Biomechanics of predator–prey arms race in lion, zebra, cheetah and impala

The fastest and most manoeuvrable terrestrial animals are found in savannah habitats, where predators chase and capture running prey. Hunt outcome and success rate are critical to survival, so both predator and prey should evolve to be faster and/or more manoeuvrable. Here we compare locomotor characteristics in two pursuit predator–prey pairs, lion–zebra and cheetah–impala, in their natural savannah habitat in Botswana. We show that although cheetahs and impalas were universally more athletic than lions and zebras in terms of speed, acceleration and turning, within each predator–prey pair, the predators had 20% higher muscle fibre power than prey, 37% greater acceleration and 72% greater deceleration capacity than their prey. We simulated hunt dynamics with these data and showed that hunts at lower speeds enable prey to use their maximum manoeuvring capacity and favour prey survival, and that the predator needs to be more athletic than its prey to sustain a viable success rate. Analysis and modelling of locomotor characteristics of two pursuit predator–prey pairs show that hunts at lower speeds enable prey to use their maximum manoeuvring capacity and favour prey survival. Nature might be red in tooth and claw, but the race is not always won by the strong. A detailed study of how lions chase down zebras and cheetahs pursue impalas shows that, although the predators in each pair had substantially more muscle power than their prey, as well as much greater capacity to accelerate and decelerate, the prey species could slip away at lower speeds, at which they are more manoeuvrable. Nevertheless, predators need to be more athletic than their prey to sustain a viable kill rate.


Muscle fibre power in predator and prey
Muscle biopsies were skinned, placed in a trehalose-glycerol mixture, frozen in liquid nitrogen in the field and transported to the United Kingdom. Peak power, velocity and stress at peak power and maximum isometric stress were determined at 25 °C for single, skinned fibres ( Fig. 2a-f). Maximum power and associated velocity and stress were then calculated (Methods).
Complete measurements were made on 37 individual skinned fibres from six cheetahs, 30 fibres from five impalas, 50 fibres from eight lions and 57 fibres from eight zebras. There was a distinct subpopulation of 'low-performance' fibres (twelve fibres from zebra, eight fibres from lions, three fibres from cheetahs and three fibres from impalas; Fig. 2d-f and Supplementary Data) with a velocity at peak power that was below 1.35 lengths s −1 and a lower peak power ( Fig. 2c and Extended Data Fig. 2g), which were either myosin heavy chain (MHC) type-I (11 of 19 fibres tested) or type-II (8 fibres) (Methods).
Linear mixed-effects models were fitted for peak power, velocity and stress at peak power and isometric stress with a factor distinguishing predator and prey, including the interaction of this factor with a categorical variable called 'fibre performance classification' . Within the factor distinguishing predator and prey, we nested a random effect by subject and fibre. The residuals of this model exhibited heteroscedasticity and so the variance of the error term was allowed to vary by subject and performance classification. Power in the high-performance fibres The fastest and most manoeuvrable terrestrial animals are found in savannah habitats, where predators chase and capture running prey. Hunt outcome and success rate are critical to survival, so both predator and prey should evolve to be faster and/or more manoeuvrable. Here we compare locomotor characteristics in two pursuit predator-prey pairs, lion-zebra and cheetah-impala, in their natural savannah habitat in Botswana. We show that although cheetahs and impalas were universally more athletic than lions and zebras in terms of speed, acceleration and turning, within each predator-prey pair, the predators had 20% higher muscle fibre power than prey, 37% greater acceleration and 72% greater deceleration capacity than their prey. We simulated hunt dynamics with these data and showed that hunts at lower speeds enable prey to use their maximum manoeuvring capacity and favour prey survival, and that the predator needs to be more athletic than its prey to sustain a viable success rate.
The values reported here are comparable to data for skinned fibres from wild rabbits at 25 °C 22 , but are high compared to published values for skinned fibres from large animals 23,24 . Muscle power is highly temperature dependent 25 and a temperature coefficient (Q 10 ; the ratiometric increase in rate with a temperature increase of 10 °C) of 2.3 is appropriate 26 , which predicts in vivo muscle power (all fibres, Extended Data Fig. 2i) of 232 (prey) and 292 (predators) W kg −1 at a body temperature of 38 °C.
Slower myosin types and muscle fibres are inherently more economical 23,25,27 , thus slower fibres confer advantages 25 , and the fast versus slow distribution of fibres reflects the opposing pressures of predation (avoidance) on one side and food and water supply, ranging distance and environmental conditions on the other 25,28 . This may partly explain why prey species have lower-power muscle fibres 25 . Therefore, muscles of desert specialists at risk of dehydration and/or starvation 29 , such as camels, vicunas and Arabian oryx, would be predicted to be biased towards economy 25 . Selection pressure for greater performance or economy could change fibre type distributions or muscle characteristics within a few generations-much more rapid than for changes in myosin contractile speed.

