance of this parameter on the model predictions. In fact, expected relative mortality under speed regulations decreases even further when the probability of avoidance is a decreasing function of vessel speed. 4. By applying encounter rate theory to the case of boat collisions with marine mammals, we gained new insights about encounter processes between wildlife and watercraft. Our work emphasizes the importance of considering uncertainty when estimating wildlife mortality. Finally, our findings are relevant to other systems and ecological processes involving the encounter between moving agents. Key-words: animal movement, effectiveness of speed zones, encounter rate, Florida manatee, marine mammals, North Atlantic right whale, protection zones, speed zones, wildlife collision Introduction The creation of protection zones and the regulation of the speed of watercraft are viewed as primary management actions taken to protect some species of marine wildlife from lethal *Correspondence author. julienmartin@usgs.gov collisions (Calleson & Frohlich 2007; Hazel et al. 2007; Vanderlaan & Taggart 2007; Van Der Hoop, Vanderlaan & Taggart 2012; Bauduin et al. 2013). Regulating speed or rerouting vessel traffic can have economic consequences, so it is important to quantify potential effects of watercraft on animal populations. This knowledge, in turn, can help identify costeffective solutions for balancing multiple objectives, such as 2015 The Authors. Methods in Ecology and Evolution 2015 British Ecological Society This article has been contributed to by U.S. Governments employees and their work is in the public domain in the USA.

2 Risk of wildlife collisions 43 maintaining sustainable populations of marine mammals while minimizing cost of regulations. Ultimately, managers are interested in linking regulations to a projected reduction in mortality or injury events. Several co-occurring processes leading to death or injury of marine wildlife from boat strikes are clearly related to boat speed. For example, the probability of avoidance between boats and marine animals may be reduced as boat speed increases (Calleson & Frohlich 2007; Hazel et al. 2007), and the severity of injuries caused by an impact is likely to increase with vessel speed (Calleson & Frohlich 2007). These processes can be synthesized into a simplified conceptual framework that links boat speed and other factors to marine mammal deaths and their effects on the overall population (Fig. 1). The influence of one of these processes in particular can be difficult to grasp intuitively: the rate at which trajectories of animals and boats intersect in space and time. We define the encounter rate as the rate at which an animal and a boat will be close enough in space and time to potentially collide (Hutchinson & Waser 2007; Gurarie & Ovaskainen 2013a). Encounters should not be confused with collisions. Collisions imply that an animal was actually struck by a boat (which may depend on other parameters, for instance the probability of avoidance by the animal or boat). Some authors have indicated that there is a negative relationship between boat speed and the probability of encounter (Vanderlaan & Taggart 2007). To date, the encounter process in the context of watercraft strikes has been approached primarily through simulations (Van Der Hoop, Vanderlaan & Taggart 2012). Although encounter rate theory (Gurarie & Ovaskainen 2013a) is relevant to understanding the process of collision between boats and marine mammals, there have been few detailed examinations of these concepts for this important application. Encounter rate theory has been developed for other applications, including naval operations research (Koopman 1956), encounter rate in the context of community structure of zooplankton and animal movement (Gerritsen & Strickler 1977; Evans 1989; Hutchinson & Waser 2007; Gurarie & Ovaskainen 2013a), and collisions between marine wildlife and renewable energy devices (Wilson et al. 2007). Here, we apply encounter rate theory to marine mam- Fig. 1. Conceptual diagram describing the relationship between key components involved in the collision process between boats and marine mammals. Number of marine mammal deaths: Death; number of marine mammals and boats: Number; habitat characteristics (e.g. waterway configuration, bathymetry, presence of seagrass, sea surface temperature) are included in Covariates; the number of encounters (as a function of animal/vessel densities and vessel speed): Encounters; the probability of the animal s being within striking depth during an encounter: P(strike depth); the number of potential collisions: Collisions; the probabilities of avoidance by boaters and/or marine mammals combined: P(avoid) which may be affected by speed of the boat; the probability of lethal injury given strike speed: P(death strike speed); boat and marine mammal Speed and Size; compliance of boaters with regulations: Compliance and Regulations. Only a fraction of the total number of deaths are reported: Carcass, which is determined by a probability of recovery: P(recover).

