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J0 + ( T - yX
which is the solution ofEq. (14) subject to the initial condition y(0) = y0.
FIGURE 2.5.6 f (y) versus y for dy/dt = — r(1 - y/ T)y.
Chapter 2. First Order Differential Equations
f,(t) = T
FIGURE 2.5.7 y versus t for dy/dt = —r(1 — y/ T)y.
If Óî < T, then it follows from Eq. (15) that y ^ 0 as t ^ro. This agrees with our qualitative geometric analysis. If y0 > T, then the denominator on the right side of Eq. (15) is zero for a certain finite value of t. We denote this value by t*, and calculate it from
Thus, if the initial population y0 is above the threshold T, the threshold model predicts that the graph of y versus t has a vertical asymptote at t = t*; in other words, the population becomes unbounded in a finite time, which depends on the initial value y0 and the threshold value T. The existence and location of this asymptote were not apparent from the geometric analysis, so in this case the explicit solution yields additional important qualitative, as well as quantitative, information.
The populations of some species exhibit the threshold phenomenon. If too few are present, the species cannot propagate itself successfully and the population becomes extinct. However, if a population larger than the threshold level can be brought together, then further growth occurs. Of course, the population cannot become unbounded, so eventually Eq. (14) must be modified to take this into account.
Critical thresholds also occur in other circumstances. For example, in fluid mechanics, equations of the form (7) or (14) often govern the evolution of a small disturbance y in a laminar (or smooth) fluid flow. For instance, if Eq. (14) holds and y < T, then the disturbance is damped out and the laminar flow persists. However, if y > T, then the disturbance grows larger and the laminar flow breaks up into a turbulent one. In this case T is referred to as the critical amplitude. Experimenters speak of keeping the disturbance level in a wind tunnel sufficiently low so that they can study laminar flow over an airfoil, for example.
The same type of situation can occur with automatic control devices. For example, suppose that y corresponds to the position of a flap on an airplane wing that is regulated by an automatic control. The desired position is y = 0. In the normal motion of the
2.5 Autonomous Equations and Population Dynamics
plane the changing aerodynamic forces on the flap will cause it to move from its set position, but then the automatic control will come into action to damp out the small deviation and return the flap to its desired position. However, if the airplane is caught in a high gust of wind, the flap may be deflected so much that the automatic control cannot bring it back to the set position (this would correspond to a deviation greater than T). Presumably, the pilot would then take control and manually override the automatic system!
Logistic Growth with a Threshold. As we mentioned in the last subsection, the threshold model (14) may need to be modified so that unbounded growth does not occur when y is above the threshold T. The simplest way to do this is to introduce
another factor that will have the effect of making dy/ dt negative when y is large. Thus
dt = -•(> — Ó)(1 — Ó) ó (17)
where r > 0 and 0 < T < Ê.
The graph of f (y) versus y is shown in Figure 2.5.8. In this problem there are three critical points: y = 0, y = T, and y = Ê, corresponding to the equilibrium solutions ô() = 0, ô2(´) = T, and ô3(³) = Ê, respectively. From Figure 2.5.8 it is clear that dy/dt > 0 for T < y < Ê, and consequently y is increasing there. The reverse is true for y < T and for y > Ê. Consequently, the equilibrium solutions ô j (t) and ô3(t) are asymptotically stable, and the solution ô2^) is unstable. Graphs of y versus t have the qualitative appearance shown in Figure 2.5.9. If y starts below the threshold T, then y declines to ultimate extinction. On the other hand, if y starts above T, then y eventually approaches the carrying capacity Ê. The inflection points on the graphs of y versus t in Figure 2.5.9 correspond to the maximum and minimum points, ó and y2, respectively, on the graph of f (y) versus y in Figure 2.5.8. These values can be obtained by differentiating the right side ofEq. (17) with respect to y, setting the result equal to zero, and solving for y. We obtain
y12 = (Ê + T ±ë/Ê2 — ^ + T2)/3, (18)
where the plus sign yields y1 and the minus sign y2.
A model of this general sort apparently describes the population of the passenger pigeon,7 which was present in the United States in vast numbers until late in the nineteenth century. It was heavily hunted for food and for sport, and consequently its
FIGURE 2.5.8 f (y) versus y for dy/dt = -r(1 — y/ T)(1 — ó/Ê)y.