Speed and acceleration of predators and prey
Stride timing and therefore frequency was derived from collar acceleration data 9 . Stride speed and accelerations were averaged over each stride; change in speed is calculated as the difference in speed between   two consecutive strides, work per stride is the change in mass-specific net horizontal kinetic energy and power per stride is the work per stride divided by stride duration. Change in heading is the angle between two consecutive stride velocity vectors 9 . Differences in the frequency of maximum effort manoeuvring between predators and prey (since predators hunt often and prey are rarely hunted) would manifest in different tails for the distributions of accelerations for each species. The predator species will have relatively heavy tails, that is, higher kurtosis, as more of their observed behaviours are associated with rapid accelerations, whereas the more sedentary (or at least steadily moving) prey have fewer such observations. Steadystate strides were removed by including a threshold on acceleration with the threshold determined for each species by the kurtosis of these distributions, which resulted in a similar distribution for all species (Methods and Extended Data Fig. 3a-c). Qualitatively, the distributions for predators and for prey are similar and the 98% percentile approximates the limit of the distribution in a reasonably consistent manner across runs of all lengths and tortuosity (Extended Data Fig. 4).
Stride parameters were grouped into non-uniform speed bins with 400 data points in each and the 98th percentile of the distribution was determined for each bin (except for stride frequency, for which data were further subgrouped on the basis of acceleration performance and a linear regression was performed on each subgroup (Methods)). The uppermost bin with fewer than 400 data points was ignored. The 98th percentile was chosen to account for the different numbers of strides in different species and to exclude occasional extreme values 9 (Extended Data Figs 5, 6). The cheetah-impala pair was more athletic than the zebra-lion pair for every metric (Extended Data Fig. 7).
Predator and prey were compared using a linear model (Methods) and test statistics computed under the null hypothesis that predator and prey are drawn from the same distribution, except for stride frequency, for which, because of species pairing differences, predator and prey pairs were compared individually. The ratio of the maximum observed performance for cheetah-impala then lion-zebra, along with the results of the test comparing predator and prey across species, are as follows: predators were 50% and 24% superior at acceleration   Figure 3 | Locomotor performance based on stride parameters. All values are averaged per stride or represent the change over a stride and where appropriate are per kg body mass (BM). a-e, Accelerating strides. a, Positive net work performed in each stride. b, Stride frequency, mean of 20% highest power strides and middle 60% of power strides. c, Average mass-specific power per stride. d, Increase in speed per stride. e, Tangential (forward) acceleration with the curved lines representing a stride mean power of 30, 60, 90, 120 and 150 W kg −1 with a limit line for a coefficient of friction (μ) of 1.3. f-j, as a-e, but for decelerating strides. k-n, Turning. k, Centripetal (lateral) acceleration. l, The relationship between speed and turn radius with limit lines for μ = 0.6 and μ = 1.3. m, Change in heading compared to horizontal speed. n, Tangential compared to centripetal acceleration with μ limits as for l. n, F, pure forward acceleration; B, deceleration; C, centripetal acceleration; ρ, polar coordinate. In each panel one line per species is shown, which (except in b, g) represents the 98th percentile for data in speed bins (each bin contains 400 data points therefore bin width varies). At the bottom of each panel, the ratio of that parameter for cheetah-impala (red circle) and lion-zebra (blue circle) is given for each speed bin, same x axis. Dataset comprised 7,509 strides for 520 runs from five cheetahs; 8,884 strides for 515 runs from seven impalas; 15,947 strides for 2,726 runs from nine lions and 14,089 strides for 1,801 runs from seven zebras.
The 98th percentile of speed was 19.9 m s −1 for cheetahs, 13.8 m s −1 for impalas, 13.9 m s −1 for lions and 10.6 m s −1 for zebras. This was 84, 78, 67 and 77%, respectively, of the maximum achieved by the third fastest individual of each species, which was 23.8 m s −1 for cheetahs, 17.7 m s −1 for impalas, 20.6 m s −1 for lions and 13.8 m s −1 for zebras. Therefore, predators were faster than their prey and all species rarely approached their maximum recorded speed (Extended Data Fig. 5).

Turning performance of predators and prey
When turning, predators were only slightly superior to prey (z = 2.93, P = 0.0034): cheetah-impala 15%, lion-zebra 10% (Fig. 3k-n). Turning does not require a change in body kinetic energy, but a centripetal acceleration of 13 m s −2 results in a 66% increase in effective weight 18 and the limbs must shorten and extend in the presence of these higher axial forces. This length change can be delivered by passive elastic structures within the limb 30,31 , but any associated muscles must deliver higher forces at that contraction velocity (equating to a higher power requirement) 19,20 . Reduced centripetal acceleration at high speed would indicate a muscle power limit rather than a grip limit for that activity 17,19 , however, we found no such evidence for a power limit 18 at these submaximal speeds. Figure 3n summarizes the capacity for maximum acceleration in any direction, relative to the track of the animal. It shows that these predators outperform their prey most markedly during deceleration (bottom) and less so during forward acceleration (top) and turning (sides). No species showed highest levels of tangential and centripetal acceleration in the same stride; the lines are elliptical, which supports a grip-type limit (as horizontal accelerations should vector sum to a limit value). Forward acceleration performance was maintained by all four species at the fastest speeds commonly used (Fig. 3e, j). Power requirements for forward acceleration increase with speed ( Fig. 3c, h and Extended Data Fig. 5), because power is the product of speed and acceleration, and if maximal acceleration was lower at the highest speeds, this would indicate a potential power constraint 15 . A reduction in manoeuvrability would result in an animal's trajectory being more predictable, which would be disadvantageous for both predator and prey.

Species differences in experimental data
Much of the difference observed between predator and prey could be attributed to predators having proportionally more muscle and/or higher muscle power (Fig. 2d), but that does not provide an explanation for the large differences that were observed between lion and cheetah and between zebra and impala (Extended Data Fig. 7). Hind limb muscle fraction of total body mass is fairly consistent across species: 17.5-19.8% (Extended Data Table 1), so muscle peak power should define the acceleration capacity of the whole animal at moderate to high speeds 9,13,15 . Athletic wild animals are, however, likely to be proportionally more muscular than the mostly sedentary domesticated animals contributing to Extended Data Table 1 and spinal, trunk and forelimb muscle will also contribute to acceleration. The predicted in vivo muscle powers of 232-292 W kg −1 are concomitant with the upper, but not lower, limit of observed whole animal powers of 30-120 W kg −1 (Fig. 3e). Carnivores hunt with empty stomachs, whereas prey carry the mass of rumen (impala) or hind gut (zebra) contents, which will impinge on any performance that is dependent on muscle power or strength (as would pregnancy). The differences within the predator and within the prey species may reflect that muscles are arranged for different roles, for example, for economical walking versus for acceleration and hunting or fighting 25,32,33 , but without contextual anatomical data, this is only speculation and the differences are too large to simply be attributed to scaling due to animal size (Extended Data Table 2a). Foot design and grip may also have a role 34 . Behavioural factors cannot be ruled out, but our data indicate that the highest values were captured (Extended Data Figs 5, 6).

