9-4 Models for Population Growth 2.
(a) dP
dt = 0.0015P (1 − P 6000) (e) A = 6000 − 1000
1000 = 5, P (t) = 6000 1 + 5e−0.0015t 4.
(a) carrying capacity:684 (b) k = 1
18
39 − 18 2 − 0 = 1
18 21
2 = 7 12
(c) exponential model: P (t) = 18e127t, logisitc model:P (t) = 684 1 + 37e−127t, where A = 684 − 18
18 = 37 (e) P (7) = 684
1 + 37e−1277
≈ 684
1 + 37e−4.08 ≈403.397 ≈ 403 6.
(a) I guess the carrying capacity for the US population is 300 million.
A= 300 − 250
250 = 0.2 ⇒ P (t) = 300 1 + 0.2e−kt (b) P (10) = 275 ⇒ 275 = 300
1 + 0.2e−10k ⇒ 275 + 55e−10k = 300 ⇒ 55e−10k = 25 ⇒ 11e−10k = 5 ⇒ ln 11 − 10k = ln 5 ⇒ k = 1
10ln11 5 (c) P (110) = 300
1 + 0.2e−101 ln115110 = 300
1 + 0.2(115)11, P (210) = 300
1 + 0.2e−101 ln115210 = 300
1 + 0.2(115)21 8.
(a) A = 10000 − 400
400 = 24, P (t) = 10000
1 + 24e−kt, P(1) = 1200 ⇒ 1200 = 10000
1 + 24e−k ⇒k = ln36
11 ⇒P(t) = 10000 1 + 24e−ln3611t (b) 5000 = 10000
1 + 24e−(ln3611)t
⇒t= ln 24
ln3611 = ln 24 ln 36 − ln 11 1
10. Omitted.
12. Omitted.
14.
(a) dy
dt = ky1+c ⇒ y−1−cdy = kdt ⇒ 1
−cy−c = kt + a, y(0) = y0 ⇒ a = 1
−cy0c ⇒ y−c = −ckt + 1
y0c = −ckty0c+ 1
y+ 0t ⇒ yc = yc0
1 − cktyc0 ⇒ y = y0
(1 − ckty0c)1c
(b) 1 − ckT yc0 = 0 ⇒ cky0cT = 1 ⇒ T = 1 cky0c (c) c = 0.01, y0 = 2, y(3) = 16 ⇒ 16 = 2
(1 − 0.03k20.01)0.011 16. Omitted.
18.
(a) dP
dt = cP lnK
P ⇒ 1
P(ln K − ln P )dP = cdt ⇒ − ln(ln K − ln P ) = ct+ a ⇒ ln(lnK
P) = −ct + a ⇒ ln(K
P ) = Ae−ct, A = ea ⇒ K P = eAe−ct ⇒P(t) = Ke−Ae−ct
(b) lim
t→∞P(t) = K (c) Omitted.
(d) d2P
dt2 = cdP dt ln(K
P )−cPP
KKP−2dP
dt = cdP dt lnK
P −cdP
dt = c[dP dt ln(K
P )−
dP
dt] = 0 ⇒ dP dt[ln(K
P ) − 1] = 0 ⇒ c ln(K
P )P [ln(K
P ) − 1] = 0 ⇒ P = KorP = 0or lnK
P = 1 ⇒ P = K e 20.
(a) dP
dt = kP cos2(rt−φ) ⇒ 1
Pdp= k cos2(rt−φ)dt ⇒ ln P = k
Z 1 + cos(2rt − 2φ)
2 dt=
k 2t+k
2sin(2rt − 2φ) 1 2r = k
2t+ k
4rsin(2rt − 2φ)
2
