MTH 264
DELTA COLLEGE
SECTION 1.1 4
Consider the population model
dP
P
P
= 0.3 1 −
−1 P
dt
200
50
(a) For what values of P is the population in equilibrium?
(b) For what values of P is the population increasing?
(c) For what values of P is the population decreasing?
Solution:
There are four factors on the right-hand side of the equation and dP/dt can only equal zero if one of these
factors equals zero. Therefore:
(a) The population is in equilibrium when dP/dt = 0 which occurs when P = 0, P = 50, or P = 200.
Furthermore, the graph of the right-hand side of the equation is a cubic that we graph as a function of P
below right.
(b) The population is increasing when dP/dt > 0 which occurs when P < 0 or when 50 < P < 200.
(c) The population is decreasing when dP/dt < 0 which occurs when 0 < P < 50 or when 200 < P .