Capture-evasion model description and predictions
A pursuit predator uses a combination of stealth and speed to get close to its prey 12 and then the prey evades capture by manoeuvring 1,4 , while the predator attempts to intercept it. The interaction has been approached analytically or numerically for continuous processes 35,36 (for example, air combat manoeuvring), but modelling the probability that the predator and prey arrive at the same location becomes increasingly complex to solve when treated as a discrete process.
In our model, predator and prey were able to accelerate in any direction up to their experimentally derived maximum during each stride, so they could go anywhere on the boundary of an approximately elliptical area that grew with the subsequent stride ( Fig. 4a-c). The predator responds to the acceleration of its prey in the preceding stride and we modelled the range of initial conditions for which the predator could catch its prey within two strides. The acceleration limits for each species and direction (impulses per stride) were the observed 98% values of centripetal, positive and negative tangential acceleration divided by the stride frequency at that speed ( Fig. 3b, e, g, j). The elliptical area prevented simultaneous maximal centripetal and tangential accelerations ( Fig. 3n). At higher speeds, acceleration and therefore manoeuvring were curtailed as the applied impulses could not cause the animal to exceed the 98% maximum speed observed for each species.
For the prey, the accelerations at the start of the first and second stride were the possible accelerations up to maximum (Extended Data  Table 2b) in any direction. The predator had zero acceleration in the first stride so its initial velocity determined its subsequent position and it could accelerate in any direction in the second stride (reacting to the previous prey acceleration). The area reached by the predator was increased by a semi-circular region with a size of half the predator body length to account for the physical size of the predator. We define capture probability as the fraction of the elliptical area of the prey that is covered by the elliptical area of the predator after two strides.
We plotted the feasible range of initial prey speeds and predatorprey spacings for capture after two strides ( Fig. 4d and Extended Data Fig. 8) and then optimized the initial predator speed for each condition to maximize the overlap in position between the predator and the prey after two strides. The predator-prey spacing at the beginning of the simulation represents less than a stride length in all cases (code used for the model written in Python can be found in the Supplementary  Information).
The model shows that the prey should avoid the predator by turning (lateral acceleration), rather than attempting to increase separation by travelling as fast as possible (Fig. 4d). If the prey is moving fast and the predator is close (Fig. 4d, bottom left), its best option requires rapid deceleration and turning, whereas turning alone becomes more beneficial if the predator is further away (and therefore closing at higher relative speed, Fig. 4d, bottom right). High prey speeds result in high capture probabilities (Fig. 4f), because the prey cannot accelerate forwards with or without turning, making its tactics highly predictable (captured by optimization of predator speed for overlap), whereas a slow moving prey (Fig. 4e, f, left) has a wider variety of escape options and is therefore less predictable. Predator and prey indeed use moderate speeds ( Fig. 4e and Extended Data Fig. 8).
The predator has the highest chance of success if it is travelling only slightly faster than the prey, which enables it to reach many of the locations the prey can move to across a broad range of starting speeds (the objective function for the optimization, relative capture area, is very flat in this region), and its advantage increases with higher prey speeds. This is reflected by the observed predator speeds ( Fig. 4e and Extended Data Fig. 8). Figure 5 shows that all species often execute a constant speed turn, but that it is rare for either of the herbivore species to accelerate or decelerate, whereas predators (especially lions) often undertake deceleration strides, either in isolation or in combination with a turn. The preferred accelerations fit with the prey using optimum escape strategies predicted by the non-overlapping areas in Fig. 4d and tactics for which they perform similar to the performance of predators (turning) rather than those for which they are outperformed (tangential acceleration and deceleration). With the same lateral acceleration, a prey that is moving more slowly than a converging faster-moving predator will have an advantageously tighter turn, because the radius is equal to v 2 / lateral acceleration. Commonly observed predator decelerations are concomitant with a faster-moving closing predator. More than one repetition of the modelled two-stride scenario can occur within a single pursuit 9 -and the overlap-derived success rates are similar to those observed for animals when hunting in the wild 9,12,37 .

Effect of athleticism on hunting success rate
We adjusted the acceleration capacity of the predator or prey and reran the simulation to obtain capture probabilities for animals of greater or lesser athleticism. Unsurprisingly, increased predator performance is beneficial, reducing the number of hunts needed to capture prey  Fig. 4g, h). Owing to the power relationship underlying Fig. 4g, h, curves steepen when the predator is below 0.8 of its actual performance (Fig. 4h), which would tend towards an unsustainably low success rate (ignoring other determinants of hunt outcome). Such a reduction could be the result of an injury or ageing, with greatest consequences for solitary animals. The data also provide insight into preferred prey and hunting style: the predicted low success rate for lions hunting impala (Fig. 4g, h) is supported by the observation that lions capture impala opportunistically rather than in an open pursuit. African wild dogs hunt impala 37 , but are less athletic than cheetahs 37 . Applying the model to a single African wild dog hunting an impala 37 predicts a success rate of 8.2%, which is lower than the actual success rate of 15.5% 37 . This would concur with African wild dogs capturing impalas during opportunistic rather than one-on-one pursuit hunts 37 .

