An Introduction to Mathematical Biology
Linda J. S. Allen
Answers to Selected Exercises and Supplementary Exercises
Chapter 1 Answers (* denotes supplementary exercises)
1 (b) first, nonlinear, autonomous
(d) second, nonlinear, nonautonomous
3 (b) xt = c1 + c2 (−1)t + c3 (−5)t
√ !t
2 3 πt πt
(d) xt = c1 sin + c2 cos
3 6 6
√ !t
2 3 √
πt
4 (a) Solution to 3 (d) xt = 3 sin
3 6
5 5 5
(b) Solution to 3 (b) xt = + (−1)t+1 + (−5)t
12 8 24
5 (b) xt = c1 2t + c2 (−2)t − 1 − 2t
(d)* Solve xt+1 − 5xt = 5t+1 . Solution: xt = c1 5t + t5t
(e)* Solve xt+1 − xt = 1 − 4t
1 1
7 (a) xt = √ λt+1
1 − √ λt+12 , where λ1 and λ2 are the roots of the characteristic equation,
5 5
λ1 > λ 2 .
xt+1
(b) Using the solution in (a) and the fact that λ1 > |λ2 | leads to lim = λ1 .
t→∞ xt
(c)* Find the number of pairs of rabbits after one year (t = 12); after 5 years. The
circumference of the earth is 24,902 miles. If the pairs of rabbits are lined end to end and
they measure one foot in length, then, after 5 years, the pairs of rabbits would encircle
the earth about 19,050 times.
8 Find the general solution, xt = c1 λt1 +c2 λt2 , where λ1 > |λ2 |. Then show xt+1 /xt approaches
λ1 as t → ∞.
0 1 0 0
0 0 1 0
10 (2) Y (t + 1) = BY (t), where B = . Show det(B − λI) = λ4 + aλ2 + b.
0 0 0 1
−b 0 −a 0
1
,
0 1 0
11 (b) Y (t + 1) = AY (t), A = 0 0 1
5 1 −5
14 ab < 1
1 3 2
15 (a) X(t) = c1 0 + c2 (−3)t 0 + c3 (2)t 1
0 −4 0
t
1 2 −1 1 1
16 At = , X(t) = c1 + c2 2t
0 2t 0 1
3a2 a3 a a2
17 (a) λ3 − λ− = 0, λ = a, ± . R0 = (a + 3).
4 4 2 4
*Show R0 > 1 iff a > 1.
(b) Apply Theorem 1.7.
19 (b) M2 is reducible and imprimitive.
20 (a) Apply Theorem 1.7.
(b) R0 = s1 b2 + s1 s2 b3 = 1 + 2s2 is never less than one and is greater than one when s2 > 0.
*Let b2 = 2, b3 = 4 and s2 = 2, then do part (b).
(c) R0 = 1 + f2 p1 .
21 Apply Theorem 1.5 or Theorem 1.7.
22 (b) 0 ≤ α < 1, 0 < α2 < 1, 0 < α3 ≤ 1, and γ, σ > 0.
(c) R0 = α1 σγ + α2 σ 2 γ(1 − α1 ) + α3 σ 3 (1 − α1 )(1 − α2 ).
23 (b) L6 > 0.
(c) In Example 1.21 when s1 is increased to one, λ ≈ 0.965. When p7 is increased to 0.95,
λ ≈ 1.002.
p p
1 − f + (1 − f )2 + 4γf 1 − f − (1 − f )2 + 4γf
25 (a) λ1 = > 0, λ2 = < 0.
2 2
R0 + M0
(c) lim Rt = .
t→∞ 1+f
26* Suppose A is an n × n matrix with a zero row or a zero column. Show that A is reducible.
27* Suppose A = (aij ) is an n × n irreducible matrix and A = D + B, where D is a diagonal
matrix whose diagonal entries are equal to those of A, D =diag(a11 , . . . , ann ). Show that B
is irreducible, i.e., the diagonal elements of an irreducible matrix do not affect irreducibility.
2
, 28* Suppose A = (aij ) is an n × n nonnegative, irreducible matrix with one positive row, i.e.,
aij > 0 for some i and j = 1, . . . , n. Show that A2 has at least two positive rows, and in
general Ak , k ≤ n has at least k positive rows. Conclude that A is primitive. (Hint: Use
the results in Exercises 25 and 26.)