3 44 J. Martin et al. mal boat collisions and present our own analytical presentation of the solution in two dimensions, the most relevant case in many ecological applications. In our model, the encounters correspond to first encounters (i.e. an animal can encounter a given boat only once per boat transit, which is applicable in most situations where the vessel moves faster than the animal). Our solution relates the number of encounters to time, area, the number of boats and marine mammals, and the speed of boats and marine mammals (Fig. 1). We used a Monte Carlo simulation approach (MCS) to incorporate uncertainty, which has seldom been accounted for in estimates of mortality rates due to watercraft collisions (see Conn & Silber (2013) for an alternative approach). We illustrate our approach with two endangered mammals, the Florida manatee Trichechus manatus latirostris and the North Atlantic right whale Eubalaena glacialis. Although we used the best available parameter values based on the scientific literature or empirical data, we caution against the use of specific values from our results for management purposes. Instead, the primary goals of our study were to gain general insights about processes involved in risk of deadly collisions and to provide an analytical framework for conducting more detailed analyses for evaluating potential management actions aimed at reducing risk of collisions. Although this framework can help improve the design of protection zones and evaluate the effectiveness of speed zones for marine wildlife, it is also applicable to other systems (e.g. road traffic in protected areas). Materials and methods ANALYTICAL APPROACH FOR ESTIMATING ENCOUNTER RATE We consider a boat of size r b andananimalofsizer m moving in an area with surface S between start time t I and final time t F. We assume that the boat moves at constant velocity v b. The number of mammals in the area is assumed to be distributed as a Poisson random variable with parameter k m. The distribution of the speed v m of the moving animals is assumed to be independent of time, the location of the animal and its orientation. Here, we define an encounter at time t if the distance between an animal and a boat is >r m + r b at time t, and r m + r b at some time during the interval [t, t + e]. We refer to the critical distance of encounter as r c = r m + r b. Some of the concepts described below are based on concepts discussed by Koopman (1956) in the context of naval operation research and Gerritsen & Strickler (1977) in the context of plankton movement. Wilson et al. (2007) applied equations developed by Gerritsen & Strickler (1977) to the problem of marine mammal collision with rotating turbines. We used a two-dimensional case of the problem (see Koopman (1956) for naval operations applications), which is more appropriate for vessel strikes. We derived an equation for the mean encounter rate k e for one boat and one animal (see Appendix S1, Supporting information for derivations): Z k e ¼ 2 r c Iðv m ; v b Þf v ðv m Þ dv m eqn 1 S vm where S is the surface area of the study region; r c is the critical distance of encounter; function I is a monotonically increasing function of the velocities (described in Appendix S1); and f v ðv m Þ is the probability distribution of the animal velocity. We now consider two limit scenarios that are relevant to our study. In the fixed time scenario, a boat will spend a fixed amount of time in an area, travelling without a specific destination; therefore, the faster the boat travels, the greater the distance it covers. This scenario is relevant to manatees because it is characteristic of some recreational boating activities (e.g. water skiing). The number of encounters during the boat s time in the area can be described by a Poisson distribution with parameter k FT. The Poisson parameter k FT is the encounter rate k e multiplied by the time interval t d (Gerritsen & Strickler 1977); specifically, t d is the fixed duration of time travelled by the boat within the study area: k FT ¼ t d k e eqn 2 In the fixed distance scenario, a boat travels to a specific destination, and the travel distance is independent of the boat s speed. This is the more relevant scenario for whales, as commercial vessels tend to go from origin to destination in the shortest distance possible. It also applies to manatees and other marine mammals whose habitat includes navigable channels. The number of encounters can be described by a Poisson distribution with parameter k FD : k FD ¼ d v b k e eqn 3 where d is the fixed distance travelled by the boat as it crosses the study area. We estimated encounter rates for the fixed time and fixed distance scenarios for manatees. For whales, we focused exclusively on the fixed distance scenario. Programming scripts written in R (R Core Team 2015) to estimate encounter rates are available on Dryad (see Data