Conclusions
The study shows that overall, the athletic capabilities of the two pursuit predators closely match their respective common prey, leading to a sustainable success rate, survival of both and reflecting an evolutionary arms race 6,7 . The predators have higher muscle power, are faster and have a greater capacity to accelerate and decelerate than their prey. The prey can match their predator's locomotor capabilities most closely through turning manoeuvrability, affording them a critical escape space. In evolutionary terms, there may be scope for further development of performance, for instance through increasing muscle power, but this specialization may be at the cost of locomotor economy, musculosketal robustness, or other ecologically relevant factors, such as prey capture ability, fighting or the capacity to adapt to a changing world.
Online Content Methods, along with any additional Extended Data display items and Source Data, are available in the online version of the paper; references unique to these sections appear only in the online paper.

MethOdS
Data reporting. No statistical methods were used to predetermine sample size. The experiments were not randomized and the investigators were not blinded to allocation during experiments and outcome assessment. Animals. All collared animals were located in northern Botswana with largely overlapping ranges (Fig. 1f). Animals were immobilized by free darting from a vehicle or helicopter mostly by A.M.W., using 80-100 mg ketamine and 2 mg medetomidine for cheetahs; 60 mg ketamine, 25 mg tiletamine hydrochloride, 25 mg zolazepam hydrochloride (as 50 mg zoletil, Virbac), 2 mg butorphanol tartrate and 6 mg medetomidine for lions; 1.5 mg thiafentanil oxalate, 2 mg butorphanol tartrate and 1,700 IU hyalase for impalas; and 7 mg etorphine hydrochloride, 50 mg azaperone and 1,700 IU hyalase for zebras. The reversal of immobilization of herbivores was done using diprenorphine or naltrexone at the end of the procedure and carnivores with atipamezole at 60 min after darting. While sedated, front and hind leg and body lengths were recorded. Collar data were downloaded by radio link every few weeks to a ground vehicle and collars were monitored. All animals were adult, nine lions (two male, seven female), five cheetahs (two male, three female), seven zebras (seven female), seven impalas (six male, one female). The lions and cheetahs were part of other ongoing projects in collaboration with the Botswana Predator Conservation Trust (http://www.bpctrust.org). Programmable drop-offs (two models, 108 g, Sirtrack Ltd or 50 g, Biotrack) were attached to the zebra and impala collars, respectively. Two drop-off units failed and collars were retrieved by re-darting.
Data Muscle fibre measurements. Biopsies were taken from the biceps femoris muscle by A.M.W., using a Bergstrom needle or conchotome forceps after collar placement. Animals were clipped, sterility was ensured and the biopsy site was treated with local antibiotics (200 mg cloxacillin, 75 mg ampicillin, Curaclox LC) and the animal was given analgesia (finadyne or metacam). Five male impala that had been killed for meat on a game ranch were dissected and provided additional muscle samples. Muscle samples were skinned by 30 min of immersion in ice-cold relaxing solution containing 2% Triton X-100 and exposed to a pH-6 relaxing solution to inactivate any foot-and-mouth disease virus. Triton X-100 was washed out with fresh relaxing solution and samples were immersed in 500 mM trehalose containing 0.5% glycerol 22 , frozen in liquid nitrogen and stored in an IATA-approved dry-shipper (Biotrek 3 Statebourne Cryogenics) for transport to the United Kingdom. In the United Kingdom, biopsies were stored at − 80 °C. Periodically, individual biopsies were thawed and had cryopreserving trehalose replaced with a relaxing solution.
Our previous work showed that biopsies stored for 20 months using this protocol showed no discernible loss of mechanical power 22 . Thawed biopsies were stored at − 20 °C in a relaxing solution made up in glycerol and used for fibre preparation and testing within four weeks.
Fibre fragments were first suspended while in relaxing solution between the motor and force transducer of a 600A permeabilized-fibre apparatus (Aurora Scientific). T-shaped aluminium clips were attached to fibre ends and used to suspend fibres from steel wire hooks that were glued with shellac to the motor and transducer. Fibres were visualized using a 900B digital camera (Aurora Scientific). The camera image was used to set the sarcomere length (SL) of a fibre fragment to between 2.5 and 2.6 μ m. Fibre length (L o ), depth and width were then measured in mm. The fibre cross-sectional area was calculated for each fibre, assuming an elliptical shape.
Single skinned fibres were activated by temperature jump, from 1 °C to 25 °C (Extended Data Fig. 2a), using approaches similar to those previously described 22 . The composition and ionic strength (200 mM) of the various solutions was as previously described 22 . To activate a fibre, it was immersed consecutively in solutions for low-temperature pre-activation (for 45 s), low-temperature activation (for 4 s), high-temperature activation (6 s) and high-temperature relaxation. The example in Extended Data Fig. 2a shows the time courses of solution changes and force responses for an impala fibre, starting from the final 3 s of cold-temperature preactivation. The force baseline at 0.7 L o was recorded in high-temperature relaxing solution before re-setting L o to the original starting length and checking SL.
The standard procedure to measure fibre power and determine peak power was modified to perform four different force-control events during each 6-s activation to deliver more data per fibre 22 . In brief, after temperature jump to 25 °C, force developed to a plateau at constant length (isometric force), and the fibre was then clamped for 20 ms to a predetermined fraction of peak isometric force-the actual force achieved in the first force clamp was calculated by the 600A based on the difference between baseline force (measured and stored within a 600A protocol just before 1 °C activation) and isometric force (measured and stored within a 600A protocol just before onset of a force clamp). The shortening velocity during force clamp was measured and used to calculate power output. The fibre was then released to slack length in order to re-measure the baseline force and it was then lengthened to L o over a period of 5 ms. This step avoids a high eccentric force transient during lengthening. The baseline measurement was saved again in the 600A protocol and used to compare with the measurement of stable isometric force achieved after lengthening the fibre to L o -the force attained in the next force clamp was based on the difference between the newly saved values of baseline and isometric force. Four different force-control events were conducted during the 6-s activation at 25 °C (see Extended Data Fig. 2a). Examples of a force clamp and of the fibre-length changes required to hold force constant are shown in Extended Data Fig. 2b, c. Relaxation and a final force baseline-check were also conducted at 25 °C (Extended Data Fig. 2a). Three activations provided up to twelve different force-clamp measurements in order to quantify a power versus force relationship and peak power for each fibre (see below).
A fibre was counted as 'tested' if it did not break on the test apparatus, and if test conditions (solution temperature and chemistry) were maintained as prescribed in the experimental design. A fibre was counted as 'included' in the mechanical tests if the maximum isometric force > 75 kPa, and during the repeated activations, isometric force for each test remained > 80% of the peak isometric force observed. For each fibre, we conducted three activations, each with four force-control, or shortening, events in order to collect, at most, 12 points for a power-force curve fit. An individual data point (that is, a single force-control event) could be rejected (either because of poor (for example, unstable/oscillating) force during fibreshortening or because of low (< 80% of maximum) isometric force), but exclusion of a data point on this basis will not necessarily have caused the fibre to have been rejected, unless the spread of usable data points was insufficient for curve fitting.
