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Øï y(t) = Ó0 K/ Ó0 = K •
Thus for each y0 > 0 the solution approaches the equilibrium solution y = ô2^) = K asymptotically (in fact, exponentially) as t ^ro. Therefore we say that the constant solution ô2(t) = K is an asymptotically stable solution of Eq. (7), or that the point y = K is an asymptotically stable equilibrium or critical point. This means that after a long time the population is close to the saturation level K regardless of the initial population size, as long as it is positive.
On the other hand, the situation for the equilibrium solution y = ô1 (t) = 0 is quite different. Even solutions that start very near zero grow as t increases and, as we have seen, approach K as t ^ro. We say that ô() = 0 is an unstable equilibrium solution or that y = 0 is an unstable equilibrium or critical point. This means that the only way to guarantee that the solution remains near zero is to make sure its initial value is exactly equal to zero.
Chapter 2. First Order Differential Equations
The logistic model has been applied to the natural growth of the halibut population in certain areas of the Pacific Ocean.6 Let y, measured in kilograms, be the total mass, or biomass, of the halibut population at time t. The parameters in the logistic equation are estimated to have the values r = 0.71/year and K = 80.5 x 106 kg. If the initial biomass is y0 = 0.25 K, find the biomass 2 years later. Also find the time ò for which ó(ò) = 0.75K.
It is convenient to scale the solution (11) to the carrying capacity K; thus we write Eq. (11) in the form
K (y0/K) + [1 - CVK)]e
Using the data given in the problem, we find that y(2) 0.25
K 0.25 + 0.75e-142 Consequently, y(2) = 46.7 x 106 kg.
FIGURE 2.5.5 y/ K versus t for population model of halibut in the Pacific Ocean.
To find ò we can first solve Eq. (12) for t. We obtain
(yjK)[1 - (y/ K)] (y/K)[1 - (y0/K)]
1, (y0/K)[1 - (y/K)]
t = — ln —
r (j/K)[1 - (y0/ K)]
Using the given values of r and y0/K and setting y/K = 0.75, we find that
0.71 (0.75)(0.75) 0.71
ln9 = 3.095 years.
A good source of information on the population dynamics and economics involved in making efficient use of a renewable resource, with particular emphasis on fisheries, is the book by Clark (see the references at the end of this chapter). The parameter values used here are given on page 53 of this book and were obtained as a result of a study by M. S. Mohring.
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
4 = - (' - J) Ó (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 ofEq. (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 ô() = 0 and ô2(´) = 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 ô() = 0 is an asymptotically stable equilibrium solution and ô2^) = 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 ofEq. (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
J = . TT ^ , (15)