Question #122579
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
1
Expert's answer
2020-06-28T18:27:12-0400

As dPdt=k1Pk2P=(k1k2)P\frac{dP}{dt}=k_1P-k_2P=(k_1-k_2)P then dPP=(k1k2)dt\frac{dP}{P}=(k_1-k_2)dt hence P=P0e(k1k2)tP=P_0e^{(k_1-k_2)t} where P0=P(0)P_0=P(0)

2.if k1>k2k_1>k_2 then as tt\to\infty , PP\to\infty (C)

3.if k1=k2k_1=k_2 then P=P0P=P_0 for every tt (E)

4.if k1<k2k_1<k_2 then tt\to\infty , P0P\to 0 (B)


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