The fundamental theorem in this paper is stated as follows about the existence of an
-periodic solution.
Theorem 3.1.
Suppose that (H) holds. Furthermore assume the following:
-
(i)
-
(ii)
,
-
(iii)
,
-
(iv)
,
then the system (1.4) has at least one
-periodic solution.
Proof.
Consider vector equation
Define
where
is the Euclidean norm. Then
and
are both Banach spaces with the above norm
. Let
. Then
and
. Since
is closed in
, then
is a Fredholm mapping of index zero. It is easy to show that
are continuous projectors such that
. Furthermore, the generalized inverse (to
)
exists and is given by
, thus
Obviously,
are continuous. Since
is a Banach space, using the Arzela-Ascoli theorem, it is easy to show that
is compact for any open bounded set
. Moreover,
is bounded, thus,
is
-compact on
for any open bounded set
. Corresponding to the operator equation
, we have
Suppose that
is a solution of (3.5) for certain
. Integrating on both sides of (3.5) from
to
with respect to
, we have
It follows from (3.5) to (3.9) that
Multiplying (3.6) by
and integrating over
gives
which yields
By using the inequality
, we have
Then
By using the inequality
, we derive from (3.17) that
Similarly, multiplying (3.7) by
and integrating over
, then synthesize the above, we obtain
It follows from (3.18) and (3.19) that
so, there exists a positive constant
such that
which together with (3.19), there also exists a positive constant
such that
This, together with (3.11), (3.12), and (3.21), leads to
Since
, there exist some points
, such that
It follows from (3.21) and (3.22) that
From (3.8) and (3.9), we obtain that
This, together with (3.12), (3.13), and (3.26), deduces
From (3.6) and (3.24), we have
From (3.7) and (3.24), it yields that
Noticing that
, from (3.8) and (3.9), deduces
There exist two points
such that
Hence,
where
. Then, this, together with (3.12), (3.13), (3.23), (3.28), (3.29), and (3.32), deduces
It follows from (3.27) to (3.33) that
From (3.34), we clearly know that
are independent of
, and from the representation of
, it is easy to know that there exist points
such that
, where
Take
, where
is taken sufficiently large such that
, and such that each solution
of the system
satisfies
if the system (3.35) has solutions. Now take
. Then it is clear that
verifies the requirement (a) of Lemma 2.11.
When
,
is a constant vector in
with
, from the definition of
, we can naturally derive
whether the system (3.35) has solutions or not. This shows that the condition (b) of Lemma 2.11 is satisfied.
Finally, we will prove that the condition (c) of Lemma 2.11 is valid. Define the homotopy
by
where
where
is a parameter. From (3.37), it is easy to show that
. Moreover, one can easily show that the algebraic equation
has a unique positive solution
in
. Note that
(identical mapping), since
, according to the invariance property of homotopy, direct calculation produces
where
is the Brouwer degree. By now we have proved that
verifies all requirements of Lemma 2.11. Therefore, (1.4) has at least one
-periodic solution in
. The proof is complete.
Corollary 3.2.
If the conditions in Theorem 3.1 hold, then both the corresponding continuous model (1.6) and the discrete model (1.7) have at least one
-periodic solution.
Remark 3.3.
If
and
in (1.6), then the system (1.6) reduces to the continuous ratio-dependence predator-prey diffusive system proposed in [17].
Remark 3.4.
If we only consider the prey population in one-patch environment and ignore the dispersal process in the system (1.4), then the classical ratio-dependence two species predator-prey model in particular of (1.4) with Michaelis-Menten functional response and time delay on time scales
where
are positive
-periodic functions,
is nonnegative constant. It is easy to obtain the corresponding conclusions on time scales for the system (3.40).
Corollary 3.5.
Suppose that (i)
, (ii)
hold, then (3.40) has at least one
-periodic solution.
Remark 3.6.
The result in Corollary 3.5 is same as those for the corresponding continuous and discrete systems.