biologia - Studia

O. RABIEIMOTLAGH, Z. AFSHARNEZHAD
Proof: Suppose that t ≥ 0
0 and ~
−
γ (t
0)∈ A . From lemma 5 and lemma 7,
for all t ≥ 0 we have w& < 0 . Therefore,
~ γ ( t ) dose not intersects G (g)
for t>0.
This means that γ ~ is contained in
76
−
A for all > t0
t , hence z (t)
and w (t)
are
decreasing for t > t0
. Suppose that lim ( z ( t ), w ( t )) = ( z 1,
w
t → +∞
1)
. Same argue as
lemma 8 shows that
z h z , w ) = 0, − α z = 0.
1
(
1 1
d 1w1
( z , w ) = 1 1
If z 0 then we have h 0 and w = 0 which means that
1 ≠
( z , w 1 1)
is a intersection point of G (g ) and z axis. If =
−
z
1
0 then (0, w
1
)∈ A
is a point on w axis with w ≤ g(0)
. This completes the proof.
Lemma 8 and 9 show that any curve
~ γ ( t ) of a compatible solution γ (t)
which
stars in A , converges to a intersection point of G (g)
and z axis or converges to a point
on w axis with w ≤ g(0)
. The case of convergence to a intersection point of G (g)
and z axis is when the DNA damage signal is damped to zero. Alternatively, If
~ γ ( t )
converges to a point on the w axis then it is the cause that the DNA damage signal
contains a permanent damage. Therefore the bifurcation of the DNA damage signal is
controlled by G (g)
as
~ γ ( t ) is entrapped by G (g)
in
±
A and leaded to one of the cases.
The map w = g(
z)
has two critical point z µ + k − k ( µ + k + k )
1
1
=
1s
1s
1s
2s
and z µ + k + k ( µ + k + k ) which are minimum and maximum point
2
=
1s
1s
1s
2s
respectively. It is easy to see that z
1
< µ < z2
and g ( z1)
> g(
z2)
, furthermore,
g (z) is increasing for z1 ≤ z ≤ z2
and decreasing for z ∉ [ z 1,
z 2]
. Hence, if z ≤ 1
0
then g (z)
has no critical point in A and intersection of G (g)
and A is a increasing
curve, but if 0 < z
1
< µ , then z
1
is the only critical point of g (z)
in A . In the later
case, if g (0), g(
z1)
< 0 then intersection of G (g)
and A is an increasing curves
which enters into A from a point on w axis, but if g ( z 1
) < 0 and g (0) > 0 then
intersection of G (g)
and A enters into A from a point on z axis and it contains two
curves which are decreasing and increasing respectively. These show that the way
which G (g)
intersects A is controlled by sing of z
1
, g ( z 1
) and g (0) . These make
six cases ( a ),( b),...,(
f ) which are listed in table A and showed in figure 5. It is easy
to see that z ≤ 1
0 if and only if Λ < 0 and g (0) ≤ 0 if and only if ∆ ≤ 0 ,
furthermore g ( z 1
) = Ω , where Λ , ∆ and Ω are bifurcation parameters introduced
by the theorem 2. Now, the lemmas 4 to 9 show that behavior of γ ~ in each case of
table A is as it is shown in figure 6.