where linear dependence on z is assumed while r
1
, r
2
and c
l
are positive constants. The system is thus considered to be a self-regulative one in a normal
subject, where basal plasma glucose level is pre- dominantly determined by insulin secretion from
normal b-cells. A decreased number of cells would maintain a normal basal plasma insulin concen-
tration by operating at an increased secretory rate per cell. This is reflected by the second term in Eq.
2, which becomes smaller in magnitude as z decreases so that x will be reduced at a slower rate
in the event that the number of cells becomes smaller.
As for the plasma glucose level y, if there is a decrease in insulin secretion due to a reduction to
1N of the normal number, n, of b-cells, the basal plasma glucose will increase until nearly normal
basal insulin levels are obtained Bajaj and Rao, 1987. Thus, the plasma glucose level is a function
of the b-cell capacity, Nn. Therefore, plasma glucose is assumed to satisfy the following rate
equation.
dy dt
= R
3
N z
− R
4
x + C
2
+ wt
3 where the first term accounts for the hyperbolic
increase in y due to a reduction in the number of b-cells, supported by the study of Turner et al.,
1979 mentioned previously. The second term accounts for the reduction in y due to the regula-
tive effect of plasma insulin, while C
2
accounts for the change in y in the absence of x and z. The last
term in Eq. 3 corresponds to the variation in the gastrointestinal absorption with time, accounting
for the absorption by the intestine and the subse- quent release of glucose into the bloodstream.
Finally, the density of pancreatic b-cells in the proliferative phase satisfies the following equation
used in the work of Rao et al. 1990: dz
dt =
R
5
y − yˆT − z + R
6
zT − z − R
7
z 4
where yˆ is the difference between the normal glucose level and its basal concentration, T is the
total density of dividing and non-dividing b-cells, while R
5
, R
6
, and R
7
are rate constants. The first term accounts for the increase in z due to the
interaction between the plasma glucose above the basal level and the non-dividing b-cells, while the
second term represents the increase in the dividing b-cells from the interaction between the dividing
and non-dividing cells. The last term is the rate of reduction in the cells density, proportional to its
current level with a rate constant R
7
. Thus, our reference model consists of Eqs. 2 –
4.
3. Single oral glucose administration
When a single oral glucose administration is utilized, a gastrointestinal absorption term wt is
assumed to have the form wt
1 p + e
a t
5 which means that the absorption by the intestine
and the subsequent release of glucose into the bloodstream takes place at an initial rate
w =
1 p + 1
6 and then falls off exponentially with time Rao et
al., 1990. On substituting Eq. 5 into Eq. 3 and differ-
entiating wt, we can eliminate the time depen- dence of the term wt by considering instead the
following system of four autonomous differential equations
dx dt
= zr
1
y − r
2
x + c
1
, x0 = x
7 dy
dt =
R
3
N z
− R
4
x + C
2
+ w,
y0 = y 8
dx dt
= R
5
y − yˆT − z + R
6
zT − z − R
7
z, z0 = z
9 dw
dt = −
a w + apw
2
w0 = w 10
4. Singular perturbation analysis
At this point, we first observe that, the con- stants a and p determine the rapidity of glucose
absorption. Thus, if a is high, the gastrointestinal absorption process equilibrates very quickly to the
steady state w = 0 of Eq. 10, which is stable. This means that as time passes w becomes infi-
nitesimally small and the system model reduces to Eqs. 7 – 9 with w = 0.
