Answer:
a) If m < P < M , then all factors > 0, P is increasing
b) If 0 < P < m , then all factors < 0, P is decreasing
Step-by-step explanation:
Given:
- the modified Logistics Equation is:
dP/dt = kP(1 - P/M)*(1-m/P)
Find:
Use the differential equation to show that any solution is increasing if m < P < M and decreasing if 0 < P < m
Solution:
- If m < P < M, then:
P/M < 1, then (1 - P/M) > 0
similarly m/P < 1, then (1-m/P) > 0
- Since all factors are positive then dP/dt > 0 , so P is increasing.
- If 0 < P < m, then:
m/P > 1, then (1 - P/M) < 0
similarly P is still < M , so
- Since all factors are positive then dP/dt < 0 , so P is decreasing.
