ordinary differential equations - vladimir igorevich arnold

24 Chapter l. Basic Concepts

variable or if we com pactify the affine x-axis to form the projective line ( cf. Chapt. 5).

Problem l. 4 Which of the differential equations x = xn determine on an affineline a phase velocity field that can be extended without singular points to the projective line?

Answer. n = 0,1,2.

8. Example: The Logistic Curve

The ordinary reproduction equation x = kx is applicable only as long as the number of individuals is not too large. As the number of individuals increases competition for food leads to a decrease in the rate of reproduction. The sim­plest hypothesis is that the coefficient k is an inhomogeneous linear function of x ( when x is not too large any smooth function can be approximated by an inhomogeneous linear function): k = a - bx.

We thus arrive at the reproduction equation taking account of competition x = (a - bx )x. The coefficients a and b can be taken as .1 by a change of scale on the t- and x-axes. We thus obtain the so-called logistic equation

x = (1- x)x.

The phase velocity vector field v and the direction field in the ( t, x )-plane are depicted in Fig. 12.

Fig. 12. The vector field and the direc­tion field of the equation x = (1 - x )x

Fig. 13. The integral curves of the equation x = (1- x)x

We conclude from this that the integral curves look as depicted in Fig. 13. More precisely, we see that

1) the process has two equilibrium positions x =O and x = 1;

2) between the points O and 1 the field is directed from O to 1, and for x > 1 to the point l.

Thus the equilibrium position O is unstable (as soon as a population arises it begins to grow), while the equilibrium position 1 is stable (a smaller popu­lation increases, and a larger one decreases ).

4 Here and in the sequel problems marked with an asterisk are more difficult than the others.

§ l. Phase Spaces 25

For any initial state x > O, as time passes the process moves toward the stable equilibrium state x = l.

It is not clear from these considerations, however, whether this passage takes place in a finite or infinite time, i.e., whether integral curves starting in the region O < x < 1 can have points in common with the line x = l.

It can be shown that there are no such common points and that these integral curves tend asymptotically to the line x = 1 as t ~ +oo and to the line x = O as t ~ -oo. These curves are called logistic curves. Thus a logistic curve has two horizontal asymptotes (x = O and x = 1) and describes the passage from one state (O) to another (1) in an infinite time.

Problem l. Find the equation of a logistic curve.

Solution. By formula (3) t = J dxj(x(l- x)) = ln(x/1- x), or x = et /(1 + et). This formula proves the asymptotic property of the logistic curve mentioned above.

Problem 2. Prove that the integral curves of the equation x = (1 - x )x in the region x > 1 tend asymptotically to the line x = 1 as t ---+ +oo and have the vertical asymptote t = const.

For small x the logistic curve is practically indistinguishable from the exponential curve, i.e., competition has little influence on reproduction. However, as x increases the reproduction becomes nonexponential, and near x = 1/2 the exponential curve diverges sharply upward from the logistic curve; subsequently logistic growth de­scribes the saturation of the system, i.e., the establishment of an equilibrium mode in it (x = 1).

U p to the middle of the twentieth century science grew exponentially ( cf. Fig. 10). If such growth were to continue, the entire population of the earth would consist of scientists by the end of the twenty-first century and there would not be enough forests on the earth to print all the scientific journals. Consequently saturation must set in befare that point: we are nearing the point where the logistic curve begins to lag behind the exponential curve. For example, the number of mathematical journal articles increased at a rate of 7% per year from the end of the Second World War until the 1970's but the growth has been slower for the past several years.

9. Example: Harvest Quotas

U p to now we ha ve considered a free population developing according to its own inner laws. As sume now that we harvest a part of the population ( for example, we catch fish in a pond or in the ocean). Let us assume that the rate of harvesting is constant. We then arrive at the differential equation for harvesting

x = (1 - x )x -c.

The quantity e characterizes the rate of harvesting and is called the quota. The form of the vector field and the phase velocity field under different values of the harvest rate e is shown in Fig. 14.

We see that for a harvesting rate that is not too large (O < e< 1/4) there exist two equilibrium positions (A and B in Fig. 14). The lower equilibrium position x = A) is unstable. If for any reason ( overharvesting or disease) the