Calculus: Differential Population Question

Question

A population is modeled by the differential equation

dP / dt = 1.2P (1 − P /5000)

(a) For what values of P is the population increasing and for what values of P is the population decreasing?

(b) If the initial population is 5500, what is the limiting population?

(c) What are the equilibrium solutions?

Answer 1

dP/dT is increasing for P in (0, 5000) and decreasing for P in (5000, ∞) and P in (-∞, 0). Note that if initial P is > 5000, P decreases to 5000, and if P is in (0, 5000), it increases to 5000 (this is the logistic function that we normally associate with this derivative). If P is less than 0, it decreases to -∞, but that does not really make sense when we think of populations.

b. The limiting population is 5000.

c) There are two equilibria, 0 and 5000. The 5000 equilibrium is stable (a perturbation eventually returns to 5000, as the derivative is negative above and positive below.

0 is an unstable equilibrium; a small perturbation above leads P to go to 5000. a perturbation below leads P to go to -∞.