Answers to Selected Exercises and Supplementary Exercises

An Introduction to Mathematical Biology Linda J. S. Allen Answers to Selected Exercises and Supplementary Exercises Chapter 1 Answers(* denotes supplementary exercises)

• (b) rst, nonlinear, autonomous

(d) second, nonlinear, nonautonomous

• (b)xt=c1+c2(1)

t +c3(5) t (d)xt= 2 p 3 3 !t c1sin 1 t 6

+c2cos 1 t 6

• (a) Solution to 3 (d)xt=

2 p 3 3 !t p

• sin

1 t 6

(b) Solution to 3 (b)xt= 5 12 + 5 8 (1) t+1 + 5 24 (5) t

• (b)xt=c12

t +c2(2) t 12t (d)* Solvext+15xt= 5 t+1 :Solution:xt=c15 t +t5 t (e)* Solvext+1xt= 14t

• (a)xt=

1 p 5

t+1 1 1 p 5

t+12, where1and2are the roots of the characteristic equation, 1> 2.(b) Using the solution in (a) and the fact that1>j2jleads to lim t!1 xt+1 xt

=1:

(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.

• Find the general solution,xt=c1

t 1+c2 t 2, where1>j2j. Then showxt+1=xtapproaches 1ast! 1.10 (2)Y(t+ 1) =BY(t);whereB=

B B B B B @

0 1 0 0

0 0 1 0

0 0 0 1

b0a0 1 C C C C C A

:Show det(B I) =

4 +a 2 +b.1 An Introduction to Mathematical Biology 1e Linda Allen (Solutions Manual All Chapters, 100% Original Verified, A+ Grade) (Answers to Selected Exercises and Supplementary Exercises) 1 / 3

11 (b)Y(t+ 1) =AY(t),A=

B B B @

• 1 0
• 0 1
• 15

1 C C C A 14ab <1 15 (a)X(t) =c1

B B B @ 1

1 C C C A +c2(3) t

B B B @ 3

4 1 C C C A +c3(2) t

B B B @ 2 1

1 C C C A 16A t =

@

• 2

t 1

• 2

t 1 A,X(t) =c1

@ 1

1 A+c22 t

@ 1 1 1 A 17 (a) 3

3a 2 4

a 3 4 = 0,=a; a 2 .R0= a 2 4 (a+ 3).*ShowR0>1 ia >1.(b) Apply Theorem 1.7.19 (b)M2is reducible and imprimitive.20 (a) Apply Theorem 1.7.(b)R0=s1b2+s1s2b3= 1+2s2is never less than one and is greater than one whens2>0: *Letb2= 2; b3= 4 ands2= 2, then do part (b).(c)R0= 1 +f2p1.21 Apply Theorem 1.5 or Theorem 1.7.22 (b) 0 <1, 0< 2<1, 0< 31, and; >0.(c)R0=1+2 2

(11) +3

3

(11)(12):

23 (b)L 6 >0.(c) In Example 1.21 whens1is increased to one,0:965. Whenp7is increased to 0.95,

1:002.

25 (a)1= 1f+ p (1f) 2

• 4f

2 >0,2= 1f p (1f) 2

• 4f

2

<0:

(c) lim t!1 Rt= R0+M0

• +f

.26* SupposeAis annnmatrix with a zero row or a zero column. Show thatAis reducible.27* SupposeA= (a ij) is annnirreducible matrix andA=D+B, whereDis a diagonal matrix whose diagonal entries are equal to those ofA,D=diag(a 11; : : : ; ann). Show thatB is irreducible, i.e., the diagonal elements of an irreducible matrix do not aect irreducibility.

2 2 / 3

28* SupposeA= (aij) is annnnonnegative, irreducible matrix with one positive row, i.e., aij>0 for someiandj= 1; : : : ; n. Show thatA 2 has at least two positive rows, and in generalA k

,knhas at leastkpositive rows. Conclude thatAis primitive. (Hint:Use

the results in Exercises 25 and 26.)

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