Journal of Applied MathematicsVolume 2012 (2012), Article ID 602679, 14 pagesdoi:10.1155/2012/602679

Research Article

Existence of at Least Two Periodic Solutions for a Competition System of Plankton Allelopathy on Time Scales

Hui Fang

Department of Mathematics, Kunming University of Science and Technology, Yunnan 650093, China

Received 3 October 2011; Accepted 15 February 2012

Academic Editor: Meng Fan

Copyright © 2012 Hui Fang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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.

1. Introduction

The allelopathic interactions in the phytoplanktonic world have been studied by many researchers. For instance, see [1–4] and references cited therein. Maynard-Smith [2] and Chattopadhyay [3] proposed the following two-species Lotka-Volterra competition system, which describes the changes of size and density of phytoplankton: where and are the rates of toxic inhibition of the first species by the second and vice versa, respectively.

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 [5] and Kuang [6], it would be of interest to study the existence of periodic solutions. This motivates us to modify system (1.1) to the form where are continuous -periodic functions.

If the estimates of the population size and all coefficients in (1.2) are made at equally spaced time intervals, then we can incorporate this aspect in (1.2) and obtain the following discrete analogue of system (1.2): where are -periodic, that is, for any (the set of all integers), is a fixed positive integer. System (1.3) was considered by Zhang and Fang [7]. However, dynamics in each equally spaced time interval may vary continuously. So, it may be more realistic to assume that the population dynamics involves the hybrid discrete-continuous processes. For example, Gamarra and Solé pointed out that such hybrid processes appear in the population dynamics of certain species that feature nonoverlapping generations: the change in population from one generation to the next is discrete and so is modelled by a difference equation, while within-generation dynamics vary continuously (due to mortality rates, resource consumption, predation, interaction, etc.) and thus are described by a differential equation [8, page 619]. The theory of calculus on time scales (see [9, 10] and references cited therein) was initiated by Hilger in his Ph.D. thesis in 1988 [11] in order to unify continuous and discrete analysis, and it has become an effective approach to the study of mathematical models involving the hybrid discrete-continuous processes. This motivates us to unify systems (1.2) and (1.3) to a competition system on time scales of the form where are rd-continuous -periodic functions.

In (1.5), let , . If (the set of all real numbers), then (1.5) reduces to (1.2). If (the set of all integers), then (1.5) reduces to (1.3).

To our knowledge, few papers have been published on the existence of multiple periodic solutions for this model. Motivated by the work of Chen [12], we study the existence of multiple periodic solutions of (1.5) by applying Mawhin’s continuation theorem of coincidence degree theory [13]. Some new results are obtained. Even in the special case when , our conditions are also easier to verify than that of [7].

2. Preliminaries on Time Scales

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 [9–11].

Definition 2.1. A time scale is an arbitrary nonempty closed subset of the real numbers .

Let . Throughout this paper, the time scale is assumed to be -periodic, that is, implies . In particular, the time scale under consideration is unbounded above and below.

Definition 2.2. We define the forward jump operator , the backward jump operator , and the graininess by respectively. If , then is called right-dense (otherwise: right-scattered), and if , then is called left-dense (otherwise: left-scattered).

Definition 2.3. Assume is a function and let . Then we define to be the number (provided it exists) with the property that given any , there is a neighborhood of (i.e., for some ) such that In this case, is called the delta (or Hilger) derivative of at . Moreover, is said to be delta or Hilger differentiable on if exists for all . A function is called an antiderivative of provided for all . Then we define

Definition 2.4. A function is said to be rd-continuous if it is continuous at right-dense points in and its left-sided limits exist (finite) at left-dense points in . The set of rd-continuous functions will be denoted by .

Lemma 2.5. Every rd-continuous function has an antiderivative.

Lemma 2.6. If , and , then (a)(b)if for all , then (c)if on , then .

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

Let where is an -periodic real function, that is, for all .

Lemma 2.7 (see [14]). Let and . If is - periodic, then

Lemma 2.8 (see [15]). Assume that is a function on such that (i) is uniformly bounded on ,(ii) is uniformly bounded on .Then there is a subsequence of which converges uniformly on .

3. Existence of Multiple Periodic Solutions

In this section, in order to obtain the existence of multiple periodic solutions of (1.5), we first make the following preparations [13].

