biologia - Studia

O. RABIEIMOTLAGH, Z. AFSHARNEZHAD
intersects z or w axis, so we conclude that the curve
~ γ ( t ) which starts from a point in
A , is contained in A and w (t)
is a decreasing map. Therefore next lemma is proven.
Lemma 6: Suppose that
~ γ ( t ) is defined on the interval [0,
T ) . Then for
all t ∈ [0,
T ) ,
~ γ ( t ) is bounded and contained in A .
Lemma 7: The compatible solution γ (t)
is defined for all t ≥ 0 and it is
positively bounded.
Proof: Suppose that maximal positive interval of existence for γ (t)
is
[0,
T ) , we makes a contradiction. If T < +∞ , then lim t→ +
T | γ ( t)
|
= +∞ , (Perko,
1991). From the lemma 6,
~ γ ( t ) is bounded and contained in A , so
lim t →
+
T | ( x(
t),
y(
t))
|
= +∞ . From the lemma 5, for all t ∈ [0,
T ) we have
+
x ( t),
y(
t)
> 0 . If x (t)
is unbounded then there exits a sequence 0 < t n
→ T
such that x ( t n
) → +∞ and x& ( ) ≥ 0 . But, from the first equation of the model
t n
(3) , x& ( t n
) → −∞ which is a contradiction. Hence x (t)
is bounded for t ∈ [0,
T ) .
+
If y (t)
is unbounded then there exits a sequence 0 < t n
→ T such that
y ( t n
) → +∞ and y& ( ) ≥ 0 . But, from the second equation of the model (3) ,
t n
y& ( t n
) → −∞ which is a contradiction. Hence for t ∈ [0,
T ) , y (t)
is bounded. This
shows that γ (t)
is bounded and therefore T = +∞ . This completes the proof.
Through next lemmas, we will see that graph of w = g(
z)
which is important
for determining behavior of z (t)
, controls the bifurcation parameters of theorem 2
i.e. Λ , ∆ and Ω . It divides A into three distinct region
−
A , down of the graph and G (g)
, the graph itself.
A
+
+
A , top of the graph,
−
= {( z,
w)
∈ A:
w > g(
z)},
A = {( z,
w)
∈ A:
w < g(
z)}.
Therefore, behavior of
~ γ ( t ) in A is determined by knowing its behavior
+ −
in A and A .
Remark: One more time, we must note that the phase space of (3) is a
four dimensional space, but here, we look at this space vertically so that we see two
dimensional plane of z and w . Therefore results and theorem for ω -limit sets of
flows of two dimensional vector fields are not valid for
~ γ ( t ) in A . Because of this
fact, we have to prove following lemmas.
Lemma 8: The curve
~ γ ( t ) which stars in
+
A intersects G (g)
or converges
to a intersection point of G (g)
and z axis.
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