In order to analyze the resulting model equations Eqs. 7 – 9 with w = 0 by a singular
perturbation technique, we scale the dynamics of the three remaining hierarchical components of
the system, namely x, y and z, by means of two small dimensionless positive parameters o and
d
. Letting
zˆ = T, r
3
= R
3
No, r
4
= R
4
o ,
r
5
= R
5
od , r
6
= R
6
od , r
7
= R
7
od , and c
2
= C
2
o , we are
led to the following system of differential equa- tions.
dx dt
= zr
1
y − r
2
x + c
1
fx, y, z 11
dy dt
= o
r
3
z −
r
4
x + c
2
n
o gx, y, z
12 dz
dt =
r
5
y − yˆzˆ − z + r
6
zzˆ − z − r
7
z od
hx, y, z 13
which means that during transitions, when the right sides of Eqs. 11 – 13 are finite but differ-
ent from zero, y; is of the order o, and z; is of
order od. Thus, we have assumed that insulin formation has faster time response than that of
plasma glucose, and the b-cells proliferation pos- sesses the slowest dynamics.
It is well known Muratori, 1991; Muratori and Rinaldi, 1992; Lenbury et al., 1996 that the sys-
tem Eqs. 11 – 13, when o and d are small, can be
analyzed with
the singular
perturbation method which, under suitable regularity condi-
tions, allows approximation of the solution of the system by a sequence of simple dynamic transi-
tions occurring at different speeds. The resulting curve, composed of these transitions, approxi-
mates the actual solution in the sense that the real trajectory is contained in a tube around the curve,
and that the radius of the tube goes to zero with o
, and d.
4
.
1
. On the manifold f = This consists of the trivial manifold z = 0 and
the nontrivial one given by the equation x =
r
1
r
2
y + c
1
r
2
14 The above equation describes a plane in the x,
y, z-space which is parallel to the z-axis. It intersects the x, z-plane along the line
x = c
1
r
2
x
1
15 as shown in Fig. 1.
4
.
2
. On the manifold h = This is the surface
y = pz
yˆ − r
6
z r
5
+ r
7
z r
5
zˆ − z 16
for which p
z = − r
6
r
5
+ r
7
zˆ r
5
zˆ − z
2
17
Fig. 1. Shapes and relative positions of the equilibrium mani- folds in the case where a limit cycle exists. Here, three arrows
indicate fast transitions, two arrows indicate transitions at intermediate speed, and a single arrow indicates slow transi-
tions.
Setting pz = 0, one obtains z = zˆ 9
r
7
zˆ r
6
Since it is necessary that z 5 zˆ, we consider only the critical point where
z = zˆ − r
7
zˆ r
6
z
1
5 zˆ
18 Moreover, since
p¦z= 2r
7
zˆ r
5
zˆ − z
3
19 we have
p¦z
1
= 2r
6
r
5
r
6
r
7
zˆ \
for positive parametric values. Thus, z is a mini- mum at the point z = z
1
, on this surface. Substi- tuting z
1
into Eq. 16, one obtains p
z
1
= yˆ − 1
r
5
r
7
− r
6
zˆ
2
y
1
20 Also, if z = 0 on this surface, then
z = zˆ which is where the manifold intersects the x,
y-plane, as shown in Fig. 1. We observe further that the slow manifold
given by Eq. 16 is parallel to the x-axis and intersects the y, z-plane along a curve on which
y becomes a minimum at the point where z = z
1
. Moreover, the manifold h = 0 intersects the
plane of the manifold f = 0 along the parabolic curve given by the equation
x = cy, z 1
r
2
[r
1
y + c
1
+ r
7
z − r
5
y − yˆzˆ − z −
r
6
zzˆ − z] 21
Substituting y
1
and z
1
, in the above equation, we obtain
x = 1
r
2
r
1
yˆ − r
1
r
7
r
5
+ 2r
1
r
5
r
6
r
7
zˆ − r
1
r
6
zˆ r
5
+ c
1
n
x
2
22 also shown in Fig. 1.
On the other hand, on substituting y = yˆ and z = 0 in Eq. 21, we obtain
x = 1
r
2
[r
1
yˆ + c
1
]
x
3
23 However, on considering Eqs. 22 and 23, we
find that x
3
\ x
2
for all positive parametric values. Now, for a constant y, c is a decreasing func-
tion of z on the interval 0, z
1
, that is, x decreases along the curve given by Eq. 21 from the point
x, y, z = x
3
, yˆ, 0, or point P in Fig. 1, to the point x, y, z = x
2
, y
1
, z
1
, or point D, along the curve PD in Fig. 1.