Of the 209 fibres initially tested with apparent success, 35 were excluded (three out of 40 cheetah fibres, 14 out of 64 lion fibres, 14 out of 71 zebra fibres, four out of 34 impala fibres).
The data for each fibre were analysed as described previously 22 .
Equation (1) describes the dependence of relative power (Q = power / (F isom × L o ); in s −1 ) on relative force (force during shortening / F isom ; units are dimensionless). Peak relative power (Q max ) was found by fitting a line to the data by adjusting three parameters: Q max , the force intercept (F o ) and force at peak power (F Qmax ). An example plot and best-fit curve is shown in Extended Data Fig. 2d. Peak power in W kg −1 is obtained by multiplying Q max by maximum isometric stress in kPa and dividing by fibre density, 1.064 g ml -1 (ref. 38). After mechanical tests, the 'low-performing' single skinned fibres were pinned onto a gelatine base in cryomolds, flooded with OCT (Tissue-Tek) and frozen in liquid nitrogen. Sections (8-μ m thick) were cut and immunostained with mouse anti-MHC fast monoclonal antibody (MY-32, 1:1,000; ab51263, Abcam) for type-II fibres, and mouse anti-MHC slow monoclonal antibody (1:50; MAB1628, Merck Millipore) for type-I fibres. Muscle data statistics. A linear mixed-effects model was fitted in R (R Foundation for Statistical Computing) for peak power, velocity at peak power, stress at peak power and isometric stress against a factor distinguishing predator and prey with the interaction of this factor with a categorical variable 'performance classification' 39 . Within the factor distinguishing predator and prey, we included a nested random effect by subject and fibre. The residuals of this model exhibited heteroscedasticity and so the variance of the error term was allowed to vary by performance classification. General linear hypothesis tests were then performed. Temperature. Muscle power is highly temperature-dependent 25 and in a previous study 26 , using their data and literature data, it was shown that for the temperature range of 20-35 °C, a temperature coefficient (Q 10; ratiometric increase in rate with a temperature increase of 10 °C) of 2.3 is appropriate 40 Ethical guidelines suggest a collar mass limit of 5-10% of the body mass 42 to minimize the effect on the animal; our collars were below that threshold at 0.3 to 1.0% (collar mass: cheetah, 340 g; lion, 970 g; zebra, 930 g and impala, 450 g. Dropoff mechanisms were 108 g (Sirtrack) and 50 g (Biotrack)). The electronics package was similar in all collar versions with almost identical functionality. Signal processing. GPS-INS processing was used to reduce noise and improve precision for the position and velocity analysis, as well as increasing the temporal resolution of the data. GPS and IMU measurements were fused 9 using a 12-state extended Kalman filter 43 in loosely coupled architecture written in MATLAB (The Mathworks). The total state formulation used propagates position, velocity and Article reSeArcH orientation states with time using the IMU measurements in a simplified form of the strap-down inertial navigation equations 44 . The associated process noise was estimated from the known error characteristics of the inertial sensors used. GPS position and velocity updates were used as measurement updates, and receiver accuracy data for each fix used to estimate measurement noise to appropriately weight the GPS to the inertial solution.
The filter was run in reverse time from the last GPS observation of each run to the beginning of the buffered inertial data. During the short time period in which only inertial data was present (delay between trigger and first GPS fix), the filter propagation was equivalent to open-loop inertial navigation. The filter was initialized using the last GPS position and velocity data, and Euler angles assumed zero with covariances appropriate for the uncertainty in that assumption. A Rauch-Tung-Striebel smoother 45 was then applied in forward time on the Kalman-filtered data. This is equivalent to combining backward and forward solutions, effectively halving the open-loop INS integration period between GPS observations. In cases for which it was not possible to reconstruct the period before the first GPS observation (time too long or GPS accuracy insufficient), runs start at medium speeds rather than very low speeds. Calculation of speed and stride times. Vertical accelerations were used to determine stride times. A zero phase band pass Butterworth filter (fourth order) was applied with cut-off frequencies of 1 Hz and 6.6 Hz (twice the maximum stride frequency in the cheetah and impala). A peak detection function was used to detect peaks with a minimum period of 0.25 s between peaks and a minimum peak height of 0.1 g.
Species-specific gait parameters, such as transition speeds and expected stride frequencies for walking and trotting (based on ref. 46), were used to remove double peaks in strides in symmetrical gaits. Horizontal stride speed was derived from the Kalman-filtered velocity averaged over strides in order to remove the effects of speed fluctuation through the stride and collar oscillation relative to the centre of mass. Tangential acceleration, change of heading and centripetal acceleration over stride. Stride times were used to calculate tangential (fore-aft) acceleration, centripetal (turning) acceleration and change in heading between strides. The displacement vectors between consecutive strides were then calculated: where P i is the two-dimensional position at stride i. Change of heading (Δ θ i ) was calculated from the angle between the two vectors: Angular velocity (ω i ) was derived by dividing the change of heading by the time between mid-stride positions Δ T: The tangential or fore-aft acceleration (a t,i ) and centripetal acceleration (a c,i ) were then computed from mid-stride speeds v i : Negative values for tangential acceleration represent deceleration. Absolute values were used for centripetal acceleration, equalling right and left turns. For visual purposes the data in Fig. 3n were mirrored around the vertical axis.
Mass-specific centre of mass (COM) stride work (net COM kinetic energy change in a stride) was calculated as change in speed over a stride multiplied by stride average speed. Mass-specific COM power was calculated as the dot product of stride-averaged tangential acceleration and stride-averaged velocity (that is, multiply forward acceleration by forward speed): Calculation of grip limits. Grip limits are shown in Fig. 3. Friction poses a limit on acceleration and is the product of friction coefficient μ and force normal to the surface (based on acceleration due to gravity, g). Therefore the maximum total horizontal acceleration a max is limited to: where a max is the resultant (combination) of tangential and centripetal acceleration: c,max t,max Maximum turning speed v max depends on friction, gravity and turning radius and is calculated based on equations (6) and (9): Calculation of stride frequency. Regression lines were fitted to stride frequency versus speed data at running speeds. Sections with running data were identified using an unsupervised clustering algorithm on three features derived from windows of accelerometer signals (4-s long) 46 . Features were chosen on the basis of domain knowledge and were the s.d. of the horizontal and vertical axis accelero meter signals and an autocorrelation estimate of the stride frequency 46 .
Features were normalized to have zero mean and unit standard deviation before they were clustered using the k-means algorithm. The number of clusters was determined using the Davis-Bouldin criterion 47 and human inspection. Subsequently, the sections identified to contain running data were isolated and vertical acceleration was used to determine stride times (see 'Calculation of speed and stride times'), stride frequency was calculated from the time between acceleration peaks. Regression lines were calculated for the subgroup from each bin representing the middle 60%, the highest 20% of positive and highest 20% of negative power (Fig. 3b, g).