accessibility). APPLICATIONS TO WHALES AND MANATEES Manatee data and analysis For our manatee example, we considered encounter rate with boats in a282 km 2 section of Lemon Bay in south-west Florida (Fig. S2). For simplicity, we considered that on average, one manatee and one boat were present at all times in the study area. The abundance process for manatees was assumed to follow a Poisson distribution with parameter k m set to one. We assumed that the manatee s swimming speed followed a Weibull distribution (shape: 072, scale: 016; mean: 020 m/s, SD = 028), estimated from manatees equipped with a DTAG (a multisensor digital acoustic tag; Johnson & Tyack 2003) and an Argoslinked GPS tag attached via a tether (for details, see Appendix S2 and Rycyk (2013)). The radius of encounter for this example was based on boat width (234 m) and an average manatee width (063 m, SD = 0046, n = 6, FWC data). Whale data and analysis We also provide an illustrative application of our model for estimating encounter rates between right whales and vessels in the southeastern USA. For this example, we summarized vessel traffic collected through the Automatic Identification System (AIS) (Silber & Bettridge 2010) in the recommended shipping route off Jacksonville, FL (Fig. S3), over a 2-week period (January 16 31, 2010). We considered the bounded shipping route as our study area (S = 7541 km 2 ). As with the manatee example, we set the expected

4 Risk of wildlife collisions 45 number of whales to one. We considered a whale with a length of 14 m, width of 35 m and a swimming speed that followed a Weibull distribution (shape: 148, scale: 043; mean: 039 m/s, SD: 027) (Fortune et al. 2012; Miller et al. 2012; Hain et al. 2013). A total of 421 vessel transits within the shipping route were recorded through AIS during the study, and we used the mean vessel length (146 m) and vessel width (22 m) for all vessels in this example. We conceptualized the radius of encounter, r c, in three ways to determine the sensitivity of the model to this parameter. (i) The areas of the whale and vessel were estimated based on their length (L) and width (W); these qffiffiffiffiffiffiffiffi areas were considered as circles whose radii were calculated as r ¼ LW p. The radius of encounter was estimated as the sum of the circles radii (r c_disk = = 3593 m). The radius of encounter was also estimated as the sum of vessel width and (ii) whale width, for the scenario in which the whale sbodyisparalleltothevessel (r c_width = = 255 m), or (iii) whale length, where the whale is perpendicular to vessel (r c_length = = 36 m). We calculated encounter rate for values of vessel speed ranging from 1 knot (185 km/h) to 33 knots (6112 km/h). For each vessel speed value, we estimated the duration of a single transit in the area by dividing the mean transit length (278 km) by the vessel speed. To estimate the total number of encounters over a day, we calculated the product of the encounter rate, duration of a transit and the number of transits per day (421/15 = 28). Wethenextrapolated toa duration of 15 or 600 days. Furthermore, we compared the expected number of encounters under the current management case (vessel speed = 10 knots (1852 km/h) (Silber & Bettridge 2010)) with that of the previous conditions (vessel speed = 157 knots (2908 km/h) (Ward-Geiger et al. 2005)); for reference, the weighted mean vessel speed from AIS data during this study period was 96 knots (1778 km/h). Encounters, collisions and expected mortality of North Atlantic right whales As shown in Appendix S3, the expected number of encounters over a period of time t d is k e t d. It is important to understand that the number of encounters is different from the number of collisions; indeed, the collision rate is as follows: k c ¼ k e PðStrike depthþð1 PðAvoidance m ÞÞ ð1 PðAvoidance b ÞÞ eqn 4a where P(Strike depth) is the probability that the whale is within the striking depth during an encounter (Fig. 1). P(Avoidance) is the probability of avoidance by the mammal m or the boat driver b and may depend on v b. Eqn 4a implies that there is no covariation between the avoidance probabilities and would require these probabilities to be estimated separately. An alternative parameterization would be as follows: k c ¼ k e PðStrike depthþð1 PðAvoidance mb ÞÞ eqn 4b where PðAvoidance mb Þ is the probability of avoidance by the animal and boater combined, which may be easier to estimate. We fixed PðAvoidance mb Þ to 0 because of lack of information, but we explored the importance of this parameter on model predictions (see Appendix S4, Fig. S1). We used a probability of being within striking depth of 06 (SD = 022) based on Hain et al. (1999). This is probably an underestimate because that study focused on the probability that whales were at the surface rather than within striking depth. The death rate is as follows: k e ¼ k c PðDeathjStrike speedþ eqn 5 where P(Death