(Z, a) space:
Ci := {(Z, a) : Z = (1 - a (Z)) n - δZ = 0},
C2 :={(Z,a):a0(Z) → ±∞}, (11)
C3 := {(Z,a):a0(Z)=0}.
The steady-state line C1 is a downward-sloping, straight line in the (Z, a) space. It intersects
the vertical axis at point (0,1) and the horizontal axis at point (n∕δ, 0). Turn to C2. Setting
the denominator in (10) equal to zero, we obtain a vertical line at point (n∕δ, 0). The locus C3
is obtained by setting the numerator in (10) equal to zero. Solving for a gives the following
locus:
r(n- 1)
1-r(n- 1)
r(n- 1) ρ+2δ
Z------Z-----τr-------Z.
1-r(n- 1) n
(12)
Using 1 - r (n - 1) > 0, (12) shows that the straight line C3 has a positive slope and a negative
intercept on the vertical axis, as shown in Figs. 1 and 2. Moreover, the point of intersection
between the straight lines C2 and C3 , labelled E , is situated in the nonnegative region of
the (Z, a) plane:
(Z a )=μn r (p+δ)(n - 1) ʌ (13)
(ZE,aE) , [1 - r (n - 1)] δ) , (13)
which is called ‘a singular point’. Note, however, that since point E may be located below or
above the resource constraint (1), the value of aE may or may not be less than 1. Depending
on this value we can draw two diagrams such as in Figs.1 and 2. Moreover, it follows from (3)
that any strategy a (Z) above line C1 implies that Z declines in time, while any strategy a (Z)
below line C1 entails an increase of Z over time.
Collecting the arguments, we can illustrate an uncountable number of the hyperbolic curves
corresponding to the solutions satisfying the HJB equation (6) in Figs. 1 and 2. These figures
display representatives of those integral curves that are divided into five types of the families
of strategies. Arrows on the families of integral curves aj, j =1,...,4, and aL illustrate the
evolution of Z over time. In particular, by direct integration of (10) and manipulating we can