Maximum performance analysis.
Extracting values that reflect maximum performance carries the risk of choosing outliers generated by non-Gaussian GPS noise rather than realistic values. Previous work reduced the risk of overestimating performance by weighting stride parameters, such as stride speed and accelerations, by the previous and following stride 9,37,48 . Here we chose a different approach, not weighting, but calculating the 98th percentile for each of a number of bins (Fig. 3) in order to also address the effect of different sample sizes and accelerations that were not sustained for three consecutive strides. In addition, obvious errors (speeds > 30 m s −1 and total stride averaged accelerations exceeding a magnitude of 20 m s −2 ) were removed from the dataset. An inherent issue with comparing the performance of different species lies in their different movement patterns, with lion and zebra having a considerably higher proportion of straight, constant low speed strides than impala and cheetah. In order to extract manoeuvring strides, a cut-off based on the magnitude of the total horizontal acceleration (combined tangential and centripetal acceleration) was performed. This cut-off could not be universal, because different animals had different amounts of low-speed steady-state behaviour in their accelerometer traces. This manifested itself in large differences between species in kurtosis of the acceleration-distribution histograms. To address these differences a species-specific cut-off was used. To ensure that this cut-off still gave comparable results for the different species, the characteristic scale of the kurtosis for each distribution was estimated using: 1 4 where s α is our characteristic scale, σ α is the standard deviation and k α is the Pearson's kurtosis, all for species α. If a cut-off of c α was used for one species, then we can calculate the cut-off for species β by: The effect of this cut-off on the distribution of total horizontal acceleration is shown in Extended Data Fig. 3a, b. In Fig. 3n, tangential acceleration is plotted against centripetal accelerations. The Cartesian coordinates were transformed into polar coordinates in order to bin the data. Calculations were performed on absolute centripetal acceleration values to boost data point numbers in bins and then mirrored on the vertical axis; the semicircle was divided into a total of six bins.
The cut-off was adjusted, so that the number of data points in a bin was at least 200 for all species. The cut-off was determined by the impala, which had the lowest number of data points.
The parameters were plotted versus horizontal speed (except for stride frequency) and the 98th percentile was calculated for each of a number of speed bins for which the width was defined by the requirement that each bin should include 400 data points. The final (highest speed) bins with less than 400 data points were discounted. In Fig. 3 a moving average of three bins was applied to all data except Fig. 3n. Data were interpolated to allow the calculation of species performance ratios (Fig. 3) at 1 m s −1 speed positions. Statistical analysis. The maximum performance of the predator and prey were compared using a set of linear models of maximum positive and negative power, positive and negative tangential acceleration and absolute centripetal acceleration.
The maximum performance of each individual was quantified by taking the 98th percentile of the positive and negative tangential acceleration and absolute centripetal acceleration of each individual.
Negative and positive power covaried with speed and was binned by speed as above and the 98th percentile within each bin was computed for each subject within a species. A linear regression was then performed and the predicted power at 8 m s −1 calculated for each subject.
Linear models were fitted to these data using restricted maximum likelihood, with the maximum powers and accelerations as dependent variables, against a factor for predator versus prey and a factor for each pairing (zero for cheetahimpala and one for zebra-lion). Models with an interaction term between these two factors were fitted, but comparing these models to those previously described (this time fitted using maximum likelihood) indicated the interaction term was superfluous (effect sizes were small and associated P values not significant). The interaction term was therefore dropped from the analysis in all models, except when comparing stride frequencies for which there was a substantial interaction term (effect sizes large and associated P values significant). Here, the model was fitted by individual species pairs. Owing to the presence of heteroscedasticity, the error term was allowed to vary for each species. Chase-evasion model. The model combines the observed acceleration capacity with a maximum speed constraint to produce possible position profiles for predators and prey in the subsequent two strides of a chase. We simulate the possible positions of the prey given the prey's initial speed. We then do the same for the predator, optimizing the predator's initial speed to give maximum overlap in final positions of the predator and prey.
We begin with the observed maximum accelerations for our subjects (Extended Data Table 2b). We approximate the possible impulsive accelerations of the animals by assuming they have a maximum tangential acceleration forward, a t , a maximum reverse tangential acceleration, a tr , and a maximum centripetal acceleration, a c . The profile of possible accelerations is assumed to be two half-ellipses with the semi-minor axis along the direction of motion. The top ellipse has semi-minor axis radius a t , the bottom ellipse has semi-minor axis radius a tr , and both have a semi-major axis of length a c .
The animals are assumed to have a maximum possible speed, v. This places a further constraint on the possible profile of accelerations as no acceleration can result in a speed above this maximum.
To find this constraint, we assume that a predator and its prey are galloping at a common stride frequency (Fig. 3b, g) and phase, and that the bulk of the impulse that they can achieve in a stride is performed in a short duration (stance). On any given stride the animal can apply an impulse to change direction, subject to the constraint that the resulting speed cannot be greater than the animal's maximum speed, v M . If the animal is at a speed v along a unit direction iˆ and an impulsive acceleration + a i a î0 1 with ĵ perpendicular to iˆ is to be applied then the resulting speed is: where f is the stride frequency. This must be less than v M . This implies a pair of quadratic relations between a 0 and a 1 subject to v and v M of the form: The simulation allows our subjects to accelerate to anywhere within the area formed by the union of the area above the negative root of this equation, below the positive root, and within the two half ellipses previously mentioned. We note that the possible acceleration profile depends both on the position of the animal and its current speed (an animal that is slow will not be constrained by its maximum possible speed, whereas one going at its maximum speed cannot accelerate forward). This means that simulating the animal's possible positions forward in time increases in complexity with each stride taken, as both the new position of the animal and the new speed must be retained. As such we confined ourselves to simulating two strides forward from our starting conditions; that is, we are only concerned with strategies for the predator and prey at the very end of a chase.