Strike speed) corresponds to the probability of death of a whale at a given striking speed. This probability was obtained from Conn & Silber (2013). The expected number of deaths follows a Poisson distribution (Appendix S3 for justification): Poisðk e t d k m N b Þ eqn 6 where t d is the time on the appropriate scale, N b is the number of boats, and k m is the parameter of the Poisson random variable for the number of mammals, with, k m = c m S, where c m is animal density. The expected number of deaths is therefore: E½Poisðk e t k m N b ÞŠ ¼ k e t k m N b eqn 7 We used a MCS approach to account for uncertainty when estimating the relative number of deaths as a function of the parameters noted in eqns 4 7. In order to make inference about new data (e.g. carcass counts), it is straightforward to convert this MCS model to a Bayesian belief network (BBN, e.g. Smid et al. 2009). In our case, we used a MCS approach to implement the BBN because we did not try to fit models to the data. Each component of the model followed a probability distribution; details about the structure of the MCS model and the distributions are provided in Appendix S5. Because we fixed the probability of avoidance (in eqn 4b) to 0 and animal abundance (k m )toone, the expected number collisions is in fact a measure of the relative number of collisions. Similarly, the number of expected deaths (or expected mortality) should be viewed as a measure of expected relative mortality. An estimate of absolute mortality would require estimating the probability of avoidance which is currently not available and may be a function of vessel speed (see Appendix S4). R scripts for this analysis are available on Dryad (see Data accessibility). Simulations For validation purposes, we compared the results from our analytical framework to simulation results using the right whale example. For the simulations, we considered a circular study area (S = 7541 km 2 )with a single vessel transiting the area in a straight line at constant speed (either 10 knots (1852 km/h) for a relevant management speed, or 118 knots (218 km/h) for an unrealistically low vessel speed) for 278 km. Vessel location was recorded every 1-s time-step for the duration of the transit. The locations of a single whale moving within the area were simulated using either a random walk, representing an arearestricted search behaviour, or a correlated random walk, representing nearly linear directed movement behaviour. The initial location of the whale was randomly selected within the study area, and subsequent locations at 30-s intervals were simulated based on draws for step length from a Weibull distribution (shape: 148, scale: 043*30; equivalent to a mean 039 m/s) and draws for turning angle from a wrapped Cauchy distribution (l = 0, q = 0001 for the random walk; l = 0, q = 0999 for the nearly linear directed movement). If the whale travelled outside the bounds of the area, its location was replaced with a new randomly generated point within the area to approximately replicate k m = 1. Simulated whale locations were interpolated to a 1-s timestep resolution to match the boat s resolution and avoid one agent s jumping over the other (Van Der Hoop, Vanderlaan & Taggart 2012) by assuming a straight path and constant speed between 30-s time-step locations. At each 1-s time-step, the distance between the boat and whale was calculated; an encounter was recorded if this distance was less than r c = 3593 m. We counted the number of encounters per simulation in two ways: (i) allowing a maximum of 1 encounter per simulation (i.e. a whale can encounter the single vessel only once, this is the case that is best approximated by the analytical approach) and

5 46 J. Martin et al. (ii) allowing a maximum of 1 encounter for each 30-s segment of the whale s path (to maintain independence among intersections; otherwise, the number of recorded encounters would approach infinity for a low vessel speed or small time-step).attheendofthevessel stransit (d = 278 km), the simulation was terminated. The simulation (1 whale and 1 boat) was repeated 1000 times for each vessel speed and whale behaviour scenario, and the average number of encounters per vessel transit was calculated. The average number of encounters per transit was then multiplied by 28*15 to estimate the number of encounters from all vessel transits over the 15-day period. We repeated this process 500 times to compute mean and 95% CI using a bootstrap approach. R scripts for this analysis are available on Dryad (see Data accessibility). Results ANALYTICAL APPROACH The manatee model shows a positive relationship between boat speed and encounter rate; note that the effect of uncertainty of manatee speed on encounter rate was small (Fig. 2a). For the fixed time scenario (time spent in the area independent of boat speed; in this case, the boat is assumed to spend 30 min in the study area), boats