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Furthermore, use your best value of R to estimate r.1980/. Discuss how that
relates to the actual figure.
˘
Exercise 2.6. The purpose of this exercise is to study the properties of the implicit
scheme for the logistic model. We recall that this scheme can be written in the form
r nC1 r n
t
r nC1
R
D ar nC1 .1
/:
For simplicity, we set a D R D 1; and get the scheme
r nC1 tr nC1 .1 r nC1 / D r n ;
(2.55)
so
1 C t C p .1 t/ 2 C 4 tr n
:
1
2t
r nC1 D
(2.56)
(a) Use (2.56) to show that if r 0
D 1; then r n
D 1,andifr 0
D 0; then r n
D 0 for
all n
>
0:
(b) Define
f.r/ D r t r.1 r/ r n
for a given value of n; and assume that 0<r n
<1:We wa n t t o fi n d r such
that
f.r / D 0;
and then set
r nC1 D r :
Show that
f.1/ > 0
and
f.r n /<0:
Use these two observations to conclude that f is zero for at least one value
between r n
and 1:
(c) Show that
f.0/ < 0
and that 25
f. 1 />0:
Use these two observations two conclude that f is zero for one value of r
between 1 and 0:
25 We use f.
1
/ as shorthand for the value of f.r/ as r tends to minus infinity.
 
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