We assume that the prey performs an evasive acceleration on the first stride, while the predator continues to chase without changing velocity. On the second stride, the prey again accelerates, and now the predator also has the ability to react to the acceleration it observes in the first stride. We ran 100 such simulations for starting separations varying from the maximal separation that makes capture possible within two strides down to half a predator length separation (cheetah = 0.66 m 49 , lion = 0.92 m 50 ). If the prey and predator are closer than this, then the predator is already close enough for prey capture.
For a given prey speed and initial predator-prey separation, we find that predator speed, which maximizes the capture probability by means of a Nelder-Mead simplex optimization, is subject to the constraint that the initial predator speed must be greater than or equal to the initial prey speeds. Owing to ambiguity in the solution space, a small penalty term encouraging faster speeds from the predator was added in the form eps × v pred , with eps = 10 −6 . This ensured that for data with a range of optimal best speeds for the predator, the fastest was selected. This had no effect on the value of the optima up to four significant digits.
To test how a change in predator or prey performance influences hunt outcome, we adjusted the performance by multiplying the maximum recorded tangential and the centripetal accelerations of the prey or predator by a number ranging from 0.6 to 1.4 to deliver values to insert into the model for animals of greater or lesser athleticism, respectively. This number is the x-axis performance adjustment in Fig. 4g, h and rerunning the simulation to obtain capture probabilities. Maximum speed was not adjusted. List of symbols. Muscle studies. F isom , fibre isometric force; F, fibre force during shortening; SL, fibre sarcomere length; L o , fibre length when the sarcomere length is set to 2.55 μ m; Q, fibre relative power; Q max , fitted maximum relative power; F o , fitted force intercept on a distribution of Q against F / F isom ; FQ max , fitted relative force at maximum power; CSA, fibre cross-sectional area; Q 10 , the ratiometric increase in rate with a 10 °C temperature change. Locomotion and model. i, stride number; P i , two-dimensional position; U i , two-dimensional position difference between subsequent strides; Δ θ i , signed change of heading; ω i , heading angular velocity; Δ T, sampling interval; a, total horizontal acceleration; a t , tangential or forward acceleration, a tr tangential reverse acceleration; a c , centripetal acceleration, a 0 and a 1 are generic accelerations; r, turn radius; v, stride-averaged horizontal speed, v max , maximal turning speed, v M , maximum speed; P t , mass-specific fore-aft power; μ , coefficient of friction; m, body mass; g, gravity; α and β , species indices; s, characteristic scale; σ, s.d.; k is the Pearson's kurtosis.  Fig. 4d with more subplots for cheetah-impala and lion-zebra, respectively. At the start of simulation, both have initial velocity towards the top of the page and initial separation. After one stride the prey can move to anywhere in the red or yellow ellipse by acceleration in the appropriate direction. Predator velocity remains unchanged, as there is no prey acceleration in the previous stride to react to. Initial positions are shown. Larger red or yellow ellipse perimeter is the area prey can reach after two strides of the chosen maximum acceleration. The blue or purple filled ellipse represents the locations the predator can occupy after its second stride (responding to the prey acceleration observed in first stride). The area of the prey ellipse that is covered by the predator ellipse line is defined as probability of capture. Predator is given a starting speed for each combination of prey speed and initial spacing that maximizes the capture probability. Rows are different initial prey speeds, values in red to the left of each row. Columns are different initial predator-prey separations at the start of the simulation with values given in red below each column. Scale for all instances is given in the bottom left plot in metres (in black). The inset black numbers in each sub-panel are the initial (optimized for maximum success) predator speeds in m s −1 . c, The optimum lion speed to maximize overlap (hotter colours indicate faster speed, key on the right) as a function of zebra speed (x axis) and starting separation (y axis). The histogram above the main plot shows the distribution of actual zebra speed at first turn of 10 degrees or more for each run (same x axis as the main plot) and the vertical histogram shows distribution of actual lion speed at first turn (scale as for heat bar). d, The proportional overlap (capture probability), as a function of zebra initial speed and starting separation. e, Modelled capacity for forward acceleration (speed increase per stride) as a function of speed (Extended Data Table 2b). Cheetah, blue; impala, red; lion, purple; zebra, yellow.
Article reSeArcH extended data table 2 | Performance parameters a, log-log slope of performance parameter versus mass for the two herbivore species and the two predator species. Stride values were extracted to represent species' performance and evaluated versus body mass to explore whether the performance difference between small and large was concomitant with effects reported across a broad animal size range 69 . Parameters with increasing values (positive and negative work and power) were represented by maximum values whereas for parameters that plateaued (positive and negative tangential acceleration, centripetal acceleration), an average value was calculated. The slope of the logarithmic coordinates (log-log slope of performance parameter versus mass) was calculated for the two predators and two herbivores. The relationship is generally consistent in predators and in prey, with most parameters dropping with increasing size, but this does not provide an explanation for the magnitude of the differences seen, as most parameters would scale weakly with animal size. b, Maximum (98%) values for stride parameters for all species. Maximum values were determined using the 98th percentile (after species-specific steady-state strides were removed from the data, positive and negative data were calculated separately). These are the parameters used in the model.