at faster speeds travel a greater distance in the area of interest, resulting in a correspondingly greater number of expected encounters (Fig. 2b). Because the relationship is nearly linear, an 80% decrease in speed (e.g. from 215knots (3982 km/h) to 43 knots (796 km/h) leads to an 80% decrease in distance travelled and in the expected number of encounters. Conversely, there is a declining relationship between boat speed and expected number of encounters for the fixed distance scenario, but this decrease is negligible for the manatee case less than 02% for the same speed range (Fig. 2c). Although the encounter rate is low when boat speed is slow (Fig. 2a), the expected number of encounters is greatest when boat speed tends towards zero in the fixed distance scenario (Fig. 2c), because time spent in the area tends to infinity. The relationship quickly reaches an asymptote when speed increases. To show the contribution of animal speed on these relationships, we also plotted the curves for the case where animal speed was increased to 31 m/s, which represents the case for the speed of bottlenose dolphins, Tursiops truncatus (Fig. 2a c, grey lines). The whale model likewise shows a positive linear relationship between encounter rate and vessel speed (Fig. 3a). The decline in the expected number of encounters is most noticeable at very slow speed (e.g. <2 knots (7 km/h) for r c_disk, Fig. 3b). The reduction in the expected number of encounters for the speed considered for management purposes was miniscule (Fig. 3b). Under case A (whale abundance in the area of interest was 1 for a short time frame (15 days) with 28 vessels per day, and vessel speed was 10 knots), there were an expected 1114 encounters, whereas under case B (same as A, but vessel speed was 157 knots), the expected number of encounters was Assuming no collision avoidance behaviour by the whale or the ship, the expected relative number of collisions was 0669 for case A and 0668 for case B. The expected relative number of deaths was 0378 for case A and 0546 for the higher Fig. 2. Relationship between boat speed and the encounter rate (a), and expected number of encounters for the fixed time (a b) and fixed distance scenario (c). The thick black solid lines correspond to the case of manatees with a mean speed of 02 m/s (SD = 028), the red dots correspond to a fixed mean speed of 02 m/s (SD = 0), and the thin greylinescorrespondtoaspeedof31 m/s (SD = 0) (based on bottlenose dolphins Tursiops truncatus mean sustainable speed, reviewed in Noren, Biedenbach & Edwards (2006)). The dashed lines in (b) correspond to the case where a boat is in the study area for 60 min instead of 30 min [solid lines; and red dots in (a)]. speed case B (see also Appendix S4). We also computed these expected numbers of encounters over a longer time frame (600 days, considering 2 months of high whale density in the south-eastern USA per year for 10 years). The expected number of encounters for the long time frame was 4457 for case A and 4453 for case B. The expected relative number of collisions for the long time frame was 2674 for case A and 2672 for case B. The expected relative number of deaths was 1513 for case A and 2185 for case B. Our results illustrate that the expected mortality process is largely driven by the probability

6 Risk of wildlife collisions 47 transits per day. For the same case and allowing a maximum of one encounter per vessel transit, the simulations resulted in an expected [mean (95% bootstrap CI)] 118 (0 294) encounters if the whale s movement followed a correlated random (nearly linear) walk and 116 (0 252) encounters if the whale s movement followed a random walk. Under the same case but allowing for multiple encounters per simulation, the simulations resulted in an expected 156 (0 378) encounters if the whale s movement followed a correlated random walk and 155 (0 378) encounters if the whale s movement followed a random walk. Under the case with an extremely slow speed (118 knots), theanalyticalapproachresultedinanexpected1213 encounters. For the same case and allowing a maximum of one encounter per vessel transit, the simulations resulted in an expected 156 ( ) encounters if the whale s movement followed a correlated random (nearly linear) walk and 127 (0 274) encounters if the whale s movement followed a random walk. Under the same case but allowing for multiple encounters per transit, the simulations resulted in an expected 590 ( ) encounters if the whale s movement followed a correlated random walk and 481 (0 1092) encounters if the whale s movement followed a random walk. Discussion Fig. 3. Relationship between boat speed and the encounter rate (a), and expected number of encounters (b) between one boat and one whale. The symbols and lines indicate the cases in which the encounter radius was based on a disc with area approximately equal to that of a whale (black line), the width