Thompson - Computing for Scientists and Engineers (523188), страница 45
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Several of the properties that we derive are important in studying ecologicalpopulation dynamics, electronic feedback, and the stability of systems under disturbances. In order to simplify matters in the following, we use the scaled population,n, and we relabel the dimensionless time variable as t rather than t'.Exercise 7.11Show that if n (0) < 1, the logistic curve n(t) has a point of inflexion (wheren” = 0) at the scaled time t = 1n l/n (0) - 1]. This behavior may be visible inFigure 7.3, and it justifies another name often given to this curve, the S curve.Show that if n (0) > 1, there is no point of inflection. nEven though for the logistic-growth curve the growth is limited, the followingexercise may convince you that prediction of the final population from observationsmade near the beginning of the growth period is not straightforward.Exercise 7.12(a) From the differential equation (7.14) show that Ne, can be predicted from thepopulations N1, N2 at times t1 and t2 and the growth rates at these times N1 andN2 according to(7.22)(b) Explain why an accurate value of Ne is difficult to obtain using this equationapplied to empirical data with their associated errors.
Consider the discussion inSection 4.3 about unstable problems and unstable methods. nThis exercise highlights the difficulties of deciding whether a natural species needsto be protected in order to avoid extinction. The question becomes especially interesting in the ecology of depleting resources, especially when one considers the following modification and application of the logistic-growth equation (7.14) to examine the question of harvesting nature’s bounty.Exercise 7.13Consider the harvesting of a biological resource, such as fish, which might follow the logistic-growth curve (7.14) if undisturbed. Suppose, however, that theresource is harvested at a rate proportional to its present numbers.7.3 NONLINEAR DIFFERENTIAL EQUATIONS: LOGISTIC GROWTH239(a) Show that the logistic-growth differential equation is then modified to(7.23)where H (positive) is the harvesting fraction rate.(b) Show that the logistic equation is regained, but with a reduced constant determining the equilibrium time(7.24)and a reduced equilibrium population(7.25)(c) Verify that the model predicts that the resource will become extinct ifExplain this result in words.
nThe logistic-growth equation and its relation to the stability of iteration methodsand to chaos is considered in Chapter 5 of Hubbard and West’s book and softwaresystem, in Chapters 4 and 8 in Beltrami’s book, and in the chapter “Life’s Ups andDowns” in Gleick’s book. A discussion of solutions of the logistic equation, theirrepresentation in a phase plane, and the effects of lags are given in Haberman’s bookon mathematical modeling. Quasi-chaotic behavior induced by discretizing the solution of (7.15) is examined in the article by Yee, Sweby, and Griffiths.Generalized logistic growthSuppose that instead of quadratic feedback to stabilize the population, as we have in(7.14), there is feedback proportional to some power of n, say the (p + 1) th power.That is, (7.14) becomes the generalized logistic equation(7.26)In mathematics this is sometimes called Bernouilli’s equation.
By following somesimple analytical steps, you can find directly the solution of this equation.Exercise 7.14Divide throughout (7.26) by n , then convert the derivative to a logarithmic derivative and introduce a new time variable(7.27)Also define(7.28)Show that np, satisfies the differential equation240INTRODUCTION TO DIFFERENTIAL EQUATIONS(7.29)which is a differential equation independent of the feedback power p. Thus, argue that, using the power factors given in (7.27) and (7.28), all the solutions ofthe generalized logistic equation are self-similar and can be obtained from that forp = 1, namely (7.21). nThe solution of the generalized logistic equation (7.26) may therefore be written(7.30)where, as in (7.18), we have n as the fraction of the equilibrium population, and t'as the scaled time, This behavior is quite remarkable in that it is the particularform of the first-order nonlinear differential equation (7.14) that allows such a generalized solution to be obtained.The behavior of the generalized logistic curve as a function of the feedbackpower p in (7.30) is shown in Figure 7.4 for n (0) = 0.5; that is, the initial population is one-half the final equilibrium population.