4
.
3
. On the manifold g = This is the cylindrical surface given by
x = fz r
3
r
4
z +
c
2
r
4
24 which is parallel to the y-axis and intersects the
x, z-plane along a hyperbola. As z tends toward infinity along this surface, x tends toward the
value c
2
r
4
. Since fz is a monotonically decreasing func-
tion of z, if z
1
\ 0 then x is decreasing for z in the
interval 0, z
1
. Substituting z = z
1
into Eq. 24, we obtain
x = r
3
r
4
z
1
+ c
2
r
4
x
4
25 Thus, by requiring that
0 B r
3
+ c
2
z
1
r
4
z
1
B r
1
r
5
yˆ − r
1
r
7
+ 2r
1
r
6
r
7
zˆ − r
1
r
6
zˆ + c
1
r
5
r
2
r
5
26 we are assured that 0 B x
4
B x
2
and the surface f = g = 0 intersects the curve f = h = 0 at the point
S which is located on the unstable portion PD of the curve. In other words, the manifold g = 0
separates the two stable submanifolds z = 0 and f = h = 0, on the former of which g \ 0 while
g B 0 on the latter. The stability of the manifolds is determined by the signs of f, g, and h. If we
further require that
yˆ \ r
7
− r
6
zˆ
2
r
5
27 and
zˆ \ r
7
r
6
28 then, on considering Eqs. 18 and 20, we are
assured that y
1
\ 0 and z
1
\ 0.
Under these conditions, the shapes and relative positions of the equilibrium manifolds f = 0, g =
0, and h = 0 will be as depicted in Fig. 1, where three arrows indicate fast transitions, two arrows
indicate intermediate ones, while a single arrow indicates slow ones.
Thus, a system initially at a generic point, say point A of Fig. 1, will make a quick transition to
the plane given by Eq. 14 on the fast manifold f = 0 point B in Fig. 1, since the signs of f assure
us that this part of the manifold f = 0 is stable. As the plane where f = 0 is approached, y has slowly
become active. A transition at an intermediate speed is made along f = 0 in the direction of
decreasing y, since g B 0 here, to the point C on a stable part of the curve f = h = 0. From the point
C, a slow transition is then made along this curve in the direction of decreasing y, since g is still
negative here.
Once the point D is reached, the stability of the manifold is lost, a catastrophic transition brings
the system to point E on the manifold z = 0 where it intersects the plane given by Eq. 14. Conse-
quently, the system will slowly develop along this line in the direction of increasing y, since g is now
positive.
At a point F on the plane z = 0, the stability will again be lost and a quick transition will bring
the system back to the point G on the stable branch of the curve f = h = 0, before repeating the
same previously described path, thereby forming a closed cycle GDEFG. Thus, the existence of a
limit cycle in the system for o and d sufficiently small is assured. The exact solution trajectory of
the system will be contained in a tube about this closed curve, the radius of which tends to zero
with o and d.
The above arguments can be summarized by the following theorem.
Theorem 1. If inequalities Eqs. 26 – 28 hold,
and o and d are sufficiently small, then the system of Eqs. 11 – 13 has a global attractor, in the
positi6e octant of the phase space, which is a limit cycle composed of a concatenation of catastrophic
transitions occurring at different speeds. A computer simulation of Eqs. 11 – 13 is
presented in Fig. 2a, with parametric values cho- sen to satisfy inequalities Eqs. 26 – 28 iden-
tified in
the above
theorem. The
solution trajectory, projected onto the y, x-plane, is seen
here to tend towards a limit cycle as theoretically predicted. The corresponding time series of insulin
x and glucose y are shown in Fig. 2b.
5. Ambulatory-fed conditions and chaotic behavior