of the whale (dashed line) or the length of the whale (white circles). of death given striking speed. The percentage reduction in the expected number of encounters based on our analyses was <01% when comparing management cases A and B. However, the slower speed case (A) resulted in a 31% reduction in the expected relative number of whale deaths relative to case B. Figure 3 shows the effect of varying values of the encounter radius, r c. Not surprisingly, using whale length led to the highest encounter rates, and width to the lowest; these represent the two extremes. When accounting for uncertainty using a MCS approach, the mean difference in expected relative mortality between cases A and B over 15 days was 014; 95% CI ( 2to2),which represents a 27% reduction in expected relative mortality. This result illustrates that despite a greater expected relative mortality at higher speeds, there is a large uncertainty associated with the projected relative numbers of mortality events and that detailed analyses of the impact of watercraft on wildlife should incorporate relevant sources of uncertainty. SIMULATIONS Under case A, the analytical framework resulted in an expected 1114 encounters for a time frame of 15 days with 28 vessel By applying encounter rate theory to the problem of boat collisions with marine mammals, we were able to gain valuable insights that may not have been as apparent using a simulation approach. We examined two limit case scenarios. Many recreational boating activities are consistent with the fixed time scenario. In the example that we considered, we found that a reduction from 215 knots (398 km/h) to nearly idle speed (~42 knots[78 km/h]) reduced the number of encounters by 80%. This is due to the nearly linear relationship between speed and encounter rate (Fig. 2a). In this scenario, a boat with greater speed will cover a greater distance for a given amount of time, resulting in more expected encounters. We also considered the case of the fixed distance scenario, where vessels go to a specific destination and the distance travelled is not affected by speed. For instance, we described the case of vessels travelling distances within shipping lanes where encounters with whales may occur. The fixed distance scenario has been discussed by Vanderlaan & Taggart (2007), who described a theoretical scenario for whales, but acknowledged that their initial analysis warranted a more detailed investigation to better elucidate the encounter process. Their analyses used a different approach (Galos, Argyrakis & Kehr 2001). Results from our new analytical solutions also showed a slight decrease in the number of encounters as boat speed increases (Figs 2c, 3b). Our mathematical framework [which builds on concepts developed and discussed by Koopman (1956) and Gerritsen & Strickler (1977)], however, is more comprehensive than that of Vanderlaan & Taggart (2007), provides more accurate estimates, and provides a mathematical rationale for extrapolation in space and time and over heterogeneous habitats. Wilson et al. (2007) applied equations developed by

7 48 J. Martin et al. Gerritsen & Strickler (1977) (who considered the problem of encounter of plankton to be 3-dimensional) to the problem of marine mammal collision with rotating turbines. In contrast, we used a 2-dimensional case of the problem (Koopman (1956) for naval operations applications), which we view as more appropriate for vessel strikes because it better approximates movement of boats. The analytical approach is more easily employed and allows for a more rapid assessment of parameter uncertainties on model output than does a simulation approach, which requires extensive computer run time and the selection of an appropriate time-step resolution (Clyde & Kennedy 2001; Van Der Hoop, Vanderlaan & Taggart 2012). Finally, our framework explicitly incorporates uncertainty, which has seldom been accounted for in estimates of mortality rates due to watercraft collisions (but see Conn & Silber (2013) for an alternative approach that also accounts for uncertainty). Although Vanderlaan & Taggart (2007) used a different approach from ours, we found a similar general pattern: the expected number of encounters was negatively related to boat speed, with a probability of encounter that approaches one when boat speed approaches zero (Fig. 3b). This finding, however, should be interpreted cautiously. Indeed, by decomposing the expected number of encounters (Fig. 3b) into the encounter rate and its time component, we can see that the encounter rate remains low at idle speed (Fig. 3a). The expected number of encounters is greater at extremely low boat speeds because as boat speed tends towards zero, time spent in the area tends to infinity and the animal thus has longer exposure to the boat. In the extreme case where the boat is stationary, if the moving animal is not actively avoiding the boat and is