differential equation

Question

Suppose that 120 deer are placed in a protected region. If the region can support a maximum deer population of 500, then the bounded growth model for the deer population gives us which differential equation?

a. dP(t)/dt= k(500-P(t))

b. dP(t)/dt=k(P(t)-120)

c. dP(t)/dt=380P(t)
d. dP(t)/dt=500-k(P(t))
e. dP(t)/dt=P(t)-380k

Answer 1

When P(t) < 500, we must have dP(t)/dt > 0 and approaching 0 as P(t) approaches 500.

 

This happens to choice "a", dP(t)/dt= k(500-P(t)) only. You can see it as 500-P(t) → 0⁺ as P(t) → 500 and multiplication of that difference by a constant k won't change this limit.

 

Be careful, choice "d" is a trap answer as it only is valid for k=1 but we want a choice that is valid for all k>0.