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where
t
n
D
nt
D
n
T
N
:
(c) Derive the following implicit scheme:
z
n
C
t.2t
nC1
t
nC1
/
z
nC1
D
:
1
t
(d) Set t
D
1=10 and compute, by hand, y
1
;y
2
;y
3
and
z
1
;
z
2
;
z
3
: Compare these
values with the analytical values given by r.t/;r.2t/,andr.3t/:
(e) Write a computer program for the implicit and the explicit schemes.
(f) Test the program for T
D
1: Evaluate the error for various values of t:
˘
Exercise 2.3.
Suppose that an isolated island is initially populated by r
0
D
1;000
rabbits. Assume an exponential growth model and suppose that the growth rate is
a
D
50:
(a) Estimate the number of rabbits at time t
D
1:
(b) Suppose the initial estimate is uncertain and that repeated counting shows that
the actual initial number is between 900 and 1;100: Provide an upper and lower
estimate for the number of rabbits at time t
D
1:
(c) Suppose also that the growth rate is uncertain but that we estimate it to be greater
than 40 and less than 60: Give a new upper and lower estimate.
˘
Exercise 2.4.
Suppose that an isolated island is initially populated by r
0
D
1;702
rabbits. Assume a logistic growth model, and suppose that the growth rate a
D
40
and that the carrying capacity is R
D
8;790:
(a) Estimate the number of rabbits at time t
D
1:
(b) Suppose the initial estimate is uncertain and that repeated counting shows that
the actual initial number is between 1;500 and 1;900: Provide an upper and
lower estimate for the number of rabbits at time t
D
1:
(c) Suppose also that the growth rate is uncertain but that we estimate it to be greater
than 30 and less than 50: Give a new upper an lower estimate.
(d) The carrying capacity is also uncertain but is in the interval ranging from 7;000
to 10;000: Give a new upper and lower estimate.
˘
Exercise 2.5.
Let us consider the population of the United States
22
between the
years 1800 and 1900. The annual growth rate can be estimated to be a
0:03:
22
This example is taken from a topic by R.B. Banks [5].