confined to an area that includes the boat, the animal will eventually cross paths with the boat. This is a theoretical result with little relevance for practical situations. As illustrated with the whale case, the ratios of the boat speed to animal speed are such that the decrease in risk of encounter due to an increase in speed is negligible. For instance, the expected number of encounters for a projected time frame of 600 days was 4457 and 4453 under cases A and B, respectively. In contrast, the expected relative mortality due to collision was 1513 under case A and 2185 under case B. This shows that the slight reduction in the number of encounters (~01%) is overwhelmed (and in fact negligible) by the increased probability of mortality due to increased vessel strike speed, resulting in a reduction of approximately 31% for the deterministic model and 27% when accounting for uncertainty with the MCS approach under the scenario with regulated vessel speed. This reduction jumps to 62% when using estimates [of probability of death given striking speed, with no uncertainty] from Vanderlaan & Taggart (2007) instead of Conn & Silber (2013). This finding emphasizes the importance of obtaining accurate estimates of the probability of death given boat striking speed, which remains a challenging task. For instance, estimation of this parameter is complicated by the fact that animals may be hit yet never seen or reported. Limitations of our analytical approach relate primarily to the estimation of parameters necessary for realistic models. In particular, the probability that marine mammals will avoid boats or that boaters will avoid animals may also be affected by speed (Vanderlaan & Taggart 2007; Gende et al. 2011; Van Der Hoop, Vanderlaan & Taggart 2012; Conn & Silber 2013). Very little is known about the avoidance process, and in our estimates of expected relative mortality, we set the avoidance probability to 0 (in fact it is because of this constraint that we report expected relative mortality instead of expected absolute mortality). In Appendix S4, we examined this issue with exploratory analyses. These results show that avoidance probabilities could have an important impact on estimates of expected mortality and that future research should focus on estimating these probabilities. Furthermore, the management scenarios that we explored were conceptual; for instance, we considered the case for one whale or one manatee. Nevertheless, the mechanistic framework that we have outlined (Fig. 1) is well-suited for incorporating new information (e.g. detailed information about animal density; Martin et al. (2015)). The MCS analyses confirmed that the expected whale mortality was greater at faster vessel speeds but indicated that large uncertainty was associated with these estimates. The model that we used for the analytical approach is based on a number of assumptions. Our approach assumes that the animals are spatially distributed according to an equilibrium distribution and that the orientation of the animal is part of the information contained in the equilibrium distribution. In addition, it assumes that the distribution of the speed of the animal is stationary, independent of time, of the animal s position and of its orientation. This analytical solution ignores the effect of boundaries of the area, which can affect the spatial distribution of the animals there. To address this specific concern, a simulation approach can be used to check the consistency of the results. The model also assumes no interactions among boats or animals (e.g. no aggregations). As explained earlier, the model estimates the rate of first encounters. Considering only first encounters can be justified because if an animal is killed, future encounters will be irrelevant; if the animal survived, it may alter its behaviour, increasing its probability of avoidance; and in most situations, the boat will be travelling in a nearly linear transit at a much greater speed than the animal, making multiple encounters unlikely. Interestingly, the selection of the radius of encounter may have an important effect on estimates of encounter rates (Fig. 3). When boats move much faster than the animal, it makes sense to consider the width of the boat and the orientation of the animal with respect to the boat s trajectory as key parameters. The relative orientation of the boat and animal (including orientation in the vertical dimension, because the animal may be diving) at the time of encounter can also influence the probability of collision and death; if known, these effects could be incorporated into the overall conceptual framework. We recommend further investigation to derive alternative metrics for encounter radius that better approximate the encounter process, which would likely lead to intermediate values between the extremes (width and length) and would account for the shape of the moving agents.

