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Solving Eq. (2) subject to the initial condition
7(0) = Óî, (3)
7 = 7oert. (4)
Thus the mathematical model consisting of the initial value problem (2), (3) with r > 0 predicts that the population will grow exponentially for all time, as shown in Figure 2.5.1. Under ideal conditions Eq. (4) has been observed to be reasonably accurate for many populations, at least for limited periods of time. However, it is clear that such ideal conditions cannot continue indefinitely; eventually, limitations on space, food supply, or other resources will reduce the growth rate and bring an end to uninhibited exponential growth.
FIGURE 2.5.1 Exponential growth: y versus t for dy/dt = ry.
Logistic Growth. To take account of the fact that the growth rate actually depends on the population, we replace the constant r in Eq. (2) by a function h(y) and thereby obtain the modified equation
dy/dt = h (y) y. (5)
4It was apparently the British economist Thomas Malthus (1766-1834) who first observed that many biological populations increase at a rate proportional to the population. His first paper on populations appeared in 1798.
Chapter 2. First Order Differential Equations
We now want to choose h(y) so that h(y) = r > 0 when y is small, h(y) decreases as y grows larger, and h(y) < 0 when y is sufficiently large. The simplest function having these properties is h(y) = r — ay, where a is also a positive constant. Using this function in Eq. (5), we obtain
dy/dt = (r — ay)y. (6)
Equation (6) is known as the Verhulst5 equation or the logistic equation. It is often convenient to write the logistic equation in the equivalent form
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 ofEq. (7) must satisfy the algebraic equation
r (1 — y/ K )y = 0.
Thus the constant solutions are y = ô() = 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
×\\\\×\\\-÷×××\\××\\×\\\\× -1 -AWWWWWWWWWWW
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