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2.5 Autonomous Equations and Population Dynamics
The graphs of y/ K versus t for the given parameter values and for several initial conditions are shown in Figure 2.5.5.
A Critical Threshold. We now turn to a consideration of the equation
t=— ('—T) y (14)
where r and T are given positive constants. Observe that (except for replacing the parameter K by T ) this equation differs from the logistic equation (7) only in the presence of the minus sign on the right side. However, as we will see, the solutions of Eq. (14) behave very differently from those of Eq. (7).
For Eq. (14) the graph of f(y) versus y is the parabola shown in Figure 2.5.6. The intercepts on the y-axis are the critical points y = 0 and y = T, corresponding to the equilibrium solutions 01(t) = 0 and 02(t) = T. If 0 < y < T, then dy/ dt < 0, and y decreases as t increases. On the other hand, if y > T, then dy/dt > 0, and y grows as t increases. Thus &1(t) = 0 is an asymptotically stable equilibrium solution and 02(t) = T is an unstable one. Further, f'(y) is negative for 0 < y < T/2 and positive for T/2 < y < T, so the graph of y versus t is concave up and concave down, respectively, in these intervals. Also, f (y) is positive for y > T, so the graph of y versus t is also concave up there. By making use of all of the information that we have obtained from Figure 2.5.6, we conclude that graphs of solutions of Eq. (14) for different values of y0 must have the qualitative appearance shown in Figure 2.5.7. From this figure it is clear that as time increases, y either approaches zero or grows without bound, depending on whether the initial value y0 is less than or greater than T. Thus T is a threshold level, below which growth does not occur.
We can confirm the conclusions that we have reached through geometric reasoning by solving the differential equation (14). This can be done by separating the variables and integrating, just as we did for Eq. (7). However, if we note that Eq. (14) can be obtained from Eq. (7) by replacing K by T and r by —r, then we can make the same substitutions in the solution (11) and thereby obtain
y = . TT , (15)
y0 + (T — y0)ert
which is the solution of Eq. (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
fo(t) = T
FIGURE 2.5.7 y versus t for dy/dt = —r(1 — y/ T)y.
If yo < 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!