Answer to Question #122579 in Differential Equations for JSE

Question

In one model of the changing population P(t) of a community, it is assumed that
dP/ dt = dB/ dt - dD/ dt, where dB/dt and dD/dt are the birth and death rates, respectively.

1. Solve for P(t) if dB/dt = k1P and dD/dt = k2P. (Assume P(0) = P0.)
P(t) =___

2. Analyze the cases k1 > k2, k1 = k2, and k1 < k2.
For k1 > k2,one has the following.

A. P → − ∞ as t → ∞
B. P → 0 as t → ∞
C. P → ∞ as t → ∞
D. P = 0 for every t
E. P = P0 for every t

3. For k1 = k2, one has the following.

A. P → − ∞ as t → ∞
B. P → 0 as t → ∞
C. P → ∞ as t → ∞
D. P = 0 for every t
E. P = P0 for every t

4. For k1 < k2, one has the following.

A. P → − ∞ as t → ∞
B. P → 0 as t → ∞
C. P → ∞ as t → ∞
D. P = 0 for every t
E. P = P0 for every t

Answer 1

As $\frac{dP}{dt}=k_1P-k_2P=(k_1-k_2)P$ then $\frac{dP}{P}=(k_1-k_2)dt$ hence $P=P_0e^{(k_1-k_2)t}$ where $P_0=P(0)$

2.if $k_1>k_2$ then as $t\to\infty$ , $P\to\infty$ (C)

3.if $k_1=k_2$ then $P=P_0$ for every $t$ (E)

4.if $k_1<k_2$ then $t\to\infty$ , $P\to 0$ (B)