Question 7.3.3: Population Growth with a Critical Threshold Draw the directi...

The direction field is particularly easy to sketch here since the right-hand side depends on P, but not on t. If P(t) = 0, then P′(t) = 0, also, so that the direction field is horizontal. The same is true for P(t) = 1 and P(t) = 2. If 0 < P(t) < 1, then P′(t) < 0 and the solution decreases. If 1 < P(t) < 2, then P′(t) > 0 and the solution increases. Finally, if P(t) > 2, then P′(t) < 0 and the solution decreases. Putting all of these pieces together, we get the direction field seen in Figure 7.13. The constant solutions P(t) = 0, P(t) = 1 and P(t) = 2 are called equilibrium solutions. The solution P(t) = 1 is called an unstable equilibrium, since populations that start near 1 don’t remain close to 1. Similarly, the solutions P(t) = 0 and P(t) = 2 are called stable equilibria, since populations either rise to 2 or drop to 0 (extinction), depending on which side of the critical threshold P(t) = 1 they are on. (Look again at Figure 7.13.)