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First published online November 28, 2008
Journal of Experimental Biology 211, 3836-3849 (2008)
Published by The Company of Biologists 2008
doi: 10.1242/jeb.024968

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Limits to running speed in dogs, horses and humans

Mark W. Denny

Hopkins Marine Station of Stanford University, Pacific Grove, CA 93950, USA

View larger version (11K): [in this window] [in a new window]   Fig. 1. Hypothetical data and the fit to them using the logistic equation (Eqn 3). The red line is the best-fit logistic model, and the black lines are the confidence limits on that fit, drawn using the best-fit values for the shape parameter k and the location parameter t and the 95% ranges for the minimum fastest speed mn and the maximum fastest speed mx. Note that the model's confidence interval does not incorporate the highest speeds.

View larger version (10K): [in this window] [in a new window]   Fig. 2. (A) The generalized Pareto equation (GPE, Eqn 6) can be used to estimate absolute maximum speed. Hypothetical measured data are shown as black dots, and the best-fit GPE fitted to these data (the red line) can be extrapolated to an exceedance probability of G=0, thereby estimating the maximum possible speed. Note from the ordinate on the right that G=1 corresponds to population size S0, and G=0 corresponds to infinite population size. (B) The information from A, presented in terms of population size rather than probability. The extrapolation of the GPE model to infinite population size gives an estimate of maximum speed, shown by the red dot.

View larger version (22K): [in this window] [in a new window]   Fig. 3. Temporal patterns of winning speeds in the Triple Crown races. Black dots are winning speeds in the years shown. Green lines are regressions for data in the plateau of each record; any slope of these regression lines is statistically insignificant. Red lines are the best-fit logistic models.

View larger version (23K): [in this window] [in a new window]   Fig. 4. Population trends in (A) US thoroughbreds, (B) Irish greyhounds and (C) humans. The red line in B is a 5 year running average of the data to emphasize the trend. The red line in C is from Eqn 7.

View larger version (18K): [in this window] [in a new window]   Fig. 5. Representative examples of variation in running speeds as a function of population size. For sufficiently large populations, there is no correlation between population size and speed in (A) thoroughbreds and (B) greyhounds. In contrast, human speeds (exemplified here by C, the men's 1500 m race) are correlated with population size throughout the historical range of population size.

View larger version (20K): [in this window] [in a new window]   Fig. 6. Temporal patterns of winning speeds in English greyhound races. Black dots are winning speeds in the years shown. Green lines are regressions for data in the plateau of each record; any slope of these regression lines is statistically insignificant. Red lines are the best-fit logistic models. Gaps in the 1970s and 1980s for the English Grand National and English Derby are due to changes in the course length in these races during that period.

View larger version (9K): [in this window] [in a new window]   Fig. 7. Temporal patterns of annual fastest speeds for humans running 100 m. Dots are winning speeds in the years shown. The green line is the regression for data in the plateau of the women's record. Red lines are the best-fit logistic models. Men's speeds appear not to have plateaued. The recent world record set by Usain Bolt (2008 Olympics) is shown as the pink dot.

View larger version (26K): [in this window] [in a new window]   Fig. 8. Temporal patterns of annual fastest speeds for humans running 200 m to 1500 m; (A) men, (B) women. Dots are winning speeds in the years shown. Women's speeds appear to have plateaued, and the green lines are the regressions for data in these plateaus. Any slope of these regression lines is statistically insignificant. Red lines are the best-fit logistic models. Men's speeds appear not to have plateaued. The recent world record set by Usain Bolt in the 200 m race is shown as the pink dot.

View larger version (22K): [in this window] [in a new window]   Fig. 9. Temporal patterns of annual fastest speeds for humans running 3000 m to 41,195 m (the marathon); (A) men, (B) women. Dots are winning speeds in the years shown. Red lines are the best-fit logistic models. Speeds in these distance races appear not to have plateaued.

View larger version (17K): [in this window] [in a new window]   Fig. 10. (A) A representative example of analysis using the population-driven model; data for the men's 1500 m race. The red line is the best-fit GPE (Eqn 6), and the black lines are the bootstrap 95% confidence limits on this model. The green line depicts the addition of the absolute maximum deviation to the best-fit GPE model. The intersection of the green line with the abscissa is the predicted absolute maximum speed. In this case, the best-fit GPE is nearly linear (=–1.024, where =–1 is precisely linear), but linearity is neither an assumption of the analysis nor a necessity of the fit. (B) Data from A translated to show speed as a function of population size. Line colors are as in A. Dots on the ordinate at infinite population size show the estimated maxima of the various lines.

View larger version (10K): [in this window] [in a new window]   Fig. 11. Summary of human running data for distances 100 m to 42,195 m. Solid lines are the current world records for men (blue) and women (red). Estimates from the no-trend approach are shown by the circles; estimates from the logistic approach are shown by the squares; and estimates from the population-driven approach are shown by the diamonds. Error bars show the 95% range of the extreme-value estimate of absolute deviations. Estimates based on 100 year return values (rather than absolute maxima) are denoted with open symbols. For clarity, symbols are staggered slightly along the abscissa.

View larger version (7K): [in this window] [in a new window]   Fig. A1. If f, the fraction of individuals that run a particular race, increases through time, the actual population of runners associated with a given measured maximum speed (open dots) is higher than that supposed by the calculations made here (solid dots). A larger population corresponds to a smaller probability of exceedance (e.g. the green arrow). Thus, if probabilities were to be adjusted for an increase in f (open dots) the estimated absolute maximum speed would be reduced (the blue arrow).

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