As you would guess, as p increases, the population approaches the equilibrium value n = 1 more rapidly.Exercise 7.15Set up a calculation to compute formula (7.30) for given n (0) and p. For asystem that starts off above equilibrium population, n (0) > 1, run this calculation for a range of t' similar to that shown in Figure 7.4. Check your resultsagainst the curve shown in Figure 7.3 for p = 1, n (0) = 1.25.
nFIGURE 7.4 Generalized logistic equation solutions (7.30) for feedback proportional to the(p + 1) power of the number present, shown for n (0) = 0.5 and four values of p.7.4NUMERICAL METHODS FOR FIRST-ORDER EQUATIONS241The population growth as a function of time in units ofis well behaved even fora non-integer power, such as with p = 1/2 in Figure 7.4, for which the negativefeedback in (7.26) is proportional to the slower n3/2 power, which is more gradualthan n2 feedback (p = 1) in the conventional logistic-growth equation (7.14).Our generalization of the logistic equation to (7.26), with its self-similar solution(7.30), does not seem to have been much investigated.
The function (7.30) hasbeen used as the Type I generalized logistic distribution, as discussed by Zeitermanand Balakrishnan in the statistics monograph edited by the latter author. For detailedapplications in statistics, use of logistic distributions is described in the book byHosmer and Eemeshow.7.4NUMERICAL METHODS FOR FIRST-ORDER EQUATIONSAs we learned in the preceding sections, if an analytical solution of a differentialequation is possible, the method of solution is often specific to each equation.Numerical methods for differential equations, by contrast, can often be applied to awide variety of problems. But, such methods are not foolproof and must be appliedcarefully and thoughtfully.
(If numerical recipes are used by those who have notgraduated from cooking school, then the dish may be quite unsavory.)Our aim in this section is to emphasize basic principles and to develop generalpurpose methods for solving first-order differential equations. In Project 7 in Section 7.5 we develop a program for the first-order Euler method. Numerical methods for differential equations are investigated further in Sections 8.3 - 8.6, wherewe emphasize second-order equations and include second-order Euler methods.In numerical methods for differential equations, one often speaks of “integratingthe differential equation.” The reason for this terminology is that since(7.3 1)we may consider solving a differential equation as an integration process.
Indeed,this is often the formal basis for developing numerical methods.In what follows we often abbreviate derivatives using the prime notation, inwhich y''(x)=Dxy(x) for the first derivative, and the number of primes indicatesthe order of the derivative. Past the third derivative, numerals are used, (3), (4), etc.Presenting error valuesErrors in numerically estimated values can be presented in two ways.
The first is topresent the actual error in the y value at x = xk, which we call ek and define as(7.32)in which the first quantity is the exact value at xk and the second is the numericallyestimated value at this point. Unlike random errors of experimental data, but liketheir systematic errors, the value of ek is usually reproduced when the calculation is242INTRODUCTION TO DIFFERENTIAL EQUATIONSrepeated. (An exception is a Monte Carlo error estimate whenever different sampling of random numbers is made from one calculation to the next.) The sign of ekis therefore significant, so we will be systematic in using definition (7.32).The second way of presenting errors is as the relative error, rk, defined by(7.33)which we use only when yk is not too close to zero. Sometimes we give 100 × rk,the percentage error, or we show errors scaled by some power of 10.
Clearly, weusually have only estimates of ek or rk, rather than exact values, as we indicate byusing thesign when appropriate. In the program Numerical DE-1 in Section 7.5 the program presents the errors in both ways.Euler predictor formulasA predictor formula for the numerical solution of a differential equation is one inwhich previous and current values are used to predict y values at the next x values.The general first-order differential equation can be written(7.34)wheref is any well-behaved (usually continuous) function of x and y, but it shouldnot contain any derivatives of y with respect to x past the first derivative.
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