Thompson - Computing for Scientists and Engineers (523188), страница 44
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These are= 0.93 s and A = 12.2 m s-2, thus= 11.0 m s-1. These are the parametersused to generate the fit shown in Figure 7.1. By running World Record Sprintswith the men’s 1972 data you may verify that there is a marginally better set of parameters for these data.Because the model is not expected to hold past about 200 m, and because theonly official race distances are now 100 m and 200 m, one should not attempt tofind a best-fit from only these two distances. Also, there is about a 0.5-s penaltyin a 200-m race if it is held on a track with a turn. In all the analysis results quotedhere 0.5 s was subtracted from the 200-m times for races on turning tracks for bothmen and women to approximate the 200-m time on a straight track, a correction justified by looking at the 200-m records for 1968.Exercise 7.7(a) Code, check, and run World Record Sprints for the men’s 1972 datagiven in Table 7.1.
Search on in order to verify that the best-fit parametersare =O.93+0.02 s, A = 11.8 m s-2 , since= 11.0 m s-1.(b) Input the men’s 1991 records for the 100-m and 200-m sprints, adjustingthe times of the latter by subtracting 0.5 s, as justified above. Use the newvalue of in order to show that the best value of A is unchanged, since stillA = 11.0 m s-1, within the validity of the model and the uncertainty in .(c) Calculate the average speeds for the mens’ 1972 records and compare themgraphically with the Keller-model prediction (7.11). Show that the men’s average speed is about 10.5 ms-l for a 200-m dash.(d) While I was preparing this book, Carl Lewis of the USA lowered the recordtime for the 100-m sprint from 9.92 s to 9.86 s.
Use this time with the otherdata for men in Table 7.1 in order to see what effect this has on the best-fitvalues of and A. n234INTRODUCTION TO DIFFERENTIAL EQUATIONSFIGURE 7.2 Average speeds for world-record sprints as a function of time for men in 1972(solid points and solid line), women in 1968 (crossed circles and dotted line), and for women in1991 (asterisks and dashed line). The lines are computed using the best fits to the Keller model.The average speeds for races that are run in a time T are shown in Figure 7.2,Note the approximately linear increase in speed (constant acceleration) for the firstsecond of the race, then the smooth approach to the steady speed (no acceleration) asthe duration of the race increases.Now that we have examined the records for the performance of male athletes, itis of interest to complement them with an analysis of the data for top female athletes.Women sprinters are getting fasterAlthough Keller did not analyse the data for women sprinters, there are adequate records for several of the distances, both in 1968 and 1991, as given in Table 7.1.Therefore we can make a similar analysis of the women’s records as for the men’srecords.
By contrast with the men’s records, which Exercise 7.7 showed have notresulted in improvements within the validity of the model, the women athletes haveimproved by about 5% over 20 years, as we now demonstrate.Exercise 7.8(a) Input to the program World Record Sprints the women’s 1968 recordsgiven in Table 7.1, subtracting 0.5 s from the time for the 200-m race to correctfor the track curvature. Show that the best value of = 0.78 s and of A is12.1 m s-2, since= 9.45 m s-l, which is about 14% slower than the malesprinters.7.3 NONLINEAR DIFFERENTIAL EQUATIONS: LOGISTIC GROWTH235(b) Input the women’s 1991 records from Table 7.1, again correcting for thecurvature of the track. Using= 0.78 s, derive the best-fit value of A as12.9 m s-2, since the speed has increased to= 10.1 m s-l, a 7% improvement in 23 years, or about 6% if scaled uniformly for 1972 to 1991.(c) Calculate the average speeds for the women’s 1968 and 1991 records, thencompare them graphically with the Keller-model prediction (7.1l), as in Figure 7.2.
Show that the women’s average speed has increased from about9.1 m s-1 to about 9.75 m s-l over a 200-m dash, a 7% improvement.(d) Estimate by linear extrapolation the decade in which women’s sprintperformances are predicted to match those of men sprinters. Predict that timesfor 100 m will match by about 2020 and for 200 m by about 2040. nAthletic improvements for women compared to men are discussed by Whipp andWard. The pitfalls of the linear extrapolations they used have been emphasized byEichtenberg.
The relatively slow increase in athletic performance of male sprinterscompared with that of female sprinters over the past two decades might suggest thatthe speed limits of human sprinting have been essentially reached by males. Thisquestion and other constraints to improving athletic performance are discussed in thearticle by Diamond.7.3NONLINEAR DIFFERENTIAL EQUATIONS:LOGISTIC GROWTHIn this section we set up a nonlinear differential equation that is often used to modelthe growth of biological populations or other self-limiting system, such as a chemical system without feedback of chemical species.
