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 ArzelaAscoli 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 ratiodependence predatorprey diffusive system proposed in [17].
Remark 3.4.
If we only consider the prey population in onepatch environment and ignore the dispersal process in the system (1.4), then the classical ratiodependence two species predatorprey model in particular of (1.4) with MichaelisMenten 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.