3
Linda J. S. Allen
Answers to Selected Exercises and Supplementary Exercises
Chapter 1 Answers (* denotes supplementary exercises)
1 (b) first, nonlinear, autonomous
(d) second, nonlinear, nonautonomous
3 (b) xt = c1 + c2 (−1)t + c3 (−5)t
√ !t
2 3 πt πt
(d) xt = c1 sin + c2 cos
3 6 6
√ !t
2 3 √
πt
4 (a) Solution to 3 (d) xt = 3 sin
3 6
5 5 5
(b) Solution to 3 (b) xt = + (−1)t+1 + (−5)t
12 8 24
5 (b) xt = c1 2t + c2 (−2)t − 1 − 2t
(d)* Solve xt+1 − 5xt = 5t+1 . Solution: xt = c1 5t + t5t
(e)* Solve xt+1 − xt = 1 − 4t
1 1
7 (a) xt = √ λt+1
1 − √ λt+12 , where λ1 and λ2 are the roots of the characteristic equation,
5 5
λ1 > λ 2 .
xt+1
(b) Using the solution in (a) and the fact that λ1 > |λ2 | leads to lim = λ1 .
t→∞ xt
(c)* Find the number of pairs of rabbits after one year (t = 12); after 5 years. The
circumference of the earth is 24,902 miles. If the pairs of rabbits are lined end to end and
they measure one foot in length, then, after 5 years, the pairs of rabbits would encircle
the earth about 19,050 times.
8 Find the general solution, xt = c1 λt1 +c2 λt2 , where λ1 > |λ2 |. Then show xt+1 /xt approaches
λ1 as t → ∞.
0 1 0 0
0 0 1 0
10 (2) Y (t + 1) = BY (t), where B = . Show det(B − λI) = λ4 + aλ2 + b.
0 0 0 1
−b 0 −a 0
1
,
0 1 0
11 (b) Y (t + 1) = AY (t), A = 0 0 1
5 1 −5
14 ab < 1
1 3 2
15 (a) X(t) = c1 0 + c2 (−3)t 0 + c3 (2)t 1
0 −4 0
t
1 2 −1 1 1
16 At = , X(t) = c1 + c2 2t
0 2t 0 1
3a2 a3 a a2
17 (a) λ3 − λ− = 0, λ = a, ± . R0 = (a + 3).
4 4 2 4
*Show R0 > 1 iff a > 1.
(b) Apply Theorem 1.7.
19 (b) M2 is reducible and imprimitive.
20 (a) Apply Theorem 1.7.
(b) R0 = s1 b2 + s1 s2 b3 = 1 + 2s2 is never less than one and is greater than one when s2 > 0.
*Let b2 = 2, b3 = 4 and s2 = 2, then do part (b).
(c) R0 = 1 + f2 p1 .
21 Apply Theorem 1.5 or Theorem 1.7.
22 (b) 0 ≤ α < 1, 0 < α2 < 1, 0 < α3 ≤ 1, and γ, σ > 0.
(c) R0 = α1 σγ + α2 σ 2 γ(1 − α1 ) + α3 σ 3 (1 − α1 )(1 − α2 ).
23 (b) L6 > 0.
(c) In Example 1.21 when s1 is increased to one, λ ≈ 0.965. When p7 is increased to 0.95,
λ ≈ 1.002.
p p
1 − f + (1 − f )2 + 4γf 1 − f − (1 − f )2 + 4γf
25 (a) λ1 = > 0, λ2 = < 0.
2 2
R0 + M0
(c) lim Rt = .
t→∞ 1+f
26* Suppose A is an n × n matrix with a zero row or a zero column. Show that A is reducible.
27* Suppose A = (aij ) is an n × n irreducible matrix and A = D + B, where D is a diagonal
matrix whose diagonal entries are equal to those of A, D =diag(a11 , . . . , ann ). Show that B
is irreducible, i.e., the diagonal elements of an irreducible matrix do not affect irreducibility.
2
, 28* Suppose A = (aij ) is an n × n nonnegative, irreducible matrix with one positive row, i.e.,
aij > 0 for some i and j = 1, . . . , n. Show that A2 has at least two positive rows, and in
general Ak , k ≤ n has at least k positive rows. Conclude that A is primitive. (Hint: Use
the results in Exercises 25 and 26.)
3
