We study a competition system of the growth of two species of plankton with competitive and allelopathic effects on each other on time scales. With the help of Mawhin’s continuation theorem of coincidence degree theory, a set of easily verifiable criteria is obtained for the existence of at least two periodic solutions for this model. Some new existence results are obtained. An example and numerical simulation are given to illustrate the validity of our results.

The allelopathic interactions in the phytoplanktonic world have been studied by many researchers. For instance, see [

Naturally, more realistic models require the inclusion of the periodic changing of environment caused by seasonal effects of weather, food supplies, and so forth. For such systems, as pointed out by Freedman and Wu [

If the estimates of the population size and all coefficients in (

In (

To our knowledge, few papers have been published on the existence of multiple periodic solutions for this model. Motivated by the work of Chen [

In this section, we briefly present some foundational definitions and results from the calculus on time scales so that the paper is self-contained. For more details, one can see [

A time scale

Let

We define the forward jump operator

Assume

A function

Every rd-continuous function has an antiderivative.

If

if

if

For convenience, we now introduce some notation to be used throughout this paper.

Let

Let

Assume that

In this section, in order to obtain the existence of multiple periodic solutions of (

Let

Mawhin’s continuation theorem of coincidence degree theory is a very powerful tool to deal with the existence of periodic solutions of differential equations, difference equations and dynamic equations on time scales. For convenience, we introduce Mawhin’s continuation theorem [

Let

In the following, we shall use the notation

We make the following assumptions.

Next, we introduce some lemmas.

Consider the following algebraic equations:
_{1}), (H_{2}) hold, then the following conclusions hold.

If

If

Assume that (H_{1})–(H_{3}) hold, then the following conclusions hold.

The proof of (i) is the same as (i) of Lemma 3.5 in [

We have

Assume that (H_{1})–(H_{3}) hold, then the following conclusions hold.

Under the conditions that _{1})–(H_{3}), we obtain that

Assume that (H_{1})–(H_{3}) hold. Then system (

Take

Define the following mappings

We first show that

It is easy to see that Ker

Corresponding to the operator equation

It follows from (

In a similar way as the above proof, one can conclude from

It follows from (

On the other hand, it follows from (

Now, let us consider

As an application of Theorem

Blue lines stand for

In addition to (H_{1}) and (H_{2}), assume further that system (

This research is supported by the National Natural Science Foundation of China (Grant nos. 10971085, 11061016).