Let , be normed vector spaces, let be a linear mapping, and let : be a continuous mapping. The mapping will be called a Fredholm mapping of index zero if and is closed in . If is a Fredholm mapping of index zero, there then exist continuous projectors and such that , . If we define as the restriction of to , then is invertible. We denote the inverse of that map by . If is an open bounded subset of , the mapping will be called -compact on if is bounded and is compact, that is, continuous and such that is relatively compact. Since is isomorphic to , there exists isomorphism .

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 [13, page 40] as follows.

Lemma 3.1 (continuation theorem). Let be a Fredholm mapping of index zero and let be -compact on . Suppose (a) for every and every ,(b) for every , and Brouwer degree Then has at least one solution in .

In the following, we shall use the notation

We make the following assumptions.(H₁). (H₂). (H₃).

Next, we introduce some lemmas.

Lemma 3.2 (see [16, Lemma 3.2]). Consider the following algebraic equations: Assuming that (H₁), (H₂) hold, then the following conclusions hold. (i)If , then (3.3) have two positive solutions: (ii)If , then (3.3) have two positive solutions:

Lemma 3.3. Assume that (H₁)–(H₃) hold, then the following conclusions hold. (i), (ii).

Proof. The proof of (i) is the same as (i) of Lemma  3.5 in [7]. We omit it.(ii)We have

Lemma 3.4. Assume that (H₁)–(H₃) hold, then the following conclusions hold. where

Proof. Under the conditions that , , , , we have Thus is increasing (decreasing) in the first variable, decreasing (increasing) in the second variable, increasing (decreasing) in the third variable. Notice that So from (3.9), (3.10), and (H₁)–(H₃), we obtain that

Theorem 3.5. Assume that (H₁)–(H₃) hold. Then system (1.5) has at least two -periodic solutions.

Proof. Take It is easy to verify that and are both Banach spaces.Define the following mappings ,, and as follows: We first show that is a Fredholm mapping of index zero and is -compact on for any open bounded set . The argument is quite standard. For example, one can see [14, 17, 18]. But for the sake of completeness, we give the details here.It is easy to see that Ker=, Im = is closed in , and . Therefore, is a Fredholm mapping of index zero. Clearly, and are continuous projectors such that On the other hand, : , the inverse to , exists and is given by Obviously, and are continuous. By Lemma 2.8, it is not difficult to show that is compact for any open bounded set . Moreover, is bounded. Hence, is -compact on for any open bounded set .Corresponding to the operator equation , one has where . Suppose is a solution of system (3.16) for some . Integrating (3.16) over the interval , we have where .It follows from (3.16) and (3.17) that That is Since , there exist , such that From (3.17), (3.20), one obtains We can derive from (3.22) that which, together with (3.21), leads to From (3.19), we have That is which, together with (3.24), leads to Therefore, we have So from (3.10), one obtains where According to (i) of Lemma 3.3, we obtain In a similar way as the above proof, one can conclude from that Noticing that , , one has According to (ii) of Lemma 3.3, one has It follows from (3.19) and (3.31) that On the other hand, it follows from (3.17) and (3.20) that that is From (3.19) and (3.38), one obtains It follows from (3.17) and (3.20) that which implies that From (3.39) and (3.41), we have which leads to By (3.39) and (3.43), we obtain that Now, let us consider with . Note that According to Lemma 3.2, we can show that has two distinct solutions Choose such that Let Then both and are bounded open subsets of . It follows from Lemma 3.2, Lemma 3.4, and (3.47) that , . With the help of (3.31), (3.35), (3.36), (3.44), and Lemma 3.4, it is easy to see that and satisfies the requirement (a) in Lemma 3.1 for . Moreover, for . A direct computation gives Here is taken as the identity mapping since . So far we have proved that satisfies all the assumptions in Lemma 3.1. Hence (1.5) has at least two -periodic solutions with . Obviously are different. The proof is complete.

Example 3.6. As an application of Theorem 3.5, we consider the following system: If , then (3.50) reduces to the following system: A direct computation gives that So according to Theorem 3.5, System (3.51) has at least two 5-periodic solutions and . The simulation results given in Figure 1 verify the above conclusion. Set , then (3.51) can be changed into the following system: Therefore, System (3.53) has at least two positive 5-periodic solutions. Similar to the proof of Theorem 3.5, we can prove the following result.

(a)

(b)

Figure 1: Blue lines stand for , , red lines stand for .

Theorem 3.7. In addition to (H₁) and (H₂), assume further that system (1.5) satisfies (H₃)'. Then system (1.5) has at least two -periodic solutions.

Acknowledgment

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

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