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where K = r/a. The constant r is called the intrinsic growth rate, that is, the growth rate in the absence of any limiting factors. The interpretation of K will become clear shortly.
We first seek solutions of Eq. (7) of the simplest possible type, that is, constant functions. For such a solution dy/dt = 0 for all t, so any constant solution of Eq. (7) must satisfy the algebraic equation
r (1 — y/ K )y = 0.
Thus the constant solutions are y = 01(t) = 0 and y = 02(t) = K. These solutions are called equilibrium solutions of Eq. (7) because they correspond to no change or variation in the value of y as t increases. In the same way, any equilibrium solutions of
FIGURE 2.5.2 Direction field for dy/dt = r (1 — y/K ) y with r = 1/2 and K = 3.
5P. F. Verhulst (1804-1849) was a Belgian mathematician who introduced Eq. (6) as a model for human population growth in 1838. He referred to it as logistic growth; hence Eq. (6) is often called the logistic equation. He was unable to test the accuracy of his model because of inadequate census data, and it did not receive much attention until many years later. Reasonable agreement with experimental data was demonstrated by R. Pearl (1930) for Drosophila melanogaster (fruit fly) populations, and by G. F. Gause (1935) for Paramecium and Tribolium (flour beetle) populations.
2.5 Autonomous Equations and Population Dynamics
the more general Eq. (1) can be found by locating the roots of f (y) = 0. The zeros of f (y) are also called critical points.
To visualize other solutions of Eq. (7), let us draw a direction field for a typical case, as shown in Figure 2.5.2 when r = 1/2 and K = 3. Observe that the elements of the direction field have the same slope at all points on each particular horizontal line although the slope changes from one line to another. This property follows from the fact that the right side of the logistic equation does not depend on the independent variable t. Observe also that the equilibrium solutions y = 0 and y = 3 seem to be of particular importance. Other solutions appear to approach the solution y = 3 asymptotically as t ^<x>, whereas solutions on either side of y = 0 diverge from it.
To understand more clearly the nature of the solutions of Eq. (7) and to sketch their graphs quickly, we can start by drawing the graph of f (y) versus y. In the case of Eq. (7), f (y) = r(1 — y/K)y, so the graph is the parabola shown in Figure 2.5.3. The intercepts are (0, 0) and (K, 0), corresponding to the critical points of Eq. (7), and the vertex of the parabola is (K/2, rK/4). Observe that dy/dt > 0 for 0 < y < K; therefore, y is an increasing function of t when y is in this interval; this is indicated by the rightward-pointing arrows near the y-axis in Figure 2.5.3, or by the upward-pointing arrows in Figure 2.5.4. Similarly, if y > K, then dy/dt < 0; hence y is decreasing, as indicated by the leftward-pointing arrow in Figure 2.5.3, or by the downward-pointing
FIGURE 2.5.4 Logistic growth: y versus t for dy/dt = r(1 — y/K)y.
Chapter 2. First Order Differential Equations
arrow in Figure 2.5.4. Further, note that if y is near zero or K, then the slope f (y) is near zero, so the solution curves are relatively flat. They become steeper as the value of y leaves the neighborhood of zero or K. These observations mean that the graphs of solutions of Eq. (7) must have the general shape shown in Figure 2.5.4, regardless of the values of r and K.
To carry the investigation one step further, we can determine the concavity of the solution curves and the location of inflection points by finding d2y/dt2. From the differential equation (1) we obtain (using the chain rule)
d2 y dy
dy = f(y) dy = f(y) f (y)- (8)
The graph of y versus t is concave up when y" > 0, that is, when f and f' have the same sign. Similarly, it is concave down when y" < 0, which occurs when f and f' have opposite signs. The signs of f and f' can be easily identified from the graph of f (y) versus y. Inflection points may occur when f'(y) = 0.
In the case of Eq. (7) solutions are concave up for 0 < y < K/2 where f is positive and increasing (see Figure 2.5.3), so that both f and f' are positive. Solutions are also concave up for y > K where f is negative and decreasing (both f and f' are negative). For K/2 < y < K solutions are concave down since here f is positive and decreasing, so f is positive but f' is negative. There is an inflection point whenever the graph of y versus t crosses the line y = K/2. The graphs in Figure 2.5.4 exhibit these properties.
Finally, recall that Theorem 2.4.2, the fundamental existence and uniqueness theorem, guarantees that two different solutions never pass through the same point. Hence, while solutions approach the equilibrium solution y = K as t ^ro, they do not attain this value at any finite time. Since K is the upper bound that is approached, but not exceeded, by growing populations starting below this value, it is natural to refer to K as the saturation level, or the environmental carrying capacity, for the given species.