Abstract and Applied Analysis

Volume 2014 (2014), Article ID 130856, 7 pages

http://dx.doi.org/10.1155/2014/130856

Research Article

Multiple Periodic Solutions of a Nonautonomous Plant-Hare Model

Yongfei Gao,¹ P. J. Y. Wong,² Y. H. Xia,¹ and Xiaoqing Yuan¹

¹Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, China

²School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, 639798

Received 27 November 2013; Accepted 24 December 2013; Published 9 January 2014

Academic Editor: Weiming Wang

Copyright © 2014 Yongfei Gao et al. 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.

Based on Mawhin's coincidence degree theory, sufficient conditions are obtained for the existence of at least two positive periodic solutions for a plant-hare model with toxin-determined functional response (nonmonotone). Some new technique is used in this paper, because standard arguments in the literature are not applicable.

1. Introduction

In the past few decades, the classical predator-prey model has been well studied. Such classical predator-prey model has, however, been questioned by several biologists (e.g., see [1, 2]). Based on experimental data, Holling [3] has proposed several types of monotone functional responses for these and other models. However, this will not be appropriate if we explore the impact of plant toxicity on the dynamics of plant-hare interactions [4]. Recently, Gao and Xia [5] considered a nonautonomous plant-hare dynamical system with a toxin-determined functional response given by

where

Here, denotes the density of plant at time , denotes the herbivore biomass at time , is the plant intrinsic growth rate at time , is the per capita rate of herbivore death unrelated to plant toxicity at time , is the conversion rate at time , is the encounter rate per unit plant, is the fraction of food items encountered that the herbivore ingests, is the carrying capacity of plant, measures the toxicity level, and is the time for handing one unit of plant. The functions , , and are continuous, positive, and periodic with period , and , , , , and are positive real constants. For any continuous -periodic function , we let

The topological degree of a mapping has long been known to be a useful tool for establishing the existence of fixed points of nonlinear mappings. In particular, a powerful tool to study the existence of periodic solution of nonlinear differential equations is the coincidence degree theory (see [6]). Many papers study the existence of periodic solutions of biological systems by employing the topological degree theory; see, for example, [7–12] and references cited therein. However, most of them investigated the classical predator-prey model or the models with Holling functional responses; see [7–10]. There is no paper studying the functional responses in model (1) except for [5]. Gao and Xia [5] have obtained some sufficient conditions for the existence of at least one positive periodic solution for the system (1). Unlike the traditional Holling Type II functional response, systems with nonmonotone functional responses are capable of supporting multiple interior equilibria and bistable attractors. Thus, for nonautonomous system (1), it is possible to find two periodic solutions of (1). However, to date there is no work done on the existence of multiple periodic solutions of (1). Therefore, in this paper we will establish the existence of at least two positive periodic solutions of (1). We will be using the continuation theorem of Mawhin's coincidence degree theory; to this end some novel estimation technique will be employed to obtain a priori bounds of unknown solutions to some operator equation, as the standard estimation techniques used in the literature are not applicable to the system (1) due to the term . We will elaborate this in Remark 3.

2. Existence of Multiple Positive Periodic Solutions

In this section, we will establish sufficient conditions for the existence of at least two positive periodic solutions of (1). We will first summarize in the following a few concepts and results from [6] that will be required later.

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

Lemma 1 (see [6]). Let be an open bounded set. Let be a Fredholm mapping of index zero and   -compact on . Assume

(a)for each , , ;(b)for each , ;(c).

Then has at least one solution in .

To proceed, we note that (1) is equivalent to

Throughout, we assume the following:

; .

We further introduce six positive numbers which will be used later as follows:

where

Under assumptions and , it is not difficult to show that

Theorem 2. In addition to () and (), suppose that

.

Then system (4) has at least two positive -periodic solutions.

Proof. Since we are concerned with positive solutions of system (4), we make use of the change of variables

Then, system (4) can be rewritten as

Take

and define

Here denotes the Euclidean norm. Then and are Banach spaces with the norm . For any , by means of the periodicity assumption, we can easily check that

are -periodic.

Set

where . Further, is defined by

Define

It is not difficult to show that

and and are continuous projectors such that

It follows that is a Fredholm mapping of index zero. Furthermore, the generalized inverse (to ) exists and is given by

Then and are, respectively, defined by

Clearly, and are continuous. By using the Arzelà-Ascoli Theorem, it is not difficult to prove that is compact for any open bounded set . Moreover, is bounded. Therefore, is -compact on for any open bounded set .

Now, we will search for two appropriate open bounded subsets in order to apply the continuation theorem.

Corresponding to the operator equation , , we have

Suppose is a solution of (20) and (21) for a certain . Integrating (20), (21) over the interval , we obtain

It follows from , (20), and (22) that

that is,

Similarly, it follows from , (21), and (23) that

Since , there exist such that

From and (23), we see that

which implies

So

This, combined with (25), gives

In particular, we have

or

In view of , we have

Similarly, it follows from and (23) that

which implies

So

This, combined with (25), gives

In particular, we have

or

It follows from that

From (25) and (34), we find

On the other hand, it follows from , (22), and (42) that

It follows from (43) that

This, combined with (26), gives

Moreover, because of , it follows from (44) that

This, combined with (26) again, gives

It follows from (46) and (48) that

Now, let us consider with . Note that

Noting , , and , we can show that the equation has two distinct solutions:

Choose such that

We are now ready to define two open bounded subsets in order to apply the continuation theorem. Let

Then both and are bounded open subsets of . It follows from (4) and (52) that and . With the help of (4), (34), (41), (42), (49), and (52), it is easy to see that and satisfies the requirement (a) in Lemma 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 1. Hence, (4) has at least two  -periodic solutions. This completes the proof of Theorem 2.

Remark 3. In the proof of Theorem 2, we have employed some new technique to obtain a priori bounds for . Here, the standard arguments in the literature (see, e.g., [7–12]) do not work. Indeed, from (23) in the proof it follows that

If we were to use the standard arguments in the literature, then we have

where and . It follows from (55) that

where and are the roots of the following equation in :

We claim that (57) has at least a negative root; that is, at least one of ,   is negative. Otherwise, if both and are positive, then from (57) we see that

which implies

On the other hand, it follows form (57) and (58) that

which contradicts the positivity of and . Therefore, at least one of is negative. However, to use the standard arguments in the literature we need both and to be positive. Hence, we have illustrated that standard arguments in the literature are not applicable to the system (4) and some new technique should be used. To see how this problem is handled, the reader may refer to (27)–(34) in the proof of Theorem 2.

Conflict of Interests

No conflict of interests exists in the submission of this paper, and the paper is approved by all authors for publication. The authors would like to declare that the work described was original research that has not been published previously and not under consideration for publication elsewhere.

Funding

This work is supported by NNSFC 11271333 and 11171090.

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