We first set up the differentialequation, then we explore its properties both analytically and numerically, beforeshowing how the equation may be generalized.The logistic-growth curveAlthough exponential-growth equations appear often in science, such growth cannotcontinue indefinitely. The corresponding differential equations must be modified sothat increasing inhibitory effects take place as time goes by. For example, in an electronic circuit this damping could be obtained by having negative feedback proportional to the output. In ecology, the population dynamics of natural species oftenhas self-generated or external constraints that limit the rate of growth of the species,as described in the introductory-level article on mathematical modeling by Tuchinsky.
Chapter 3 of the classical treatise by Thompson provides a wealth of materialon growth equations.One interesting differential equation that models such inhibitory effects is thenonlinear logistic-growth differential equation(7.14)236INTRODUCTION TO DIFFERENTIAL EQUATIONSin which both and are positive. The first term on the right-hand side gives thefamiliar rate of increase proportional to the number present at time t, N(t), and thusto exponential increase of N with time, since > 0.
The second term on the righthand side is an inhibitory effect since> 0, and at any time t it increases as thesquare of the current value, N(t). The competition between increase and decreaseleads to a long-time equilibrium behavior. In the biological sciences (7.14) is oftencalled the Verhulst equation, after the nineteenth-century mathematician.Exercise 7.9(a) Show that the equilibrium condition DtN = 0 is satisfied by setting theright-hand side of (7.14) to zero to obtain the solution for the equilibrium Nvalue as Ne =, so that (7.14) can be rewritten as(7.15)(b) Noting that at early times such that N(t)/Ne << 1 we have the equation forexponential growth, and that for long times the slope tends to zero, assume asolution of the logistic-growth equation (7.14) of the form(7.16)By using the immediately preceding arguments, show that A = Ne.(c) Show that the assumed form of the solution satisfies the original differentialequation (7.14) for any value of B.(d) If the number present at time zero is N (0), show that this requiresB = Ne/N(0) - 1.
nIf we assemble the results from this exercise, we have that the solution of the differential equation for a quadratically self-inhibiting system has the number present attime t, N(t), is given by(7.17)where Ne is the equilibrium population and N(0) is the initial population. Equation(7.17) describes logistic growth.The function N(t) comes in various guises in various sciences. In applied mathematics, and ecology it is called the logistic, sigmoid, or Verhulst function. Instatistics it is the logistic distribution, and in this context t is a random variable. Asdiscussed in the book edited by Balakrishnan, its first derivative resembles theGaussian distribution introduced in Section 6.1.
In statistical mechanics, t usuallyrepresents energy, and N(t) is called the Fermi distribution. It is called this (or theWoods-Saxon function) also in nuclear physics, where t is proportional to the distance from the center of the nucleus and the function measures the density of nuclearmatter. In this context, many of the properties of N(t) are summarized in the monograph by Hasse and Myers.In order to explore the properties of logistic growth, it is convenient to recast(7.17) so that only dimensionless variables appear.7.3 NONLINEAR DIFFERENTIAL EQUATIONS: LOGISTIC GROWTH237Exercise 7.10(a) Define the number as a fraction of the equilibrium number as(7.18)and change the time unit to(7.19)Show that the logistic differential equation can now be written in dimensionlessform as(7.20)in which there are no parameters.(b) Show that the solution to this equation can be written as(7.21)in which n (0) is obtained from (7.18).
nGraphs of n (t’) against t’ are shown in Figure 7.3 for several values of n(0).If the system is started at less than the equilibrium population, that is if n(0) < 1 ,then it grows towards equilibrium, and the opposite occurs for starting above equilibrium, as for n(0) = 1.25 in Figure 7.3. By contrast, if there is no feedback thenthe system population increases exponentially for all times, as shown in Figure 7.3for an initial population of 0.125.FIGURE 7.3 Solutions to the logistic differential equation (7.20) for various initial numbers asa fraction of the equilibrium number n (0). The dotted curve shows exponential growth withoutfeedback for an initial number of 0.125.238INTRODUCTION TO DIFFERENTIAL EQUATIONSNow that we have understood how to model feedback by a nonlinear differentialequation, it is interesting to explore logistic growth in more detail.Exploring logistic-growth curvesCurves of logistic growth have many interesting properties, many of which are notself-evident because of the nonlinear nature of the differential equations (7.14) and(7.20).
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