8 Risk of wildlife collisions 49 When considering the encounter process, the ratio of speeds between the two moving agents should also be considered instead of just absolute speeds for each moving agent independently. As shown in eqn 1 and Fig. 2b c, with everything else being equal (e.g. animal and boat density in the area) at a given boat speed, the encounter rate will be higher for faster-moving animals than for slowly moving animals (Koopman 1956; Evans 1989; Anderson, Gurarie & Zabel 2005; Gurarie & Ovaskainen 2013a,b in the contexts of zooplankton interactions and animal movement). An appealing property of our approach is that it is scalable and can be applied to heterogeneous landscapes or can be extrapolated in time (see Appendix S3 for mathematical justification). For instance, North Atlantic right whales in the south-eastern USA tend to occur at higher densities in cooler, nearshore waters (Keller et al. 2006). Observed data or predictive models can be incorporated into this analytical framework to estimate the number of collisions in habitats with different animal or vessel densities. Thus, our framework should facilitate the creation of risk maps through straightforward integration in GIS which would be more cumbersome with a simulation approach. The simulations of animal boat collisions led to encounter rates that were reasonably consistent with the analytical results for the speeds that we considered, particularly for speeds relevant to regulations. But we found substantial discrepancies at very slow vessel speeds when allowing multiple encounters per whale; this is due to the fact that the analytical approach that we used assumes a maximum of one encounter between an animal and a boat during a transit. As expected based on the assumptions of the analytical approach, if we allowed only one encounter per whale, the encounter rates from the simulated and analytical approaches did not differ substantially. Here, we set most parameters to fixed values because the primary purpose of this paper was to provide general insights based on our solutions rather than to focus on detailed applications. Although we used empirically based values for our parameters when possible to make the analyses more meaningful, we intend to extend this framework to account for additional details and sources of uncertainties. In fact, the framework we have presented can be modelled with a MCS model or BBN to describe these sources of variation. As a case in point, we computed estimates of mortality events for whales that accounted for uncertainty, mostly as a way to illustrate the approach. Such a network would also be appropriate for incorporating some of the processes we have not included, such as mammals ability to avoid vessels, vessel operators ability to avoid collisions and compliance with regulations. The benefit of such an approach is that it is well-suited for combining the estimation and statistical methods advocated by Conn & Silber (2013) with more mechanistic approaches described by Gerritsen & Strickler (1977) and Van Der Hoop, Vanderlaan & Taggart (2012). In fact, carcass recovery data could be incorporated into the analyses to assess and inform predictions. Finally, our analytical approach could be coupled with a population projection model (e.g. Runge et al. 2015), allowing us to infer impacts of regulations on the population of interest. To conclude, our work constitutes a significant advance in understanding the process of encounter rates and risk of collisions between marine mammals and vessels. It provides an interesting case where theoretical work can be used to help improve management decisions and policies. The generality of our framework makes it applicable to other marine animals (e.g. sea turtles) and other animal groups and systems, including wildlife collisions involving motor vehicles on roads, snowmobiles and wind turbines. Finally, because many ecological processes involve the encounter between moving agents (e.g. predator-prey interactions) our findings are also relevant to these ecological topics (e.g. Gerritsen & Strickler 1977; Hutchinson & Waser 2007; Gurarie & Ovaskainen 2013a). Acknowledgements We thank L. Ward-Geiger, R. Muller, R. Hardy, B. Zoodsma, B. Bassett, B. Crowder, H. Edwards, R. Flamm, F. Johnson, V. Engel and two anonymous reviewers for their insights and contributions. We are grateful to J. 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