Performance parameter
Predator Herbivore Pos. tangential acceleration (ms -2 ) -0. somewhat higher than that for lion published previously (Kohn and Noakes J Exp Biol, 216:960), although the very different assay temperature and arbitrary correction for fibre swelling in that study make direct comparisons difficult.
Chase Simulation: Simulation, no experiment replicated.

Randomization
Describe how samples/organisms/participants were allocated into experimental groups.
Muscle Physiology: Randomization was not used, per se. Once the muscle biopsies were transported to the SML facility we purposely began fibre tests with sample 1 of the first species available to us. Three experimenters were involved with the testing; Diack, Lorenc and West. All used the same fibre-testing protocol. A range of fibres sizes (diameters) were tested, for each individual of each species.
Chase Simulation: No experiment, therefore no randomisation

Blinding
Describe whether the investigators were blinded to group allocation during data collection and/or analysis.
Muscle Physiology: Each of the experimenters knew the biopsy origin.
Chase Simulation: No experiment, therefore no blinding.
Note: all studies involving animals and/or human research participants must disclose whether blinding and randomization were used.

Statistical parameters
For all figures and tables that use statistical methods, confirm that the following items are present in relevant figure legends (or in the Methods section if additional space is needed).

n/a Confirmed
The exact sample size (n) for each experimental group/condition, given as a discrete number and unit of measurement (animals, litters, cultures, etc.) A description of how samples were collected, noting whether measurements were taken from distinct samples or whether the same sample was measured repeatedly A statement indicating how many times each experiment was replicated The statistical test(s) used and whether they are one-or two-sided (note: only common tests should be described solely by name; more complex techniques should be described in the Methods section) A description of any assumptions or corrections, such as an adjustment for multiple comparisons The test results (e.g. P values) given as exact values whenever possible and with confidence intervals noted A clear description of statistics including central tendency (e.g. median, mean) and variation (e.g. standard deviation, interquartile range)

Clearly defined error bars
See the web collection on statistics for biologists for further resources and guidance.

Software
Policy information about availability of computer code 7. Software Describe the software used to analyze the data in this study.
Muscle Physiology: Matlab was used for processing raw data files to calculate peak power, maximum isometric stress, and velocity and stress at peak power. Data then analyzed in R. Figures showing data inter-relationships and box-plots with error bars were generated in Sigma-plot.
Chase Simulation: Custom python code was written using the scipy library.
For manuscripts utilizing custom algorithms or software that are central to the paper but not yet described in the published literature, software must be made available to editors and reviewers upon request. We strongly encourage code deposition in a community repository (e.g. GitHub). Nature Methods guidance for providing algorithms and software for publication provides further information on this topic.

Materials and reagents
Policy information about availability of materials

Materials availability
Indicate whether there are restrictions on availability of unique materials or if these materials are only available for distribution by a for-profit company.
Muscle Physiology: No for profit restrictions.
Chase Simulation: No restriction.

Antibodies
Describe the antibodies used and how they were validated for use in the system under study (i.e. assay and species).
Muscle Physiology: N/A unless it is relevant here that we are asked to expand on our one statement about muscle fibre-types, which were based on antibody binding to MHC. Animals were collared and biopsied as part of an ongoing collaborative program of ecology field work as described in the paper. Muscle was also taken from male Impala shot for human consumption on a game reserve in Zimbabwe.
Policy information about studies involving human research participants

Description of human research participants
Describe the covariate-relevant population characteristics of the human research participants. N/A