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Based on the classic Lotlk-Volterra cooperation model, we establish a time-delay model of which a species cannot survive independently. By continuation theorem, we discuss existence of positive periodic solutions of the model.

The plants can survive independently and insect pollination can improve the growth rate of plants in [

There is still less research work of the model which cannot exist independently. The existing researches basically are the autonomous models (see [

Suppose that there are two plant populations (A and B) living in their natural environment, which are free from other interference factor. Let

The main aim of the paper is to discuss existence of periodic positive solution for the model.

Assume X and Z are normed vector space,

Lemma 1 (Continuation theorem) [

Lemma 2

Proof: By formula (2), we have

Since formula (2) is always true for

For the convenience of discuss, we give following notations.

In order to apply Continuation theorem to system (3), we define

and

then X, Z is Banach space under the norm

Let

Since

and

are continuous function with the periodicity

Let

So that we get

It is obvious that

We assume that

The corresponding operator equation is

We assume that

To move term from one side of an algebraic equation to the other side, reversing its sign to maintain equality, we get the following

From formula (5), formula (6) and formula (7), we have

and

From formula (8) and formula (9) we can get

From formula (10) and formula (11) we get

So that

Similarly, we have

Using formula (7) we get

Thus

Similarly, we have

From formula (8) and formula (13) we get

From formula (9) and formula (12) we get

Since

By formula (12), formula (13) and formula (14), we get

So that

It is obvious that

Let

condition for Lemma 1. When

So that

Let

Theorem If

This research was financially supported by the National Science Foundation of Zhejiang Province (LY12A01010) and by the College Students’ Scientific and Technological Innovation of Zhejiang Province (2015R411035).

Binbin Wang,Hailiang Zhang, (2016) Existence of Positive Periodic Solutions for a Time-Delay Biological Model. Journal of Applied Mathematics and Physics,04,1300-1304. doi: 10.4236/jamp.2016.47137